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## 2.3 Multiscale Simplicial Flat Norm and Integer Lin- ear Programming 

The problem of finding the multiscale simplicial flat norm of the d -chain t (Equation 2.5) can be cast formally as the following optimization problem. 

<img src='content_image/299689.jpg'>

The objective function is piecewise linear in the integer variables x and s . Using standard modeling techniques from linear optimization [7, pg. 18], we can reformulate the problem as the following integer linear program (ILP). 

<img src='content_image/299688.jpg'>

The objective function coefficients need to be nonnegative for this formulation to work – indeed, we have w$_{i}$, v$_{j}$ , and λ nonnegative. Integer linear programming is NP-complete [37]. The linear programming relaxation of the ILP above is obtained by ignoring the integer restrictions on the variables. 

<img src='content_image/299686.jpg'>

