The behavior of a minimizing point when an objective function is tilted by adding a small linear term is studied from the perspective of second-order conditions for local optimality. The classical condition of a positive-definite Hessian in smooth problems without constraints is found to have an exact counterpart much more broadly in the positivity of a certain generalized Hessian mapping. This fully characterizes the case where tilt perturbations cause the minimizing point to shift in a lipschitzian manner.
Abstract. Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function d C is continuously differentiable everywhere on an open "tube" of uniform thickness around C. Here a corresponding local theory is developed for the property of d C being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of d 2 C being locally of class C 1+ or such that d 2 C + σ| · | 2 is convex around x for some σ > 0. Prox-regularity of C at x corresponds further to the normal cone mapping N C having a hypomonotone truncation around x, and leads to a formula for P C by way of N C . The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.
Abstract. The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-C 2 functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.
Necessary and sufficient conditions are obtained for the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting. Properties of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the key to these results. A previous characterization of tilt stability comes out as a special case.
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