Be anything but crabby after enjoying this sunny citrus bisque. You'll net half your daily folate needs per bowl.
Reserve 1 tablespoon lime juice. Blend remaining lime juice, zest, corn kernels, chile, adobo sauce, broth and 1/4 teaspoon salt in a blender until smooth. Divide soup among 4 bowls. Toss crabmeat, avocado, cilantro, cumin seeds, reserved lime juice and remaining 1/4 teaspoon salt in a bowl. Sprinkle soup with crab mixture and serve with tortilla chips, if desired.
Just OK. Not good enough for the jumbo lump crab.
followed the recipe exactly, but was disappointed with the results. The flavors just never melded. Won't be making this again.
I will eat almost anything and I thought this soup was gross. My husband, on the other hand, liked it. He didn't LOVE it. Just liked it. He said it was WAY too limey and he loves lime. I cut the recipe (after reading other reviews) to the juice of two limes and a tiny bit of lime zest (not even a whole lime)... and still he said too limey. I almost think the lime should be added at the end. Oh... and I tried it cold and it wasn't good, so I heated it (again after reading the other reviews). It was just okay.
I have made this recipe twice and we love it. For a summer recipe, I loved the fact that you do not have to cook anything. The second time I used a whole chipolte chili (seeded) and it really had some kick. I made it for my book club and everyone asked for the recipe. It is unique!
I agree that the recipe was incomplete. The base should be really pureed! Not clear about what should be cooke. I cooked the corn altho I thought you could use frozen cooked corn instead. Will try it again without quite so much lime juice. I also agree the avocado should be in slices. The crabmeat should not be overpowered by the lime. I'm going to try it warmed also.
The flavors were lovely with the corn, crab, avocado and chipotle. However, I did have to do some monkeying around to get a recipe I liked though. I believe its supposed to be cold and I decided to make it warm instead as I'm not a fan of cold soups. I made a standard cream soup base with a flour roux, milk and broth and added the pureed corn and chipotle and heated. Then I ladled into bowls and added the toppings. It ended up being more of a bisque, but still tasty.
sounded like it would be good. failed to notice the recipe was no-cook before purchasing the ingredients. didn't impress me too much. i'm not really against cold soups but this REALLY didn't do it for me. sick to my stomach. cheeseburger please.
Every recipe has someone's "taste" in it and some of this fit mine. Too much ancho for me but I liked most of the pairings. I will make it again and make a few changes but isn't that what we do to most recipes? Adapt it to our taste? I will find "my" version and make it for my customers this summer.
recipe calls for far too much chile chilpotle. Ground chili ancho is the recommended for seasoning.A bisque should be delicate not over powering the seafood. Your guests should be able to savour the soup. Chicken stock? No, fish stock. Also, one should never use cumin seeds, they sell ground cumin, use it. Add the avocado in strips just before serving. What's with the chopped coriander? Obviously written by someone with absolutely no knowledge of la cocina mexicana trying to put in every mexican food cliche there is. If you wish to serve chile chilpotle, then put it on the table in a dish. I served seafood bisque every Friday in my restaurant, "Sombreros" and it sold out every night. Retired Mexican chef.
I think the confusion is that the recipe doesn't state that this is s cold soup, but it is. There is nothing unsafe about not cooking this soup, as mentioned below. Broth has already been cooked as well as the crab and is perfectly safe to eat.
I made this soup because I love all the ingredients involved: lime, avocado, crab, cilantro, chipolte peppers. But I agree with the previous reviewer; the lime and zest is WAY too overpowering. I will try one zest and two limes next time. I also recommend to puree the soup for quite a while to break down the corn kernels. I did not previously cook anything, and although it may be nice warmed, it was also refreshing cold. The crab mixture was very good!
I have not tried this yet, but I am also confused by the absence of cooking. I am going to give it a go and let you know.
Like the previous reviews state, this recipe is incomplete. Not sure who reviews these before they are posted, but you are not a food person. For starters the recipe does not call for cooking the broth. Not a food safe method. Second, the zest and juice of 4 limes is overkill, completely overpowers the other ingredients. I halved the zest and it was still to much. I would recommend the zest of one lime/juice of two and replacing some of the chicken stock with whipping cream or half and half. The broth will not hold the corn hulls in suspension.
I definitely would prepare in advance. After blending the ingredients indicated, cover and chill for a minimum of one hour. Fill chilled bowls and garnish with crab meat, etc.as recipe suggests.
Would you recommend preparing in advance?
Fresh, light and delicious chilled. Serve with warm tortilla chips for a lovely contrast.
This sounds luscious but when and what do you heat? It goes from blender to bowl and the rest is added to the bowl. Is it meant to be served cold? That would be really yucky, it seems to me.
Should this be served heated, chilled, or at room temperature? Seems as though these instructions were overlooked.


Today's real estate markets have certainly heated up in recent months. It is a seller's market on the used residential side and a buyer's market with substantial new construction being outpaced by the market of buyers. Even though it seems like the commercial market is flush with opportunities, we are seeing an uptick in new construction, sales and leasing.
We often receive inquiries to review and comment on leases, sales contracts, construction contracts, offers to purchase, closing documents title insurance and environmental indemnifications. Many of the documents presented to us are prepared by non-lawyers or prepared by lawyers who do not adequately represent the interests of our clients. Most often, we receive pre-printed forms used by real estate brokers. Wisconsin law prohibits any person, including a licensed real estate broker, from practicing law without a law license. To help brokers from violating the law, the Wisconsin Department of Regulation and Licensing developed preprinted and pre-approved forms which may be used by brokers without leaving them sideways with the law.
The terms and conditions of these documents are easy to misunderstand. Even the basic Commercial Offer to Purchase form is nine pages long and contains obligations on both buyer and seller that are often overlooked. By way of example, a common misunderstanding by buyers is that their only risk in entering into a purchase contract is losing their earnest money; however, in reality the Residential Offer to Purchase has language that allows a seller to sue the buyer to force the purchase of the property. Other topics contained in the forms that require a more in-depth understanding of the law include, without limitation, estoppels, title, surveys, representations and warranties, contingencies and default provisions.
In addition to examining and understanding the issues these forms address, it is also important to consider the issues these forms overlook. As beneficial and convenient as standardized forms are, one of the greatest downsides to the forms is that buyers, sellers, and brokers sometimes mistakenly believe that these standardized forms adequately cover all issues that might arise in connection with a real estate transaction. To be sure, in more complex real estate transactions, these forms should be considered a starting point for negotiations leading to a contract. The benefit of involving an attorney in a real estate transaction is that we are able to use addenda or riders to address issues about which the form is silent or tailor the form to the transactions at hand.
Leases, Mortgages, Closing documents and Title Insurance also contain provisions, which are many times confusing or misconstrued. With most real estate transactions involving significant capital investment, we greatly suggest that, for our clients' protection, we have the opportunity to review these documents and provide valuable feedback to ensure the transaction goes smoothly – a modest expense given the investment at hand.
For more information please contact a member of the Commercial Real Estate, Development & Leasing Team at the Law Firm of Conway, Olejniczak & Jerry, S.C.


Suzanne Young is the Director of Graduate and Postdoctoral Teaching Development. She earned her Ph.D. in English Literature from the University of Virginia. She has taught literature and composition for over 20 years, including a course on "Paris in the '20s" that focuses on transatlantic modernist artists. Her research interests include composition, modernist literature, digital humanities, and online learning. At the CTL, she works with graduate students and faculty on course design, teaching innovation, technology-enhanced teaching, and student engagement.


The United Arab Emirates, sometimes simply called the Emirates or the UAE, is an Arab country situated in the southeast of the Arabian Peninsula in Western Asia on the Persian Gulf, bordering Oman to the east and Saudi Arabia to the south, as well as sharing sea borders with Qatar to the west and Iran to the north. The UAE is a federation of seven emirates (equivalent to principalities), each governed by a hereditary emir, with a single national president. The constituent emirates are Abu Dhabi, Ajman, Dubai, Fujairah, Ras al-Khaimah, Sharjah, and Umm al-Quwain. The capital is Abu Dhabi, which is also the state's center of political, industrial, and cultural activities.
The UAE's political system is based on its 1971 Constitution, which is composed of several intricately connected governing bodies. As a federation of seven monarchies, whose rulers retain absolute power within their emirates, but with a UAE president, it is neither a constitutional monarchy nor a republic. The emirs choose one of their members to be the president of the federation, but this does not alter the monarchical character of the government of the individual emirates. The constitution is concerned solely with the relations between the emirates as members of the federation, and does not prescribe a constitutional system of government. Islam is the official religion of the UAE, and Arabic is the official language.


An energy efficiency consultant could be your first step to saving money on your electricity bills. There is a guide below that is going to give you tips that will help you know what to look out for in the consultant. The company that is in a position to identify opportunities through which your company can actually save energy costs is the one you should go with. It is also important you ensure that the services being offered by the consultant are in accordance with the law. Getting quotes from different consultants is the best thing to do because you will get to see how they compare against each other for you to choose the best.
It is always advisable for you to get an experienced consultant so that you can ask for references and get testimonials as well. The reason you ask for references is so that you see if they have been successful at the implementation. The consultant is going to generate a report and you should know what to expect so that you know what to look for in the quotes that you get from the different consultants. It is important that you are aware of the resources that are available for use in the project and within what time as well. Confidentiality is crucial when it comes to projects like these and the consultant should be in a position to uphold it and confidently answer any questions you have on the project.
The next course of action after you have hired the energy consultant is to get an energy audit. This is how they will find out how you could reduce your energy use and electricity bills. The audit is supposed to identify how and where energy is being used and how much it is costing the company. They will then get the energy cost and usage control measures that could be effectively implemented. Some of the measures they can implement include turning off certain appliances and installing new technologies. There is also the possibility of having low cost initiatives such as installing sensors.
It is always reassuring to work with a provider that truly understands what it is like to be a business owner. The business does not interruptions just because they are upgrading and this is the reason why you need an energy consultant that gets that. The reason you go with a provider that understands the business operations is so that they do not interrupt them. The consultant that has a team dedicated strictly to research and development is the best. Energy efficiency industry is growing at a fast rate and the products along with it and this is why you ought to be careful. Therefore, the company you go with should ensure that these claims are true on your behalf.


Vote for a strong voice for Rideau-Rockcliffe!
Rawlson King is an experienced, result-based community leader.
He has lived in Ward 13 for 15 years and has been an active volunteer for 10 years on a wide range of civic issues.
Recognized for his volunteer work, Rawlson won a 2018 United Way Ottawa Community Builder of the Year Award, conferred during National Volunteer Week.
He successfully advocated for a youth strategy for Ward 13.
Led the organization when it raised $42,000 to hold a highly-acclaimed musical.
Helped obtain heritage designation of a former neighbourhood public school.
Assisted in piloting a post-incident response plan to serious crime, now used City-wide.
Supported the creation and opening of a new community park in the Ward.
Was a vocal leader in the fight to keep Rideau High School open.
He's Treasurer of a $1.8 million facility that provides social services to Ward 13.
Advocated for the establishment of a new community hub at the former Rideau High.
Improved contact between residents, communities & organizations.
Supported community development & public safety efforts in the Ward.
Builds relationships with police, Indigenous, faith-based & racialized communities.
Advocates for equality in the police service & social investments in neighbourhoods.
Supportive of local visual and performative arts, Rawlson served as President of Gallery 101, a prominent Ottawa artist-run centre for many years. To encourage more local educational options, he currently serves as a Board member at AlivEducation, an innovative and non-traditional evening and weekend after-school program for youth that is offered at the Overbrook Community Centre.
Educated in Ottawa, Rawlson holds both a Bachelor of Journalism (with concentration in Legal Studies), and a M.A. in Communication from Carleton University.


