Lipschitz Behavior of the Robust Regularization
LIPSCHITZ BEHAVIOR OF THE ROBUST REGULARIZATION ADRIAN S. LEWIS AND C.H. JEFFREY PANG ABSTRACT. To minimize or upper-bound the value of a function “ro- bustly”, we might instead minimize or upper-bound the “ǫ-robust regu- larization”, defined as the map from a point to the maximum value of the function within an ǫ-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 com- putations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small ǫ > 0, even if the original function is nonlipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such func- tions have graphs that are finite unions of sets defined by finitely-many polynomial inequalities, and are commonly encountered in applications. CONTENTS 1. Introduction 2 2. CalmnessasanextensiontoLipschitzness 4 3. Calmness and robust regularization 7 4. Robust regularization in general 9 5. Semi-algebraic robust regularization 13 6. Quadratic examples 21 7. 1-peaceful sets 22 8. Nearly radial sets 27 arXiv:0810.0098v1 [math.OC] 1 Oct 2008 References 31 Key words: robust optimization, nonsmooth analysis, locally Lipschitz, regularization, semi-algebraic, pseudospectrum, robust control, semidefi- nite representable, prox-regularity. AMS subject classifications: 93B35, 49K40, 65K10, 90C30, 15A18, 14P10. Date: October 30, 2018. 1 LIPSCHITZBEHAVIOROF THEROBUST REGULARIZATION 2 1. INTRODUCTION In the implementation of the optimal solution of an optimization model, one is not only concerned with the minimizer of the optimization model, but how numerical errors and perturbations in the problem description and implementation can affect the solution.
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