  The paper addresses parametric inequality systems described by polynomial
functions in finite dimensions, where state-dependent infinite parameter sets
are given by finitely many polynomial inequalities and equalities. Such systems
can be viewed, in particular, as solution sets to problems of generalized
semi-infinite programming with polynomial data. Exploiting the imposed
polynomial structure together with powerful tools of variational analysis and
semialgebraic geometry, we establish a far-going extension of the \L ojasiewicz
gradient inequality to the general nonsmooth class of supremum marginal
functions as well as higher-order (H\"older type) local error bounds results
with explicitly calculated exponents. The obtained results are applied to
higher-order quantitative stability analysis for various classes of
optimization problems including generalized semi-infinite programming with
polynomial data, optimization of real polynomials under polynomial matrix
inequality constraints, and polynomial second-order cone programming. Other
applications provide explicit convergence rate estimates for the cyclic
projection algorithm to find common points of convex sets described by matrix
polynomial inequalities and for the asymptotic convergence of trajectories of
subgradient dynamical systems in semialgebraic settings.

Optimization and Control