The NFIB Research Foundation has collected Small Business Economic Trends data with quarterly surveys since the 4th quarter of 1973 and monthly surveys since 1986. Survey respondents are drawn from NFIB's membership. The report is released on the second Tuesday of each month. This survey was conducted in March 2019.
The NFIB Small Business Optimism Index increased 0.1 points to 101.8 in March, a historically strong level and an indication that small businesses continue to power the economy after being briefly shaken by January's government shutdown. Overall, the Index anticipates solid growth, keeping the economy at "full employment" with no signs of a recession in the near term. The Uncertainty Index dropped six points to 79, returning to a more normal level for recent years.
Job creation was solid in March with a net addition of 0.50 workers per firm (including those making no change in employment), close to February's record of 0.52, and up from 0.33 in January. One percent (down 2 points) reported reducing employment an average of 2.4 workers per firm (seasonally adjusted), the lowest percentage of owners reporting reductions in survey history. Twelve percent (unchanged) reported increasing employment an average of 2.7 workers per firm. Owners are trying to hold on to the employees that they have in a highly competitive labor market. Sixty percent reported hiring or trying to hire (up 3 points), but 54 percent (90 percent of those hiring or trying to hire) reported few or no qualified applicants for the positions they were trying to fill (up 5 points). Twenty-one percent of owners cited the difficulty of finding qualified workers as their Single Most Important Business Problem, only 4 points below the record high. Thirty-nine percent of all owners reported job openings they could not fill in the current period, up 2 points from February and equal to the record high set in December. Twelve percent of all owners reported using temporary workers (down 2 points). Labor markets are tight for both skilled and unskilled workers, and it will be difficult to fulfill job creation plans. Thirty-three percent have openings for skilled workers (up 2 points) and 14 percent have openings for unskilled labor. Thirty-four percent of owners reported few qualified applicants for their open positions (up 5 points) and 20 percent reported none. Hiring is an "investment," and clearly owners look for an economy that will provide a good return on their labor investments.
A net 5 percent of all owners (seasonally adjusted) reported higher nominal sales in the past three months, a 6-point improvement, rebounding back from the "shut down, slow down" in February sales. The net percent of owners expecting higher real sales volumes rose 3 points to a net 19 percent of owners, a solid reading. For perspective, in the 12 months prior to the 2016 election results, the average was a net negative 3 percent, 22 points worse.
The net percent of owners reporting inventory increases rose 3 points to a net 5 percent (seasonally adjusted), consistent with improved expectations for retail sales. The net percent of owners viewing current inventory stocks as "too low" lost 4 points to a net negative 6 percent, suggesting that inventories are still viewed as excessive. The net percent of owners planning to expand inventory holdings fell from 1 percent to a negative 1 percent, a 2-point decline.
Sixty percent reported capital outlays, up 2 points. Of those making expenditures, 44 percent reported spending on new equipment (up 1 point), 29 percent acquired vehicles (up 1 point), and 17 percent improved or expanded facilities (unchanged). Six percent acquired new buildings or land for expansion (unchanged) and 16 percent spent money for new fixtures and furniture (up 2 points). Twenty-seven percent plan capital outlays in the next few months, unchanged. Plans to invest were most frequent in wholesale trades (36 percent), manufacturing (34 percent), retail (33 percent), and transportation (32 percent).
The net percent of owners raising average selling prices fell 1 point to a net 12 percent, seasonally adjusted. Firms in construction most frequently reported raising their average prices (a net 29 percent). Seasonally adjusted, a net 24 percent plan price hikes (down 2 points). The net percent planning hikes is above 30 percent in construction and manufacturing, three fewer industry groups than last month. Overall, a very quiet report for inflation watchers.
Reports of higher worker compensation rose 2 points to a net 33 percent of all firms. Plans to raise compensation rose 2 points to a net 20 percent. Overall, reports of rising compensation are holding at historically high levels. Twenty-one percent selected "finding qualified labor" as their top business problem, more than cited taxes or regulations. The frequency of reports of positive profit trends improved 1 point to a net negative 8 percent reporting quarter on quarter profit improvements. Thirty-one percent of those reporting weaker profits blamed sales (down 10 points), 12 percent blamed labor costs, and 6 percent cited lower selling prices. For those reporting higher profits, 56 percent credited sales volumes. Six percent credited higher prices. The cost of materials was not an issue.
Three percent of owners reported that all their borrowing needs were not satisfied, unchanged and historically very low. Thirty-three percent reported all credit needs met (down 1 point) and 51 percent said they were not interested in a loan, unchanged. Six percent reported their last loan was harder to get than the previous one, unchanged and historically low. Only one percent reported that financing was their top business problem (down 1 point). The percent of owners reporting paying a higher rate on their most recent loan was unchanged at 17 percent. Thirty-four percent of all owners reported borrowing on a regular basis (up 1 point). The average rate paid on short maturity loans fell 10 basis points to 6.1 percent after a decline in February of 70 basis points. Overall, credit markets have been very supportive of growth and will not likely become an impediment this year.
Economic growth hit a small pothole in the first quarter with a government shutdown and bad weather all around. The most recent economic data indicate that the economy is headed back toward stronger growth – retail sales revised up and housing showing some real energy. Capital spending is turning out to be better than first thought.
The percent of owners citing labor costs as their Single Most Important Business Problem hit a 45-year record high in February reaching 10 percent, falling to 8 percent in March. However, the larger problem is finding someone to pay, not the actual wage cost. In construction, 6 percent cited labor cost as their top business problem, but 34 percent said finding a qualified worker to pay was their top business problem. Sixty-seven percent reported few or no qualified applicants. In manufacturing, 5 percent cite labor cost, but 29 percent reported the lack of qualified workers as their biggest problem. Hiring a worker is a major investment. Consequently, hiring decisions are strong indicators about the level of confidence owners have in the economy. The NFIB labor market indicators are at record levels, a good sign that on Main Street, the economy is still strong.


text:
Juncus georgianus är en tågväxtart som beskrevs av Frederick Vernon Coville. Juncus georgianus ingår i släktet tåg, och familjen tågväxter. Inga underarter finns listade i Catalogue of Life.

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georgianus

redpajama_set_name: RedPajamaWikipedia


Keratosis pilaris is harmless (benign), does not get worse over time, and often disappears with age. It is more common in patients who have very dry skin, or who have atopic dermatitis (eczema). It seems to run in families.
In mild cases, small bumps, similar in appearance to "goose bumps," are found on the backs of the upper arms. The texture is that of very coarse sandpaper.
Bumps may also appear on the buttocks and thighs. Less commonly, bumps appear on the face and may be mistaken for acne.
Individual bumps are small, skin-colored papules that form within hair openings (follicles). The condition is generally worse in winter and often clears in the summer.
Physical examination is usually all that is needed for your health care provider to make this diagnosis. Testing is usually not necessary.
Moisturizing lotions are often soothing and may help the appearance of the skin. Skin creams with medications containing urea, lactic acid, glycolic acid, salicylic acid, tretinoin, or vitamin D may be recommended by your physician. However, improvement often takes months and the bumps are likely to come back.
Call for an appointment with your health care provider (or discuss the condition during a routine visit) if you suspect that you have keratosis pilaris and the condition does not respond to use of over-the-counter moisturizing lotions.


We are the oldest dealer having more than 50 years experience in Force Motors dealership in Kolkata. We are dealing with MUV, LCV.
Easy finance: Available from major Nationalized Bank & Foreign Bank.
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text:
\section{Introduction}
Minimizing nonsmooth nonconvex Lipschitz functions $f \colon \Rn \to \R$ is in general intractable.  A proxy approach, dating back to the 1970's \cite{gold_eps_stat}, can be viewed as seeking a point in $\Rn$ around which $f$ is differentiable on some cluster of nearby points at which the gradients have a small convex combination --- a small ``Goldstein subgradient'', in the language of \cite{gold_eps_stat}.  One algorithm accomplishing this goal, but without complexity analysis, appeared in \cite{mahdavi}.

A 2020 breakthrough \cite{mit} proposed an algorithm for this Goldstein subgradient problem with a complexity analysis depending on the radius $\delta$ of the cluster and the size $\epsilon$ of the subgradient, but {\em independent of the dimension} $n$.  The algorithm assumes directional differentiability of the objective $f$, and, relying on an associated directional subgradient oracle, uses an innovative randomized line search to achieve an efficiency guarantee of essentially $O(\epsilon^{-3}\delta^{-1})$.  Two subsequent papers \cite{somit,ddmit} point out that small random perturbations instead allow a standard subgradient oracle.

Two recent developments \cite{kornowski-shamir,jordan-lin-zampetakis}\footnote{We became aware of these concurrent independent works after completing the initial draft of this manuscript.} raise the question of {\em deterministic} algorithms for this problem.  While both prove positive results in the smooth case, and \cite{jordan-lin-zampetakis} thereby develops a ``white-box'' deterministic smoothing approach to the nonsmooth problem, both manuscripts also prove negative results for the general dimension-independent question.

Our contribution is to show, notwithstanding the negative results of \cite{kornowski-shamir,jordan-lin-zampetakis}, that a simple deterministic black-box version of the algorithm of \cite{mit} achieves, up to a {\em nonconvexity modulus} for the objective, a dimension-independent complexity of $O(\epsilon^{-4}\delta^{-1})$.  Our analysis covers interesting objectives such as piecewise linear functions, which are not even weakly convex.  We relate the nonconvexity modulus of the objective with its distributional second derivative, hinting at an intriguing relationship between such derivatives and algorithmic complexity in general optimization.

\section{The optimization problem and oracle}
Primarily to emphasize the elementary nature of our development, we adopt a rudimentary setting for our optimization problem.  On a real inner product space $\X$ with corresponding norm $|\cdot|$, we consider the problem of minimizing a function $f \colon \X \to \R$.  The objective $f$ may be neither smooth nor convex, and the space $X$ may be neither finite-dimensional, nor even complete.

The method we develop, following \cite{mit}, relies on an underlying {\em directional subgradient map} $G \colon \X^2 \to \X$ associated with the objective $f$:  for all points $x \in \X$ and directions $e \in \X$, the G\^ateaux directional derivative
\[
f'(x;e) ~=~ \lim_{t \downarrow 0} \frac{1}{t}\big(f(x+te) - f(x)\big)
\]
exists and satisfies
\[
\ip{G(x,e)}{e} ~=~ f'(x;e).
\]
We say that $G$ is {\em $L$-bounded} for some constant $L>0$ if its norm $|G(x,e)|$ is never larger than $L$.
In applications, the objective function $f$ is $L$-Lipschitz and the vector $G(x,e)$ is a subgradient of some kind for $f$ at the point $x$ associated with the direction $e$, so we loosely refer to $G(x,e)$ as a ``subgradient''.  Nonetheless, we choose this rudimentary setting to emphasize again the elementary nature of our development, which makes no recourse to variational or Lipschitz analysis.

\begin{exa}[differentiable functions]
{\rm
If the function $f$ is $L$-Lipschitz and has a G\^ateaux derivative $\nabla f(x)$ at every point $x \in \X$, then the equation
\[
G(x,e) ~=~ \nabla f(x)
\]
defines an $L$-bounded directional subgradient map.
}
\end{exa}

\begin{exa}[convex functions]
{\rm
For an $L$-Lipschitz convex function $f$ with convex subdifferential $\partial f$, any map $G$ satisfying
\[
G(x,e) ~\in~ \mbox{argmax}\{\ip{g}{e} : g \in \partial f(x) \} \qquad \mbox{for all}~ x,e \in \X
\]
is an $L$-bounded directional subgradient map.
}
\end{exa}

More generally, we have the following example, which covers many interesting objectives, including the weakly convex case.  For a Lipschitz function $f \colon \Rn \to \R$, the {\em Clarke subdifferential}
$\partial^c f(x)$ is the convex hull of the set of all limits of the form
$\lim \nabla f(x^r)$ for points $x^r \to x$ in $\Rn$.  The function $f$ is {\em subdifferentially regular} when its G\^ateaux directional derivative satisfies
\[
f'(x;e) ~=~ \max\{\ip{g}{e} : g \in \partial^c f(x) \} \qquad \mbox{for all}~ x,e \in \Rn .
\]

\begin{exa}[subdifferentially regular functions]
{\rm
For an $L$-Lipschitz subdifferentially regular function $f \colon \Rn \to \R$, any map $G$ satisfying
\[
G(x,e) ~\in~ \mbox{argmax}\{\ip{g}{e} : g \in \partial^c f(x) \} \qquad \mbox{for all}~ x,e \in \X
\]
is an $L$-bounded directional subgradient map.
}
\end{exa}

Notwithstanding the generality of this example, we emphasize that our framework is not restricted to objectives that are subdifferentially regular, as the following result shows.

\begin{prop}[directional Clarke subgradient maps] \label{clarke}
Any locally Lipschitz function $f \colon \Rn \to \R$ that is directionally differentiable has a directional subgradient map $G \colon \Rn \times \Rn \to \Rn$ satisfying $G(x,e) \in \partial^c f(x)$ for all $x,e \in \Rn$.
\end{prop}

\pf
We just need to prove that if $f$ has a G\^ateaux directional derivative at the point $x \in \Rn$ in the direction $e \in \Rn$, then there exists a Clarke subgradient $g \in \partial^c f(x)$ satisfying $\ip{g}{e} = f'(x;e)$.  For $r=1,2,3,\ldots$, by the nonsmooth mean value theorem, there exists a point
$x_r \in [x,x+\frac{1}{r}e]$ and a subgradient $g_r \in \partial^c f(x_r)$ satisfying
\[
f\Big(x+\frac{1}{r}e\Big) - f(x) ~=~ \ip{g_r}{\frac{1}{r} e}.
\]
Since the subdifferential $\partial^c f$ mapping is closed and locally bounded, any limit point $g$ of the sequence $\{g_r\}$ has the desired property.
\finpf

Minimizing nonconvex functions is in general intractable.  Instead we seek a point $x \in \X$ that is, in some sense, approximately critical.  To this end, we fix a constant $\delta > 0$, and consider what we call {\em Goldstein subgradients} at $x$:  vectors of the form
\[
\sum_{i=1}^k \lambda_i G(x_i,e_i)
\]
for a positive integer $k$, positive weights $\lambda_i$ summing to $1$, points
$x_i$ in $B_{\delta}(x)$ (the closed ball of radius $\delta$ and center $x$) and directions $e_i \in \X$ for $i=1,2,\ldots,k$.  We denote the set of all Goldstein subgradients by $\partial_\delta f(x)$:  loosely, it consists of all convex combinations of subgradients at nearby points.  Strictly speaking, our notion is potentially smaller than the standard definition for Lipschitz $f \colon \Rn \to \R$, namely the closed convex hull of the set $\partial^c f\big(B_\delta(x)\big)$.

We can now state our goal, which relies on a second constant $\epsilon > 0$.
\bigskip

\noindent
{\bf Aim:}  Find a point $x \in \X$ and a Goldstein subgradient $g \in \partial_\delta f(x)$ such that $|g| \le \epsilon$.
\smallskip

\noindent
The development of \cite{mit} accomplishes this goal, explicitly in the setting of Proposition \ref{clarke}), and assuming that $f$ is directionally differentiable in the (stronger) Hadamard sense.  It relies on the following oracle.

\begin{oracle}[directional subgradient] \label{oracle}
{\rm
\mbox{} \\
{\bf Input:}
\begin{itemize}
\item
a point $x \in \X$
\item
a direction $e \in \X$.
\end{itemize}
{\bf Output:}
\begin{itemize}
\item
the objective value $f(x)$
\item
the directional derivative $f'(x;e)$
\item
a subgradient-like vector $G(x,e)$.
\end{itemize}
}
\end{oracle}

In this work we rely on the same directional subgradient oracle, which we should emphasize is stronger than the standard subgradient oracle typically assumed.  In generic practice, for a Lipschitz objective $f$, we may expect that the algorithm never encounters points $x$ where $f$ is nondifferentiable, in which case any subgradient oracle simply returns $g = \nabla f(x)$.  More formally, however, we must consider nonsmooth points.  Undeterred, the authors of \cite{mit} argue that the stronger oracle is often a reasonable assumption.  However, our aim here is a fully deterministic algorithm. Accordingly, we use this same stronger oracle, but instead use a deterministic line search, much as in \cite{kornowski-shamir}.  To ensure termination, we make a mild assumption about the the directional behavior of the objective function $f$, similar in spirit to the idea of {\em semismoothness} \cite{mifflin-semismooth} common in nonsmooth computation, but simpler and weaker.

We call the objective $f$ {\em directionally semismooth} if all points $x \in \X$ and directions $e \in \X$ satisfy
\[
\lim_{t \downarrow 0} f'(x+te;e) ~=~ f'(x;e).
\]
For comparison, following the definition in \cite{fin_2}, a Lipschitz function $f \colon \Rn \to \R$ is {\em semismooth} if it is directionally differentiable and for all points $x \in \Rn$ the directional derivative satisfies the stronger property
\[
f'(x+d;d) - f'(x;d) ~=~ o(d) \quad \mbox{as}~ d \to 0.
\]
Most Lipschitz functions in practice are semismooth, including in particular difference-of-convex functions and semi-algebraic functions \cite{semismooth}.  As we shall see, directional semismoothness suffices to guarantee termination of our algorithm, but before describing it, we focus first on the line search.

\section{A simple line search}
We pose the line search problem as a self-contained question.
Consider points $p<q$ in $\R$ and a function $h \colon [p,q] \to \R$ satisfying
$h(p)>h(q)$.  Suppose that $h$ is right-differentiable on the interval $[p,q)$, and an oracle returns, for any input $t \in [p,q)$, the value $h(t)$ and the right derivative
\[
h'_+(t) ~=~ \lim_{s \downarrow t} \frac{h(s) - h(t)}{s-t}
\]
(possibly extended-valued).
How difficult is it to find a point $t$ satisfying $h'_+(t) < 0$?

When $h$ is Lipschitz, the most basic randomized strategy --- uniformly sampling random points $t$ in the interval --- solves this problem with high probability.  Denoting the Lipschitz constant by $L$, the right derivative $h'_+$ always lies in the interval $[-L,L]$.  Denote the measure of the set where $h'_+ \ge 0$ by $\lambda$.  Then, providing the average slope satisfies
\[
\frac{h(q)-h(p)}{q-p} ~=~ -\sigma ~<~ 0,
\]
the fundamental theorem of calculus implies
\[
-\sigma ~=~ \frac{1}{q-p} \int_p^q h'_+ ~\ge~
L\Big(\frac{\lambda}{q-p} - 1\Big).
\]
Hence the probability that a uniformly distributed random point $t \in [p,q]$ fails to satisfy $h'(t) < 0$ is no larger than $1-\frac{\sigma}{L}$.  Thus for small $\sigma$, using at least $\frac{L}{\sigma}$ independent samples, the probability of success is at least $\frac{1}{2}$.

However, we seek a deterministic algorithm, so we instead consider the following simple method, similar in spirit to one described in \cite{ddmit}.  We repeatedly bisect the interval $[p,q]$, each time discarding the subinterval over which the function $h$ decreases the least. The algorithm checks whether the right derivative at the midpoint of the current interval is negative, in which case it terminates.

\begin{alg}[bisection]
\label{bisection}
{\rm
\begin{algorithmic}
\STATE
\IF{$h'_+(p) < 0$}
\RETURN{$p$}
\ENDIF
\STATE	$l=p$
\STATE	$r=q$
\WHILE{not done}
\STATE  $m = \frac{1}{2}(l+r)$
\IF{$h'_+(m) < 0$}
\RETURN{$m$}
\ELSIF{$	2h(m) < f(l) + f(r)$}
\STATE	$r=m$
\ELSE
\STATE	$l=m$
\ENDIF
\ENDWHILE
\end{algorithmic}
}
\end{alg}

\noindent
Notice that the algorithm initially calls the oracle at the left endpoint $p$ of the given interval, calculates the function value at the right endpoint $q$, and then calls the oracle once during each bisection.

In general, this algorithm may fail to terminate.  It is easy to construct a Lipshitz function $h$ satisfying $h(p) > h(q)$ and yet the derivative of $h$ at the initial endpoint $p$ and at every midpoint $m$ is positive.  To rule out such oscillatory examples, we can rely on directional semismoothness of $h$, which in this univariate setting simply means that the right derivative exists and is right continuous,
\[
\lim_{t \downarrow \bar t} h'_+(t) ~=~ h'_+(\bar t),
\]
and that the left derivative also exists and is left continuous:
\[
h'_-(t) ~=~ \lim_{s \uparrow t} \frac{h(s) - h(t)}{s-t}
\qquad
\mbox{satisfies}
\qquad
\lim_{t \uparrow \bar t} h'_-(t) ~=~ h'_-(\bar t).
\]

For Lipschitz functions $h$, these two properties amount exactly to {\em semismoothness}.  Most Lipschitz functions in practice are semismooth, including in particular, convex and concave functions, and piecewise smooth functions.  Furthermore, any linear combination of semismooth functions is semismooth.  When the function $h$ is semismooth, the Clarke subdifferential is given by
\[
\partial^c h(t) ~=~ \mbox{conv}\{h'_-(t)\, ,\, h'_+(t)\},
\]
and the following property also holds \cite[Lemma 2]{mifflin-semismooth} and \cite[Lemma 2.2]{henrion-outrata}:
\[
\lim_{t \downarrow \bar t} h'_-(t) ~=~ h'_+(\bar t)
\qquad
\mbox{and}
\qquad
\lim_{t \uparrow \bar t} h'_+(t) ~=~ h'_-(\bar t).
\]

Semismoothness is more than enough to prove termination of the line search.  The simple argument below also applies to nonlipschitz functions.

\begin{prop}[Termination of the line search]~~
\label{terminates}
Suppose that the function $h \colon [p,q] \to \R$ satisfies
$h(p) > h(q)$, and that its left and right derivatives satisfy the semismoothness conditions
\begin{eqnarray*}
\lim_{t \downarrow \bar t} h'_+(t) ~=~ h'_+(\bar t) \qquad \mbox{for}~ \bar t \in [p,q) \\
\lim_{t \uparrow \bar t} h'_+(t) ~=~ h'_-(\bar t) \qquad \mbox{for}~ \bar t \in (p,q].
\end{eqnarray*}
Then Algorithm~\ref{bisection} terminates.
\end{prop}

\pf
If the algorithm does not terminate, then it generates monotonic sequences $l_k \uparrow$ and $r_k \downarrow$, satisfying $r_k-l_k \to 0$,
\[
h'_+(l_k) ~\ge~ 0, \qquad \mbox{and} \qquad
\frac{h(r_k) - h(l_k)}{r_k-l_k} ~\le~ -\sigma ~<~ 0
\]
for each iteration $k=0,1,2,\ldots$.  Denote the two sequences' mutual limit by $\bar m$.  Semismoothness ensures $h'_+(l_k) \to h'_-(\bar m)$, so $h'_-(\bar m) \ge 0$.

