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Convex Analysisの二,三の進展について

The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided.

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Implicit Functions and Solution Mappings

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Functions Defined Implicitly by Equations.- Implicit Function Theorems for Variational Problems.- Regularity properties of set-valued solution mappings.- Regularity Properties Through Generalized Derivatives.- Regularity in infinite dimensions.- Applications in Numerical Variational Analysis.

Implicit Functions and Solution Mappings: A View from Variational Analysis

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

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Local differentiability of distance functions

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.

Stability of Locally Optimal Solutions

We consider parametric families of constrained problems in mathematical programming and conduct a local sensitivity analysis for multivalued solution maps. Coderivatives of set-valued mappings are our basic tool to analyze the parametric sensitivity of either stationary points or stationary point-multiplier pairs associated with parameterized optimization problems. An implicit mapping theorem for coderivatives is one key to this analysis for either of these objects, and in addition, a partial coderivative rule is essential for the analysis of stationary points. We develop general results along both of these lines and apply them to study the parametric sensitivity of stationary points alone, as well as stationary point-multiplier pairs. Estimates are computed for the coderivative of the stationary point multifunction associated with a general parametric optimization model, and these estimates are refined and augmented by estimates for the coderivative of the stationary point-multiplier multifunction in the case when the constraints are representable in a special composite form. When combined with existing coderivative formulas, our estimates are entirely computable in terms of the original data of the problem.

Coderivatives in parametric optimization

We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0źM(p,x), whereM is a multifunction. Our basic implicit multifunction theorem provides an approximation for a generalized derivative of the implicit multifunction in terms of the derivative of the multifunctionM. Our primary focus is on three special cases of inclusions 0źM(p,x) which represent different kinds of generalized variational inequalities, called "variational conditions". Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivatives of the solutions to each kind of variational condition. We characterize a well-known generalized Lipschitz property in terms of generalized derivatives, and use our implicit multifunction theorems to state sufficient conditions (and necessary in one case) for solutions of variational conditions to possess this Lipschitz, property. We apply our results to a general parameterized nonlinear programming problem, and derive a new second-order condition which guarantees that the stationary points associated with the Karush-Kuhn-Tucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.

Implicit multifunction theorems for the sensitivity analysis of variational conditions

Generalized equations are common in the study of optimization through nonsmooth analysis. For instance, variational inequalities can be written as generalized equations involving normal cone mappings, and have been used to represent first-order optimality conditions associated with optimization problems. Therefore, the stability of the solutions to first-order optimality conditions can be determined from the differential properties of the solutions of parameterized generalized equations. In finite-dimensions, solutions to parameterized variational inequalities are known to exhibit a type of generalized differentiability appropriate for multifunctions. Here it is shown, in a Banach space setting, that solutions to a much broader class of parameterized generalized equations are "differentiable" in a similar sense.

https://www.ams.org/tran/1994-345-02/S0002-9947-1994-1260203-5/S0002-9947-1994-1260203-5.pdf

Sensitivity analysis of solutions to generalized equations

Attouch's Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch's Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch's Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.

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Adam B. Levy

Papers

Stability of Locally Optimal Solutions

Coderivatives in parametric optimization

Implicit multifunction theorems for the sensitivity analysis of variational conditions

Sensitivity analysis of solutions to generalized equations

Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings