R. A. Poliquin

Local differentiability of distance functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC 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 dC being locally of class C 1+ or such that dC + σ| · |2 is convex around x for some σ > 0. Prox-regularity of C at x corresponds further to the normal cone mapping NC having a hypomonotone truncation around x, and leads to a formula for PC by way of NC . 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.

Prox-regular functions in variational analysis

The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-C2 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.

Tilt Stability of a Local Minimum

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.

Stability of Locally Optimal Solutions

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 keys to these results. A previous characterization of tilt stability arises as a special case.

Integration of subdifferentials of nonconvex functions

IN NONSMOOTH analysis and optimization, subgradients come in many different flavors, e.g. approximate, Dini, proximal, (Clarke) generalized; see [2, 5, 6, 12, 14, 191. These subgradients are important and valuable tools. However, many questions remain unsolved concerning the exact link between the function and its subgradients. For instance, can two functions, not differing by an additive constant, have the same subgradients? In this paper, we study the fundamental problem of determining functions that can be recovered, up to an additive constant, from the knowledge of their subgradients. This “integration” problem is not very well understood, and very few functions or classes of functions are known to be recoverable from their subgradients. In Section 4, we show that if the “basic constraint qualification” holds at R, then the composition of a closed (i.e. lowersemicontinuous) proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients (actually in this case all above-mentioned subgradients are the same). Beside the obvious theoretical interest of this integration problem, it is our hope (or perhaps our long-term goal) that once this problem is better understood, we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives. An example of such an equation that well deserves study is the extended Hamilton-Jacobi equation used in optimal control; see Clarke [2]. Let us also mention that a problem similar to the integration problem is the one of determining the set-valued mappings that are in fact subgradient set-valued mappings (uniqueness is not mandatory); for a contribution to this problem see Janin [8]. Before we look at some of the known cases, where the function can be recovered from its subgradients, let us look at some negative examples. It is clear that not every function can be recovered, up to an additive constant, from its subgradients. We only need to look at the following two functions: 0 x50 0 x10 f(x) = 1 x>o g(x) = 2 x>o.

Generalized Hessian Properties of Regularized Nonsmooth Functions

We take up the question of second-order expansions for a class of functions of importance in optimization, namely, Moreau envelope regularizations of nonsmooth functions

Characterizing the Single-Valuedness of Multifunctions ?

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An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions

. It is shown that when

Proto-derivative formulas for basic subgradient mappings in mathematical programming

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Second-Order Nonsmooth Analysis in Nonlinear Programming

is prox-regular, which includes convex functions and the extended real-valued functions representing problems of nonlinear programming, the many second-order properties that can be formulated around the existence and stability of expansions of the envelopes of