If $r_k = \bar m$ for all large $k$, then
\[
\frac{h(r_k) - h(l_k)}{r_k-l_k} ~=~ \frac{h(\bar m) - h(l_k)}{\bar m -l_k} ~\to~ h'_-(\bar m),
\]
which is a contradiction.  Hence for all large $k$ we have $q > r_k > \bar m$, so $h'_+(r_k) \ge 0$, and
semismoothness ensures
$h'_+(r_k) \to~ h'_+(\bar m)$, so $h'_+(\bar m) \ge 0$.  We deduce
\[
h(\bar m) - h(l_k) ~\ge~ -\frac{\sigma}{2}(\bar m - l_k)
\qquad \mbox{and} \qquad h(r_k) - h(\bar m) ~>~ -\frac{\sigma}{2}(r_k - \bar m).
\]
Adding now gives a contradiction.
\finpf

\section{The optimization algorithm}
To minimize a locally Lipschitz function $f \colon \X \to \R$ using the directional subgradient oracle described above, we study the following algorithm.  The method we describe is essentially that of \cite{mit}, but with the deterministic line search described in the preceding section.  Given a current point $x \in \X$ and a current Goldstein subgradient,
the method proceeds as follows.

\begin{alg}[nonsmooth minimization]
\label{minimization}
{\rm
\begin{algorithmic}
\STATE
\WHILE{not done}
		\IF{$|g| \le \epsilon$}
			\RETURN{$x$}								\hfill \COMMENT{{\em Small subgradient so stop.}}
		\ENDIF
		\STATE $\hat g = \frac{g}{|g|}$					\hfill \COMMENT{{\em Normalize subgradient.}}
		\STATE  $x' = x - \delta \hat g$				\hfill \COMMENT{{\em Trial step of fixed length}.}
		\IF{$f(x) - f(x') \ge \frac{\delta\epsilon}{3}$}
			\STATE $x=x'$								\hfill \COMMENT{{\em Sufficient decrease so update point.}}
			\STATE $g = G(x,0)$			\hfill \COMMENT{{\em Re-initialize subgradient.}}
		\ELSE
			\STATE										\hfill \COMMENT{{\em Insufficient decrease so update subgradient.}}
			\STATE Define $h$ on $[0,\delta]$ by
			\begin{eqnarray*}
			x(t) &=& x + (t-\delta) \hat g \\
			h(t) & = & f\big( x(t) \big) ~-~ \frac{\epsilon t}{2}.
			\end{eqnarray*}
			Apply Algorithm \ref{bisection} (bisection), using the formula
			\[
			h'_+(t) ~=~ \ip{G(x(t),\hat g)}{\hat g} ~-~ \frac{\epsilon}{2}
			\]
			to find $t \in [0,\delta]$ satisfying
			$h'_+(t) < 0$.
			\STATE $g = \mbox{shortest vector in}~ [g,G(x(t),\hat g)]$
		\ENDIF
\ENDWHILE
\end{algorithmic}
}
\end{alg}

When the objective $f$ is directionally semismooth, Algorithm \ref{bisection} terminates by Proposition \ref{terminates}, which in turn guarantees termination of Algorithm \ref{minimization}, as we shall now prove.  We use the following simple tool, following \cite{mit}.

\begin{lem}
If two vectors $g,g' \in B_L(0)$ satisfy $\ip{g'}{g} \le \frac{1}{2}|g|^2$, then the shortest vector $g''$ in the line segment $[g,g']$ satisfies
\[
|g''|^2 ~\le~ |g|^2 \Big(1 - \frac{|g|^2}{16L^2} \Big).
\]
\end{lem}

\pf
For all $t \in [0,1]$ we have
\[
|g''|^2
~\le~
|g + t(g'-g)|^2
~=~
|g|^2 + t^2|g'-g|^2 + 2t \ip{g}{g'-g}
~ \le ~
(1-t)|g|^2 + 4L^2t^2.
\]
Setting $t = \frac{|g|^2}{8L^2}$ proves the result.
\finpf

We can now prove the validity of the algorithm, again imitating parts of the argument in \cite{mit}, which we reproduce for ease of reading.

\begin{thm}[finite termination] \label{finite}
Suppose that we apply Algorithm \ref{minimization} to a directionally semismooth function
$f \colon \X \to \R$ that is bounded below.  Suppose that the directional subgradient map in Oracle \ref{oracle} is $L$-bounded.  Then the algorithm terminates with a point $x \in \X$ and a subgradient $g \in \partial_\delta f(x)$ satisfying\ $|g| \le \epsilon$.
\end{thm}

\pf
Suppose that the current subgradient $g$ is {\em inadequate}, in the sense that, in the terminology of the algorithm description, it is neither small, nor generates sufficient decrease.
We then apply the bisection method, Algorithm \ref{bisection}, to the given function $h$.  The initial interval is $[p,q] = [0,\delta]$, and the average slope satisfies
\[
\frac{h(q)-h(p)}{q-p} ~=~
\frac{1}{\delta} \Big( f(x) - \frac{\delta\epsilon}{2} - f(x')\Big) ~<~ - \frac{\epsilon}{6}.
\]
We thus arrive at a subgradient $g' \in \partial_{\delta} f(x)$ satisfying
$\ip{g'}{\hat g} < \frac{\epsilon}{2}$.  We deduce $\ip{g'}{g} < \frac{|g|^2}{2}$.  The algorithm replaces the current subgradient $g$ by the shortest vector $g''$ in the line segment $[g,g']$.
Since $g'' \in \partial_\delta f(x)$, we can repeat this shortening process, providing that $g''$ is also inadequate.  Suppose that the subgradient $g$ remains inadequate after completing $k$ such shortening steps.  Let
$\rho_i$ denote the quantity $\frac{|g|^2}{16L^2}$ after $i=0,1,2,\ldots,k$ steps.
Then $\rho_0 \le \frac{1}{16}$ and for each $i$ we have
\[
0 ~<~ \rho_{i+1} ~\le~ \rho_i(1-\rho_i),
\]
so
\[
\frac{1}{\rho_{i+1}} ~\ge~ \frac{1}{\rho_i} + \frac{1}{1 - \rho_i} ~>~ \frac{1}{\rho_i} + 1.
\]
Consequently,
\[
\frac{1}{\rho_k} ~\ge~ 16 + k,
\]
so we deduce
\[
\frac{\epsilon^2}{16L^2} ~<~ \frac{|g|^2}{16L^2} ~\le~ \frac{1}{16+k}.
\]
Hence after no more than
\[
16 \Big(\frac{L^2}{\epsilon^2} - 1 \Big)
\]
shortening steps, each requiring one line search, we arrive at an adequate subgradient
$g \in \partial_\delta f(x)$.  To summarize, starting at any point with an inadequate subgradient, we require no more than $\frac{16L^2}{\epsilon^2}$ line searches before finding an adequate subgradient $g$.

There are now two possibilities.  Either the subgradient $g$ satisfies $|g| \le \epsilon$, in which case we stop, or we perform a reduction step, replacing the current point $x$ by $x - \delta \frac{g}{|g|}$, thereby decreasing the objective value by at least the quantity $\frac{\delta\epsilon}{3}$.  Since the objective is bounded below, beginning from the initial point $x_0$, this procedure terminates after no more than
$\lceil \frac{3}{\delta\epsilon}\big(f(x_0) - \min f)\big)\rceil$ reduction steps
and hence
\bmye \label{searches}
\left\lceil \frac{3(f(x_0) - \min f)}{\delta\epsilon} \right\rceil \cdot \frac{16L^2}{\epsilon^2}
\emye
line searches.  Proposition \ref{terminates} ensures that each line search requires only finitely many oracle calls, completing the proof.
\finpf

\section{Complexity of the line search}
To complete our complexity analysis for the minimization algorithm, we simply need to bound the number of oracle calls needed by each line search, and multiply by our bound (\ref{searches}) on the number of line searches.  Consider, therefore, the bisection method.  When the function $h \colon [p,q] \to \R$ is convex, the problem is trivial: since
\[
h(p) > h(q) \quad \Rightarrow \quad h'_+(p) < 0,
\]
the algorithm terminates at the first oracle call.  More generally, we proceed by correcting any lack of convexity in $h$ by adding a convex perturbation $s \colon [p,q] \to \R$.

Recall that, for any interval $J$, a function $h \colon J \to \R$ is {\em difference-of-convex} when there exists a convex function $s \colon J \to \R$ such that $h+s$ is also convex.  For such functions, we have the following tool.

\begin{lem}
\label{estimate}
Consider a function $h \colon [p,q] \to \R$, and a convex function $s \colon [p,q] \to \R$ such that $h+s$ is also convex.
For any points $x < y$ in the interval $[p,q]$, if $h'_+(x) \ge 0$, then
\[
\frac{h(y) - h(x)}{y-x} ~\ge~ s'_+(x)-s'_-(y).
\]
\end{lem}

\pf
Denote the convex function $h+s$ by $r$.
The convex functions $r$ and $s$ satisfy
\[
r'_+(x) ~\in~ \partial r(x) \qquad \mbox{and} \qquad
s'_-(y) ~\in~ \partial s(y).
\]
Hence
\begin{eqnarray*}
s'_-(y)(x-y)
&\le&
s(x)-s(y) ~=~ r(x)-h(x)-r(y)+h(y) \\
&\le&
h(y)-h(x) + r'_+(x)(x-y) ~\le~  h(y)-h(x) + s'_+(x)(x-y),
\end{eqnarray*}
and the result follows.
\finpf

We can then use the change in derivative of the necessary perturbation $s$ to bound the number of iterations in the line search.

\begin{thm} \label{time}
Consider a function $h \colon [p,q] \to \R$, and a convex function $s \colon [p,q] \to \R$ such that $h+s$ is also convex.  If the bisection method, Algorithm \ref{bisection}
evaluates $h'_+$, the right-derivative, $k \ge 1$ times without terminating, then
\[
\frac{h(q) - h(p)}{q-p} ~\ge~ \frac{s'_+(p)-s'_-(q)}{k}.
\]
\end{thm}

\pf
We proceed by induction on the number of evaluations $k=1,2,3,\ldots$.  The case $k=1$ follows immediately from Lemma \ref{estimate}, by setting $x=p$ and $y=q$.

Suppose that the result holds for any points $p<q$ and for any number of evaluations no larger than $k$.  Now consider an instance of the algorithm that completes $k+1$ evaluations.  After the first evaluation, consider the midpoint $m = \frac{1}{2}(p+q)$.  There are two possible cases, depending on whether or not
\bmye \label{dichotomy}
2h(m) ~<~ h(p)+h(q).
\emye
We consider them in turn.

