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Lipschitz behavior of the robust regularization

Adrian S. Lewis(aslewis***at***orie.cornell.edu)
C.H. Jeffrey Pang(cp229***at***cornell.edu)

Abstract: To minimize or upper-bound the value of a function ``robustly'', we might instead minimize or upper-bound the ``epsilon-robust regularization'', defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius of a matrix leads to pseudospectral computations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small epsilon > 0, even if the original function is nonlipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such functions have graphs that are finite unions of sets defined by finitely-many polynomial inequalities, and are commonly encountered in applications.

Keywords: robust optimization, nonsmooth analysis, locally Lipschitz, regularization, semi-algebraic, pseudospectrum, robust control, semidefinite representable, prox-regularity.

Category 1: Robust Optimization

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 3: Applications -- Science and Engineering (Control Applications )


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Entry Submitted: 10/01/2008
Entry Accepted: 10/01/2008
Entry Last Modified: 10/01/2008

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