Suppose first that inequality (\ref{dichotomy}) holds.  After the first evaluation, the algorithm makes $k$ further evaluations, beginning with the initial interval $[p,m]$.  Hence, by the induction hypothesis,
\[
\frac{h(m) - h(p)}{m-p} ~\ge~ \frac{s'_+(p)-s'_-(m)}{k}.
\]
Since the algorithm did not terminate during the first two evaluations, we know
$h'_+(m) \ge 0$.  Applying Lemma \ref{estimate} with $x=m$ and $y=q$ shows
\[
\frac{h(q) - h(m)}{q-m} ~\ge~ s'_+(m)-s'_-(q).
\]
Hence
\begin{eqnarray*}
s'_+(p)-s'_-(q)
&\le& \big(s'_+(p)-s'_-(m)\big) ~+~ \big(s'_+(m)-s'_-(q)\big) \\
&\le&
k \frac{h(m) - h(p)}{m-p} ~+~ \frac{h(q) - h(m)}{q-m} \\
&=&
(k-1) \frac{h(m) - h(p)}{m-p} ~+~ \Big( \frac{h(m) - h(p)}{m-p} ~+~ \frac{h(q) - h(m)}{q-m} \Big) \\
&=&
(k-1) \frac{h(m) - h(p)}{m-p} ~+~  2\frac{h(q) - h(p)}{q-p} \\
&<&
(k-1) \frac{h(q) - h(p)}{q-p} ~+~  2\frac{h(q) - h(p)}{q-p} ~=~ (k+1) \frac{h(q) - h(p)}{q-p},
\end{eqnarray*}
as required.

The case where inequality (\ref{dichotomy}) fails is similar.  After the first bisection, the algorithm makes $k$ further bisections, beginning with the initial interval $[m,q]$.  Hence, by the induction hypothesis,
\[
\frac{h(q) - h(m)}{q-m} ~\ge~ \frac{s'_+(m)-s'_-(q)}{k}.
\]
Since the algorithm did not terminate during the first bisection, we know
$h'_+(p) \ge 0$.  Applying Lemma \ref{estimate} with $x=p$ and $y=m$ shows
\[
\frac{h(m) - h(p)}{m-p} ~\ge~ s'_+(p)-s'_-(m).
\]
Hence
\begin{eqnarray*}
s'_+(p)-s'_-(q)
&\le& \big(s'_+(p)-s'_-(m)\big) ~+~ \big(s'_+(m)-s'_-(q)\big) \\
&\le&
\frac{h(m) - h(p)}{m-p} ~+~ k\frac{h(q) - h(m)}{q-m} \\
&=&
\Big( \frac{h(m) - h(p)}{m-p} ~+~ \frac{h(q) - h(m)}{q-m} \Big)
 ~+~ (k-1) \frac{h(q) - h(m)}{q-m} \\
&=&
2\frac{h(q) - h(p)}{q-p} ~+~ (k-1) \frac{h(q) - h(m)}{q-m}  \\
&\le&
2\frac{h(q) - h(p)}{q-p} ~+~ (k-1) \frac{h(q) - h(p)}{q-p} ~=~ (k+1) \frac{h(q) - h(p)}{q-p},
\end{eqnarray*}
as required.
\finpf

\begin{defn}
{\rm
Given any interval $J \subset \R$, the {\em concave deviation} of a function $h \colon J \to \R$ is the infimum of the Lipschitz constants of convex functions $s \colon J \to \R$ such that the sum $h+s$ is also convex.
}
\end{defn}
Consider, for example, a $\rho$-{\em weakly convex} function $h$, for some constant
$\rho  \ge 0$, meaning that the function
$t \mapsto h(t) + \frac{\rho}{2}t^2$ is convex.

\begin{prop}
\label{weakly}
Any \mbox{$\rho$-weakly} convex function $h \colon [p,q] \to \R$, for $\rho \ge 0$, has concave deviation at most $\frac{\rho}{2}(q-p)$.
\end{prop}

\pf
The function $s(t) = \frac{\rho}{2}(t-\frac{p+q}{2})^2$ is convex, with Lipschitz constant $\frac{\rho}{2}(q-p)$, and $h+s$ is also convex.
\finpf

The concave deviation for functions that are not weakly convex may shrink more slowly than the length of the interval.  For example, on the interval $[-\delta,\delta]$, the function $h(t) = -|t|^\frac{3}{2}$  has concave deviation $\frac{3}{2}\sqrt{\delta}$, and the piecewise linear function $-|\cdot|$ has concave deviation $1$.

The concave deviation of a function $h \colon [p,q] \to \R$ that is difference-of-convex may not be finite.  An example is the concave function $\sqrt{\cdot}$ on the interval $[0,1]$.  However, if $h$ extends to a difference-of-convex function on an open interval containing the interval $[p,q]$, then its nonconvexity bound on $[p,q]$ must be finite, since we can write $h$ as a difference of convex functions, each of which must be Lipschitz on $[p,q]$.

Consider any continuous semi-algebraic function $h \colon \R \to \R$.  We can partition $\R$ into finitely-many closed intervals $J$ such that each restriction $h|_J$ is either convex or concave.  In general, such a partition may not guarantee that $h$ is difference-of-convex: an example is the function $x^{\frac{1}{3}}$.  However, if $h$ is also Lipschitz, then each convex or concave ingredient $h|_J$ extends to a corresponding convex or concave Lipschitz function on $\R$, and from these we can easily decompose $h$ into a difference of convex Lipschitz functions.  Thus all semi-algebraic functions on $\R$ are difference-of-convex, with finite concave deviation on any bounded interval.

\begin{cor}[Line search complexity]~
\label{line}
If a function $h \colon [p,q] \to \R$ has finite concave deviation $M$ and average rate of decrease
\[
\sigma ~=~ -\frac{h(q)-h(p)}{q-p}  ~>~ 0,
\]
then the number of evaluations of $h'_+$, the right-derivative, required before the bisection method, Algorithm \ref{bisection}, terminates is no more than $1 + \big\lfloor \frac{2M}{\sigma} \big\rfloor$.
\end{cor}

\pf
Suppose that the bisection method evaluates the left-derivative
$k \ge 1$ times without terminating.  Fix any value $M' > M$.
By assumption, there exists a convex function $s$ with Lipschitz constant less than $M'$ such that the sum $h+s$ is also convex.  From Theorem \ref{time}, we deduce the inequalities
\[
-\sigma ~\ge~ \frac{s'_+(p)-s'_-(q)}{k} ~>~ \frac{-2M'}{k}
\]
so $k < \frac{2M'}{\sigma}$.  Since $M'$ was arbitrary, we deduce $k \le \frac{2M}{\sigma}$,
and hence $k \le \lfloor \frac{2M}{\sigma} \rfloor$.  The result follows.
\finpf

Given an open interval $I$, consider a difference-of-convex function $h \colon I \to \R$.  As observed in \cite{hartman}, such functions are characterized by having left and right derivatives everywhere, which furthermore are of bounded variation on every compact interval in $I$.  Any such function $h$ also has a second derivative $D^2 h$ in the distributional sense:  in general it is a signed Radon measure on the interval $I$ (see \cite{dudley}).

We review briefly the underlying construction.  Consider any convex function  $s \colon I \to \R$.  Its right derivative $s'_+$ is nondecreasing and right-continuous, and hence defines a nonnegative Radon measure $D^2 s$ on the interval $I$ via the property
\[
(D^2 s)(p,q] ~=~ s'_+(q) - s'_+(p) \qquad \mbox{for all}~ p<q ~\mbox{in}~ I.
\]
More generally, for any difference-of-convex function $h \colon I \to \R$, consider any convex function
$s \colon I \to \R$ such that $h+s$ is also convex.  The second derivative $D^2h$ is just the signed measure $D^2(h+s) - D^2 s$,
which is independent of the choice of $s$.  Convexity of $h$ is characterized by the property
$D^2 h \ge 0$.  More generally, the Hahn decomposition decomposes $D^2 h$ uniquely into a difference of nonnegative Radon measures supported on disjoint sets.  These positive and negative parts satisfy
\[
D^2 h ~=~ (D^2 h)^+ ~-~ (D^2 h)^-.
\]

\begin {thm}
If the function $h \colon [p,q] \to \R$ is difference-of-convex, then its concave deviation is
\[
\frac{1}{2} (D^2 h)^- (p,q).
\]
\end{thm}

\pf
For any convex function $s : [p,q] \to \R$ such that $h+s$ is also convex, the second derivatives satisfy
\[
0 ~\le~ D^2 s ~=~ D^2 (h+s) - D^2 h ~\ge~ -D^2 h,
\]
so $D^2 s \ge (D^2 h)^-$.
If $s$ is $M$-Lipschitz on $[p,q]$, then
\[
M
~\ge~
\max \{ -s'_+(p) , s'_-(q) \} ~\ge~
\frac{s'_-(q) - s'_+(p)}{2} \\
~=~
\frac{1}{2} (D^2 s) (p,q) ~\ge~ \frac{1}{2} (D^2 h)^- (p,q).
\]

If the right-hand side is infinite, this completes the proof, so suppose that it is finite.  Define a function $g \colon (p,q] \to \R$ by
\[
g(t) ~=~ (D^2 h)^- (p,t).
\]
Then $g$ is a nonnegative nondecreasing left-continuous function that is bounded above and
$g(t) \downarrow 0$ as $t \downarrow p$.  Now define a convex function $s \colon [p,q] \to \R$ by
\[
s(t) ~=~ \int_p^t g(\tau)\,d\tau.
\]
Then $s'_-(t) = g(t)$ for all $t \in (p,q]$ and $s'_+(p) = 0$.  Furthermore we have
\[
D^2(h+s) ~=~ D^2 h + D^2 s ~=~ D^2 h + (D^2 h)^- ~=~ (D^2 h)^+ ~\ge~ 0,
\]
so the sum $h+s$ is also convex.  The function
\[
t ~\mapsto~ \tilde s(t) ~=~ s(t) - \frac{1}{2}s'_-(q) t
\]
is also convex, as is $h+\tilde s$, and the function $\tilde s$ has Lipschitz constant
\[
-\tilde s'_+(p) ~=~ \tilde s'_-(q) ~=~ \frac{1}{2} s'_-(q) ~=~ \frac{1}{2} (D^2 h)^- (p,q).
\]
This completes the proof.
\finpf

As an illustration, we have the following result.

\begin{cor}[Piecewise linear functions] \label{pw}
Consider any continuous piecewise linear function $h \colon [p,q] \to \R$, with $m$ derivative discontinuities $t_1 < t_2 < \cdots < t_m$ in the interval $(p,q)$. Define $t_0 = p$ and $t_{m+1}=q$, and let $g_i$ be the value of the derivative on the  interval $(t_i,t_{i+1})$ for $0 \le i \le m$.  Then $h$ has concave deviation
\[
\frac{1}{2}\sum_{i=1}^m (g_{i-1} - g_i)^+.
\]
If $h$ is $L$-Lipschitz, then this bound is no larger than $\big\lceil \frac{m}{2} \big\rceil L$.
\end{cor}

\pf
Denoting a unit point mass at the point $t$ by $\delta_t$, we have
\[
D^2 h ~=~ \sum_i (g_i - g_{i-1})\delta_{t_i},
\]
and hence
\[
(D^2 h)^- ~=~ \sum_i (g_{i-1} - g_i)^+ \delta_{t_i},
\]
from which the claimed equation follows.  The inequality is an easy consequence, using $|g_i| \le L$ for each $i$.
\finpf

As an illustration, we present an example that underlines why the line search complexity estimate in Corollary \ref{line} is the best that we can expect in general.  In outline, while the functions $h \colon [p,q] \to \R$ that we consider satisfy $h(p) > h(q)$, their derivatives may often be positive.

\begin{exa}[Optimality of the line search]
{\rm
Consider any constant $M > 0$, and a set $T \subset [0,1)$ of cardinality strictly less than $2M$.  Then there exists a function $h \colon [0,1] \to \R$ with concave deviation less than $M$, that satisfies $h(0)=0$ and $h(1)=-1$, and has strictly positive derivative throughout $T$.

To see this, suppose first $T \subset (0,1)$.  (The case when $T$ contains zero is an easy modification.)  Enumerate the points in increasing order:
\[
t_1 < t_2 < \cdots < t_k,
\]
where $k<2M$.  Define $h(0) = 0$ and $h(1)=-1$.
Fix any small $\gamma > 0$, and define
\[
h(t_i - \gamma) ~=~ -t_i - \gamma^2 \quad \mbox{and} \quad
h(t_i + \gamma) ~=~ -t_i + \gamma^2 \quad \mbox{for}~ i=1,2,\ldots,k.
\]
At intermediate points in $[0,1]$, define $h$ by linear interpolation.  A quick calculation, using Corollary \ref{pw}, shows that $h$ has concave deviation
\[
\frac{1}{2}
\sum_{i=1}^{k-1}
\Big(
\frac{t_{i+1} - t_i + 2\gamma^2}{t_{i+1} - t_i - 2\gamma} + \gamma
\Big)
~+~
\frac{1}{2}
\Big(
\frac{1 - t_k + \gamma^2}{1 - t_k - \gamma} + \gamma
\Big)
~=~ \frac{k}{2} + O(\gamma) ~<~ M
\]
providing that $\gamma$ is sufficiently small.

Now consider any line search method applicable to functions $h \colon [0,1] \to \R$ satisfying $h(0)=0$ and $h(1) = -1$, relying on evaluations of the value $h$ and the right derivative $h'_+$ at points chosen one-by-one, and terminating once a derivative is negative.  Suppose that the method is guaranteed to terminate after at most $k$ queries providing that the underlying function $h$ has concave deviation strictly less than some given value $M > 0$.  The example above proves $k \ge 2M$.
}
\end{exa}

\section{Multivariate functions}
To understand the complexity of Algorithm \ref{minimization} (nonsmooth minimization), we apply our analysis in the previous section to restrictions of multivariate objectives $f$ to line segments.
For any convex set $C \subset \X$, consider a function $f \colon C \to \R$.  Given any length
$\delta > 0$, let $\Lambda(\delta)$ denote the supremum over all points $x,y \in C$ with
$|x-y| \le \delta$ of the concave deviation for the function $h \colon [0,\delta] \to \R$ defined by
\bmye \label{1D}
h(t) ~=~ f\Big(x + \frac{t}{\delta}(y-x)\Big).
\emye
We call the function $\Lambda \colon \R_{++} \to [0,+\infty]$ the {\em nonconvexity modulus} for $f$.   The following illustration follows immediately from Proposition \ref{weakly}.

\begin{prop}
The nonconvexity modulus of any $\rho$-weakly convex function (for $\rho \ge 0$) satisfies
\[
\Lambda(\delta) ~\le~ \frac{\rho\delta}{2}.
\]
\end{prop}

More generally, the function $f$ is {\em difference-of-convex} when there exists a convex function $q \colon C \to \R$ such that $f+q$ is also convex.

\begin{prop}
Consider a convex set $C \subset \X$ and functions $f,q \colon C \to \R$ with both $q$ and $f+q$ convex.  If $q$ is $M$-Lipschitz, then the nonconvexity modulus of $f$ satisfies $\Lambda(\delta) \le M$ for all
$\delta > 0$.
\end{prop}

\pf
Consider any points $x,y \in C$ with $|x-y| \le \delta$, and the function $h$ defined by equation (\ref{1D}).  The function
$s \colon [0,\delta] \to \R$ defined by
\[
s(t) ~=~ q\Big(x + \frac{t}{\delta}(y-x)\Big)
\]
is convex, and $M$-lipschitz, and $h+s$ is convex, so the concave deviation of $h$ is no larger than $M$.  The result follows.
\finpf

\noindent
As a consequence, we deduce the following result.

\begin{cor}
Consider any convex sets $C \subset C' \subset \Rn$, where $C$ is nonempty and compact and $C'$ is open, and any difference-of-convex function $f \colon C' \to \R$.  Then the nonconvexity modulus of the restriction $f_C$ is uniformly bounded: there exists a finite constant $M$ such that
$\Lambda(\delta) \le M$ for all $\delta > 0$.
\end{cor}

\begin{cor}
If a function $f \colon \X \to \R$ is the difference $p-q$ between a convex function $p \colon \X \to \R$ and a polyhedral convex function $q \colon \X \to \R$, then the nonconvexity modulus of $f$ is no larger than any Lipschitz constant for $q$.
\end{cor}

More generally, consider a continuous function $f \colon \X \to \R$ that is {\em semi-linear}, in the sense that $\X$ is a finite union of polyhedra, on each of which the function $f$ is affine.  Any such function has a Lipschitz constant $L$, and furthermore a uniform upper bound $m$ on the number of possible gradient discontinuities in any function of the form (\ref{1D}).  By Corollary \ref{pw}, we deduce that the nonconvexity modulus
$\Lambda(\delta)$ is no larger than $\big\lceil \frac{m}{2} \big\rceil L$.

Returning to our analysis of Algorithm \ref{minimization}, we are ready for our main result.

\begin{thm}[complexity of minimization]
Given two constants $\delta,\epsilon > 0$, consider a convex set $C \subset \X$, a function
$f \colon C \to \R$ that is bounded below, an associated $L$-bounded directional subgradient map
$G \colon \X^2 \to \X$, and an initial point $x_0 \in C$ such that
\[
f(x) \le f(x_0) \quad \mbox{and} \quad |y| \le \delta \quad \Rightarrow \quad x+y \in C.
\]
Suppose that $f$ has finite nonconvexity modulus $\Lambda(\delta)$.  Then Algorithm \ref{minimization} (nonsmooth minimization) requires at most
\[
\left\lceil \frac{3(f(x_0) - \min f)}{\delta\epsilon}\right\rceil \cdot \frac{16L^2}{\epsilon^2} \cdot
\Big(1 + \Big\lfloor \frac{12\Lambda(\delta)}{\epsilon} \Big\rfloor \Big)
\]
calls to Oracle \ref{oracle} (directional subgradient)  to find a point $x \in \X$ and a Goldstein subgradient $g \in \partial_\delta f(x)$ satisfying\ $|g| \le \epsilon$.
\end{thm}

\pf
Corollary~\ref{line} with
$\sigma = \frac{\epsilon}{6}$ shows that the bisection method requires at most
\[
1 + \Big\lfloor \frac{12\Lambda(\delta)}{\epsilon} \Big\rfloor
\]
oracle calls to terminate.  Then, one further call returns the desired subgradient
$g' \in \partial_\delta f(x)$ satisfying $\ip{g'}{g} < \frac{|g|^2}{2}$.  Multiplying by the bound (\ref{searches}) on the number of line searches completes the proof.
\finpf

\section{Appendix:  distributional second derivatives}
We saw previously that the concave deviation of a univariate function $h$ is determined by the negative part of its second distributional derivative $D^2h$.  In our application, we consider functions $h$ that are restrictions of the underlying objective $f$ to line segments of fixed length $\delta$.  We would therefore expect the nonconvexity modulus of $f$ to be related to its own distributional second derivative.  Here, we explore that relationship informally.

Consider a locally Lipschitz function $f \colon \Rn \to \R$.  The distributional derivative of $f$ is an $n$-vector $Df$, entries of which are distributions --- linear functionals on the space of smooth, compactly supported functions $g \colon \Rn \to \R$ ({\em test functions}), that are continuous with respect to uniform convergence on compact sets.  We can define $Df$ through the relationship
\[
\ip{u^T (Df)}{g} ~=~ - \int  f (u^T \nabla g)
\]
for all vectors $u \in \Rn$ and test functions $g \colon \Rn \to \R$.  However, by a suitable version of Rademacher's Theorem \cite[Section 6.2, Theorem 1]{evans-gariepy}, the gradient $\nabla f$ exists almost everywhere and is essentially bounded, and satisfies
\[
\ip{u^T (Df)}{g} ~=~ \int  g (u^T \nabla f).
\]
In standard terminology \cite{evans-gariepy}, we can identify the classical gradient $\nabla f$ with both the distributional derivative $Df$ and the ``weak'' derivative of $f$.

The second distributional derivative of $f$ is an $n$-by-$n$ matrix $D^2 f$, entries of which are distributions.  We can define $D^2 f$ through the relationship
\[
\ip{u^T (D^2 f) v}{g} ~=~ - \int (u^T \nabla f) (v^T \nabla g)
\]
for all vectors $u,v \in \Rn$ and test functions $g$.
If $f$ is smooth, then $D^2 f$ is just the matrix-valued measure with density $\nabla^2 f$.  More generally we must consider $D^2 f$ as a distribution, but at least for {\em convex} functions we can be more specific:  it is a positive-semidefinite-valued Radon measure (\cite{dudley} and
\cite[Section 6.3]{evans-gariepy}).

\begin{exa}[A piecewise linear function]~~ \label{hinge}
{\rm
Consider the convex function $f \colon \R^2 \to \R$ defined by $f(x) = x_1^+$.  For any vectors $u,v \in \R^2$ and smooth, compactly supported function $g \colon \R^2 \to \R$, we have
\begin{eqnarray*}
\ip{u^T (D^2 f) v}{g}
&=&
 - \int_{x_1 > 0} u_1 v^T \nabla g(x)\, dx
~=~ - u_1 \int_{x_1 > 0} \mbox{div}(g(x) v)\, dx \\
&=& -u_1 \int_{\R} (-e_1)^T (g(\bigl[ \begin{smallmatrix} 0 \\ y \end{smallmatrix} \bigr]) v)\, dy ~=~ u_1 v_1 \int g(\bigl[ \begin{smallmatrix} 0 \\ y \end{smallmatrix} \bigr])\, dy,
\end{eqnarray*}
by the Gauss-Green formula.
Thus $D^2 f$ is the matrix $\bigl[ \begin{smallmatrix} \mu & 0 \\ 0 & 0 \end{smallmatrix} \bigr]$, where the measure $\mu$ is related to Lebesgue measure $\lambda$ via
\bmye \label{measure}
\mu(S) ~=~ \lambda\{ s \in \R : \bigl[ \begin{smallmatrix} 0 \\ s \end{smallmatrix} \bigr] \in S \},
\emye
for all measurable subsets of $S \subset \R^2$.
}
\end{exa}

For a more general understanding, we begin with the univariate case.
\begin{exa}[Univariate convex functions] \label{univariate}~~~
{\rm
Consider a convex function $f \colon \R \to \R$.
For $0 < \gamma < 1$, we can construct a smooth approximation $h_\gamma \colon \R \to [0,1]$ of the standard step function, with the following properties:
\[
h_\gamma (t) ~=~
\left\{
\begin{array}{ll}
0 			& (t \le 0) 	\\
\gamma^2	& (t=\gamma^2)	\\
1-\gamma^2	& (t=\gamma - \gamma^2) \\
1			& (t \ge \gamma),
\end{array}
\right.
\]
$h$ is convex on $[0,\gamma^2]$, linear on $[\gamma^2,\gamma-\gamma^2]$, and concave on
$[\gamma-\gamma^2,\gamma]$.  For any interval $(p,q] \subset \R$, the test function
$r_\gamma \colon \R \to [0,1]$ defined by
\[
r_\gamma (t) ~=~
\left\{
\begin{array}{ll}
h_\gamma(t-p)		& (t \le p+\gamma^2 )	\\
1					& (p+\gamma^2 \le t \le q) \\
1 - h_\gamma(t-q)	& (t \ge q)
\end{array}
\right.
\]
converges pointwise to the characteristic function $\chi_{(p,q]}$ pointwise as $\gamma \downarrow 0$.  By  Dominated Convergence we deduce
\[
\int r_\gamma\,d(D^2 f) ~\to~ \int \chi_{(p,q]}\, d(D^2 f) ~=~ (D^2 f)(p,q].
\]
But the left-hand side is
\begin{eqnarray*}
- \int r'_\gamma f'
& = &
- \int_p^{p+\gamma} \Big(\frac{1}{\gamma} + O(1)\Big) f' ~-~
  \int_q^{q+\gamma} \Big(-\frac{1}{\gamma} + O(1)\Big) f' \\
&=&
\frac{f(q+\gamma) - f(q)}{\gamma} ~-~ \frac{f(p+\gamma) - f(p)}{\gamma} ~+~ O(\gamma) \\
& = &
f'_+(q) - f'_+(p) + O(\gamma)
\end{eqnarray*}
as $\gamma \downarrow 0$.  We thus reproduce our earlier definition:
\[
(D^2 f)(p,q] ~=~ f'_+(q) - f'_+(p).
\]
Clearly, this fact also holds for any difference-of-convex function $f \colon \R \to \R$.
}
\end{exa}

The modulus of nonconvexity for a function $f \colon \Rn \to \R$, which we denoted $\Lambda(\delta)$ (for $\delta > 0$) is the supremum of the concave deviation of the restriction of $f$ to line segments of the form
\[
S ~=~ \{ z+tw : 0 \le t \le \delta \}
\]
for some point $z \in \Rn$ and unit direction $w \in \Rn$.
That concave deviation is the measure of $S$ under the negative part of the measure $D^2 (f|_S)$.
We would therefore like to compare the distributional second derivative of this restriction with the distributional second derivative $D^2 f$.  As we see in the next result, we should focus specifically on the {\em directional distributional second derivative} $w^T (D^2 f)w$.

A simple approach is furnished by mollification.  We fix a {\em mollifier} $\phi \colon \Rn \to \R$:  a test function satisfying $\int\phi = 1$ and with the property that, as $\gamma \downarrow 0$, the function
$\phi_{\gamma}(x) = \gamma^{-n}\phi(\frac{1}{\gamma} x)$ converges as a distribution to the Dirac delta function.  Given a Radon measure $\mu$ on $\Rn$ and any test function $g \colon \Rn \to \R$, we can define the convolution $g \star \mu \colon \Rn \to \R$ by
\[
(g \star \mu)(y) ~=~ \int g(y-x)\,d\mu(x).
\]

\begin{thm}[Chain rule via mollification]
Consider a locally Lipschitz function $f \colon \Rn \to \R$, a mollifier $\phi \colon \Rn \to \R$, and a direction $w \in \Rn$.  Then for almost all points $z \in \Rn$, the function $f$ is differentiable almost everywhere on the line $z+\R w$, and the distributional second derivative $D^2 h$ of the function defined by
\[
h(t) = f(z + tw) \qquad (t \in \R)
\]
is the distributional limit, as $\gamma \downarrow 0$, of the convolution
\[
t ~\mapsto~ \Big(\phi_\gamma \star \big(w^T(D^2 f)w\big) \Big)(z+tw).
\]
\end{thm}

\pf
By Rademacher's Theorem and standard properties of convolutions
\cite[Section 4.2, Theorem 1(iv)]{evans-gariepy}, there exists a full measure set $\Omega \subset \Rn$ on which $f$ is differentiable and the convolution $\phi_\gamma \star (w^T \nabla f)$ converges pointwise to the essentially bounded function $w^T\nabla f$.  By Fubini's Theorem, for almost all points $z \in \Rn$, we have $z+tw \in \Omega$ for almost all $t \in \R$.  Restricting attention to such $z$, consider any test function $g \colon \R \to \R$.  Fubini's Theorem implies
\begin{eqnarray*}
\lefteqn{
\int_{t \in \R} g(t) \int_{x \in \Rn} \phi_\gamma(z+tw-x)\, d\big(w^T(D^2 f)w\big)(x)\, dt}
\hspace{3cm} \mbox{}
\\
& & =  \int_{x} \Big( \int_{t} g(t) \phi_\gamma(z+tw-x)\, dt \Big) \, d\big(w^T(D^2 f)w\big)(x)
\\
& & = \int_{x} w^T \nabla f(x) \Big( w^T\int_{t} g(t) \nabla \phi_\gamma(z+tw-x)\, dt \Big) \, dx.
\end{eqnarray*}
(We can interchange the order of differentiation and integration since the test functions $g$ and $\phi_\gamma$ are well behaved.)  Rewriting, integrating by parts, and using Fubini's Theorem and Dominated Convergence again, we obtain
\begin{eqnarray*}
\lefteqn{
\int_{x} w^T \nabla f(x) \int_{t} g(t) w^T \nabla \phi_\gamma(z+tw-x)\, dt \, dx}
\hspace{3cm} \mbox{}
\\
& & =
\int_{x } w^T \nabla f(x) \int_{t} g(t) \frac{d}{dt} \phi_\gamma(z+tw-x)\, dt \, dx
\\
& & =
 - \int_{x } w^T \nabla f(x) \int_{t} g'(t) \phi_\gamma(z+tw-x)\, dt \, dx
 \\
& & =
 - \int_{t} g'(t) \int_{x }\phi_\gamma(z+tw-x)  w^T \nabla f(x) \, dx\, dt \\
& & =
 - \int_{t} g'(t) \big(\phi_\gamma \star (w^T \nabla f)\big)(z+tw)\, dt \\
& & \to~
- \int_{t} g'(t) w^T \nabla f(z+tw)\, dt ~=~ - \int g' h' ~=~ \int g\,d(D^2 h),
\end{eqnarray*}
as desired.
\finpf

\noindent
In this result, the effect of the convolution is to focus attention on the line through the point $z$ in the direction $w$.  In informal language, we deduce that {\em the modulus of nonconvexity $\Lambda(\delta)$ is determined by the concentration of the negative parts of the measures $w^T (D^2 f)w$ around line segments of length $\delta$ in unit directions $w$}.

For more intuition on directional distributional second derivatives of the form $w^T(D^2f)w$, let us consider a convex function $f \colon \R^2 \to \R$.  After a suitable choice of basis we can suppose that
$w$ is the first unit vector $e^1$ and therefore consider the Radon measure $(Df)_{11}$.  To understand this measure, consider the integral
\[
\int_{\R^2} r_\gamma(x_1) g(x_2)\, d(D^2 f)_{11}(x)
\]
for the function $r_\gamma$ of Example \ref{univariate} and any test function $g \colon \R \to \R$.  As $\gamma \downarrow 0$, we observe
\begin{eqnarray*}
\int_{\R^2} r_\gamma(x_1) g(x_2)\, d\big(e_1^T(D^2 f)e_1\big)(x)
&=&
-\int_{\R^2} \big(e_1^T \nabla f(x)\big) \cdot \Big(e_1^T \nabla\big(r_\gamma(x_1) g(x_2)\big)\Big)\,dx
\\
&=&
-\int_{\R^2} \frac{\partial f}{\partial x_1} r'_\gamma(x_1) g(x_2)\,dx_1\,dx_2 \\
& \to &
\int
\Big(
f'(\left[\begin{smallmatrix} q \\ t \end{smallmatrix}\right] ;
\left[\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right]) -
f'(\left[\begin{smallmatrix} p \\ t \end{smallmatrix}\right] ;
\left[\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right])
\Big)
 g(t)\,dt.
\end{eqnarray*}
More generally, this argument suggests, loosely, that $w^T (D^2 f)w$ measures the variation of the directional derivative $f'(\cdot;w)$ along the direction $w$.

\bibliographystyle{plain}
\small
\parsep 0pt

\def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$}

redpajama_set_name: RedPajamaArXiv


Rosendahl, Saskia Seiffert, Rachel Seiffert, Rachel.
Domestic fiction. Drama Feature films. Fiction films. Film adaptations. Historical fiction. Thriller Thrillers (Motion pictures) Video recordings for the hearing impaired. War stories.
Summary: Growing up in a Glasgow community haunted by Ireland's Troubles and religious divides, young Stevie finds his precarious family stability and sense of self threatened by his father's marching band's loyalist paramilitary ties.
Summary: After her parents are arrested for war crimes after World War II, a fourteen-year-old girl leads her four siblings across war-torn Germany and encounters a Jewish man willing to help them complete their journey.


The Department of Educational Leadership and Policy Studies would like to introduce you to Rachel Hodges. Rachel is a secondary English teacher and classroom support coach at Union County High School, where she has taught since 2013. She has served as a professional learning committee leader for the past few years, and sits on the school leadership committee. Hodges has a bachelor's degree in political science from UT and earned her teacher licensure through a post-baccalaureate program at Lincoln Memorial University.
Rachel is one of 15 fellows chosen for the 2017-18 Center for Educational Leadership's Leadership Academy based in the Department of Educational Leadership and Policy Studies. During this full-time, intensive 15-month fellowship program, Rachel will spend four days a week at a school with an experienced mentor principal and a fifth day in coursework and seminars with professors and expert practitioner partners learning the research and theory behind great school leadership. Next, all the fellows will lead an action research capstone project that integrates their learning across the many curricular areas of the academy, and complete an electronic portfolio documenting proficiency in school leadership. Upon successful completion of the program, Rachel will be awarded a master's or educational specialist degree and will be eligible to earn a license to be a school principal in Tennessee.
Congratulations Rachel for being selected as one of the 2017-18 Leadership Academy Fellows!


text:
Dig Updates /
Dig Updates Dig Updates 2017
Jamestown Rediscovery archaeologists continued excavations in the 1907 Memorial Church this March, focusing on the foundations of the 1617 church. For decades visitors have been able to look through glass portals in the floor to see the cobblestone foundations of Jamestown's "third church." It was here that the first General Assembly met on July 30, 1619. Referenced by the Ancient Planters in their Brief Declaration, this church was described as "of timber being fifty foot in length and twenty foot breadth."
The Association for the Preservation of Virginia Antiquities (APVA) found the north and south foundations of the 1617 church during their excavations in 1901. They describe a cobblestone footing with a layer of bricks set on top that made the wall nine inches thick. Upon completion of their excavations, the APVA poured concrete in between surviving sections of wall, and then filled the narrow gap between this foundation and the 1640s brick foundation, which was built just outside. Archaeologists started by cleaning the north wall, which had three sections of wall separated by gaps of several feet. Convinced it was necessary to remove the concrete, archaeologists were confident evidence overlooked by the early 20th century excavators would be revealed. "We figured it was very likely that the APVA poured concrete into the areas where they found no cobblestones or bricks like those in the undamaged portions of the foundation. However, there was hope that some of the yellow clay, which secures those cobbles, might survive under the concrete," explained, field supervisor Mary Anna Hartley.
Their assessment of the concrete's placement was accurate. Not only did the concrete conceal intact clay related to the foundation, but it also hid the clues as to why the wall was so fragmented. The team identified the outlines of approximately six burial shafts along the wall's interior that cut through the 1617 church's north foundation. These graves, likely related to the 1640s church, had undercut and obliterated several feet of the 1617 wall. The removal of the first five feet of concrete revealed an additional four feet of the 1617 wall, which was indicated by intact yellow clay with impressions where cobblestone had once been placed.
Simultaneously, while working on the 1617 foundations, archaeologists opened up a test unit in the back northwest corner of the church. The goal was to find evidence of the 1617 church's west wall, as the location of the east and west ends had not been determined. Shown in 1901 photographs, excavations in that corner were deeper than in other parts of the church. Thus far, the unit is full of the APVA backfill and contains fragments of burned plaster, some nails, brick, and mortar. The only ceramics uncovered in the area were fragments of banded yellow ware and blue shell-edge white ware. Curator Merry Outlaw says, "the ceramics included sherds of a jug and a platter, both dating to about the mid-19th century and relating to food serving. Perhaps they are remnants of ceramic objects broken during picnics in the area, or they may relate to 1860s food consumption in the nearby Civil War fort. "
Finally, discussions began about the conservation of one of Jamestown's most iconic artifacts–a ledger stone known as the "Knight's Tomb." This rectangular black stone slab is approximately six feet long by three feet, and it bears carved depressions in which monumental brass plates were once attached . It is the only surviving ledger stone of its kind in this country.
In 1901 this ledger stone, along with the fragment of another for Reverend John Clough, were found in the chancel aisle, just inside the south entrance of the church. They most likely relate to Jamestown's last colonial church, which was built in the 1680s following the Bacon's Rebellion fire that destroyed the 1640s church.
When the stone was uncovered in the early 20th century, the brasses were missing and it is unknown whose grave it originally marked. However, two knights were buried in the 1617 church on this site. The first, Thomas West, the Lord De La Warre, was buried at Jamestown after dying on the voyage over in 1618. The second, Sir George Yeardley, was buried here in 1627. Research into the ledger stone and to whom it belongs is ongoing.
Early 20th-century reports and photos illustrate that today the Knight's Tomb in approximately the same location where it was uncovered just over a hundred years ago. The APVA moved it only a foot to the south, placed it on a brick base, and then repaired the pre-existing breaks with Portland cement. John Tyler, Jr., an engineer who worked at Jamestown in May and June of 1901 (and also the grandson of President John Tyler) wrote this about the Knight's Tomb discovery:
"Immediately upon entering this door was found a tomb, lying north and south along the aisle. This tomb had been robbed of the brass tablets with which it had been inlaid. The stone, however, bears the channels in which the brass was, as well as the brass bolts, leaded in the stone. These bolts held the tablets, consisting of a rectangular border 2 inches wide, and enclosing in the northeast corner of the stone and a shield, and in the northwest a scroll, and down the middle of the tomb a knight in armor standing upon a rectangular plate which evidently bore the inscription."
Jamestown Rediscovery archaeological conservators consulted with well-known Monuments Conservator Jonathan Appell about moving and conserving the stone. As noted by senior conservator Michael Lavin, "Monuments conservation is completely different from archaeological conservation. There are different methods, tools, and chemicals involved when working with stone." Appell will begin working with the team on Phase I of the restoration of this significant artifact in April. In the meantime, the Reverend John Clough tombstone north of the Knight's Tomb was moved. Like the Knight's Tomb, the APVA had placed it upon a brick base. Both stones will be replaced in the chancel when excavations are completed.
Check out the video update on our Jamestown Rediscovery YouTube channel for a behind the scenes update from our lab!
Dig Update Archive, 2004-present
Mary Ann Richardson

redpajama_set_name: RedPajamaCommonCrawl


SOLAR WARNING BECON LED LIGHT.
Multipurpose solar warning becon LED light designed to luminate and attract attention. Solar panel and light sensitive switch ensures automatic lumination in the dark. Comes in 5 colours and is fully sealed to ensure durability against dust and water. Can be mounted using the mounting bracket to operate either vertically or horizontally.


India is only the fourth country to acquire such specialised and modern capability after the US, Russia and China.
New Delhi: Prime Minister Narendra Modi on Wednesday announced that India had demonstrated anti-satellite missile capability by shooting down a low-orbit satellite, describing it as a rare achievement that puts the country in an exclusive club of space super powers.
Quoting sources, news agency ANI reported that the target was a decommissioned Indian satellite which was eliminated within three minutes. The anti-satellite (A-SAT) weapon was launched at 11:16am, the report added.
India is only the fourth country to acquire such specialised and modern capability after the US, Russia and China. The "entire effort is indigenous. India's stands tall as a space power. It will make India stronger, even more secure and will further peace and harmony", Modi said.


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The 'QUEENS' Web Series Proves to be a Hilarious Portrait of Asian American Millennials
May 20, 2020 May 13, 2020 Lauren Espejo
As a Queens, New York native, I jumped at the chance to review the comedy series QUEENS for the YLWRNGR blog. Directed by Nicole Gomez Fisher and created by Cindy Chu, QUEENS is a fresh, hilarious show that pays homage to my favorite borough with vibrancy and care. Following the Asian American best friend trio of Sarah (Cindy Chu), Kim (Spring Inés Peña), and Gia (Carolina Do), QUEENS brings you shocking twists, a ton of laughs, and a lot of heart. If you threw Insecure and Girls in a blender and added a dash of Asian Americans into the mix, you'd get QUEENS. I'm also obsessed with the fact that it's brought to you by a cast and crew made up of 75% people of color.
I had the opportunity to watch the pilot and second episode while sheltering in place during the coronavirus pandemic, so the distraction was welcome. I'm already going into this viewing experience with the anticipation that I might have something in common with these characters. I'm introduced to Sarah as she recites a warm, heartfelt speech to an older gentleman as they are both perched on the park bench. Once you realize she's practicing vows, it sets the tone for what ensues. I'm then introduced to her fiancé, David. Slender, tall, and long-haired, I thought to myself "hmm, not bad, Sarah!" As time goes on I start to question certain things about their relationship, like why he won't let her around his "basketball club." I brush it off as I'm onto the next and soaking in the boho-chic, social media-hooked whirlwind that is Sarah's friend Gia. As Gia sips on bubble tea simultaneously filming an Instagram story, I'm getting the feeling that there's more than meets the eye with this character. I'm intrigued to see where she'll take us next, as I can tell she's the type to view life as a straight-up adventure.
In comes Kim, with her light pink power suit and stick-straight, long black hair. She's in front of a mirror in a public bathroom rehearsing a job-related exchange in Spanish. You can tell this professional suffers no fools. As Kim goes on to interact with her superiors at the law firm, I'm cringing hard as they throw several racial microaggressions her way. Meanwhile, Sarah enters her home and stumbles across a scene that stuns her and leaves her infuriated. She hightails it out of here (barefoot) and flees to Kim's workplace interrupting a super important meeting. The story unfolds as Sarah desperately picks up the pieces, Gia navigates the trials and tribulations of being a social media influencer, and Kim deals with woes of being the token Asian in a less than POC-friendly work environment.
Needless to say, I'm completely sucked in. Sarah, Gia, and Kim are characters that I could see myself being friends with. Each woman brings something different to the table to create a sort of chaotic harmony. Cindy Chu, Carolina Do, and Spring Inés Peña have effortless chemistry together making their characters' plight relatable. Carolina Do brings panache with ease to Gia. Inés Peña has a composed ferocity that you don't encounter often. Chu's performance is spectacular, honest, and genuine. You want to see Chu's character Sarah and her friends win. As Sarah struggles with her conservative Asian family, you find yourself thinking of your own family strife. Since creating the YLWRNGR blog in 2016, representation of Asian-Americans in the media has been a huge topic of discussion for me. So when a show like QUEENS comes your way, it feels like a breath of fresh air. Even seeing the B-roll footage of Queens, NY in between scenes was exhilarating. I even found myself calling out familiar locations to the screen! This show is truly a love letter to the borough and I can't wait to read the rest of it.
Does this web series sound like your cup of tea? You're in for a treat. QUEENS is hosting a virtual premiere watch party on June 11 at 8pm EST! For tickets, head to bit.ly/queenspremiere. They'll be screening the first two episodes, followed by video Q&A with the cast afterward. The series will launch right after the screening and will be available on the series' website: www.queensseries.com, Amazon Prime, and Vimeo. Make sure to stream the show and show it some love!
Here's the QUEENS series trailer for your viewing pleasure:
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What makes the movie smart is its refusal to cast Troy, a difficult role well-played by Epino, as strictly a villain. Instead, Mendoza delves into the cycle of violence that can be passed down through generations.
The complex tonal, textural and thematic mix here doesn't always work, but it's always interesting and often invigorating.


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AMLO and the Outlook for North American Free Trade
Wed. Oct. 10 2018
NAFTA/USMCA
After thirteen months of intense negotiations, the United States, Mexico and Canada announced they had reached a new trade deal on September 30, 2018. The United States-Mexico-Canada Agreement, or USMCA, is set to replace the 25-year-old North American Free Trade Agreement.
The Mexico Institute hosted Jesús Seade, Mexican President-elect Andrés Manuel López Obrador's chief NAFTA negotiator, for a conversation on the outlook for trade in North America under Mexico's new administration.
Prior to his position as Chief NAFTA Negotiator, Jesus Seade was Vice-President of two leading universities in Hong Kong and China and was a member of the Advisory Board of the Ministers of Trade and Financial Services in Hong Kong. He is a former Ambassador to the General Agreement on Tariffs and Trade (GATT) and served as Deputy Director General of the World Trade Organization (WTO). He has also been a senior adviser to the International Monetary Fund (IMF), the World Bank, and the WTO in Geneva. Seade is the author of several influential publications on public finance, market structure, and international trade. He holds a Bacherlor's Degree in Chemical Engineering from the National Autonomous University of Mexico (UNAM) and a Ph.D. in Economics from the University of Oxford.
Selected Quotes
Jesús Seade
"My mandate was [to] support the process, don't create problems. If there is an issue where you don't quite agree, it has to be a very serious concern before you open the issue, before you create the problem. You will intervene only when you can help, not hinder."
"In my reading, every free trade agreement consists of two parts: one is one sentence that says, 'Hereby, we create free trade.' The other is a thousand pages explaining, 'not quite,' all the blocks and limitations to free trade."
"We had to have an awful week of exploration of alternatives and how to say it. It was what I call my darkest week, without any doubt. For a couple of days, I woke up in the morning shaking – straight from sleep to shaking, feeling that the country was falling apart and it was my fault because I'm not finding a way to bring things together."
"[When] the walls around North America rise a little bit, now Mexico, as the cost-effective, emerging market economy in that group, next to the big economical United States, has a competitive advantage created artificially by the situation, compared to Germany or Korea or China or Brazil. Commercially, there is a benefit."
Mexican President-elect Andrés Manuel López Obrador's Chief NAFTA Negotiator
Duncan Wood
Director, Mexico Institute
NAFTA and USMCA Resource Page
The New NAFTA: Congratulations in Order, More Dealmaking Ahead
The Grinding Gears of North America
Rhetoric Meets Reality at the U.S.-Mexico Border: Immigration and NAFTA in a New Era of Bilateral Relations