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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

We present a generic approach for the sensitivity analysis of solutions to parameterized finite-dimensional optimization problems. We study differentiability and continuity properties of quasi-solutions (stationary points or stationary point-multiplier pairs), as well as their existence and uniqueness, and the issue of when quasi-solutions are actually optimal solutions. Our approach is founded on a few general rules that can all be viewed as generalizations of the classical inverse mapping theorem, and sensitivity analyses of particular optimization models can be made by computing certain generalized derivatives in order to translate the general rules into the terminology of the particular model. The useful application of this approach hinges on an inverse mapping theorem that allows us to compute derivatives of solution mappings without computing solutions, which is crucial since numerical solutions to sensitive problems are fundamentally unreliable. We illustrate how this process works for parameterized nonlinear programs, but the generality of the rules on which our approach is based means that a similar sensitivity analysis is possible for practically any finite-dimensional optimization problem. Our approach is distinguished not only by its broad applicability but by its separate treatment of different issues that are frequently treated in tandem. In particular, meaningful generalized derivatives can be computed and continuity properties can be established even in cases of multiple or no quasi-solutions (or optimal solutions) for some parameters. This approach has not only produced unprecedented and computable conditions for traditional properties in well-studied situations, but has also characterized interesting new properties that might otherwise have remained unexplored.

Solution Sensitivity from General Principles

Subgradient mappings have many roles in variational analysis, such as the formulation of optimality conditions and, in the special case of normal cone mappings, the statement and analysis of variational inequalities and related expressions. Central to the study of perturbations of solutions to systems of conditions in which subgradient mappings appear are concepts of generalized differentiation such as proto-differentiability, which is unhampered by a solution mapping’s potential multivaluedness. This paper extends the known examples of proto-differentiability by showing that a large and important class of “partial” subgradient mappings have this property. Until now the main examples have been the complete subgradient mappings associated with fully amenable functions. The extension, made difficult by the need to deal geometrically with projections of graphs, relies on the notion of a fully amenable function having additional variables that provide a “compatible parameterization.” The results are applied to the sensitivity analysis of generalized variational inequalities in which the underlying set need not be convex and can vary with the parameters.

Variational conditions and the proto-differentiation of partial subgradient mappings

We analyze the sensitivity of parameterized variational inequalities for convex polyhedric sets in reflexive Banach spaces. We compute a generalized derivative of the solution mapping where the formula for the derivative is given in terms of the solutions to an auxiliary variational inequality. These results are distinguished from other work in this area by the fact that they do not depend on the uniqueness of the solutions to the variational inequalities. To obtain our results, we use second-order epi-derivatives to analyze the second-order properties of polyhedric sets. We apply our results to sensitivity analyses of stationary points and KKT pairs associated with constrained infinite-dimensional optimization problems.

Sensitivity of Solutions to Variational Inequalities on Banach Spaces

We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their 'premonotonicity' and lower semicontinuity. This result completes the well-known fact that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is premonotone and has a generalized Lipschitz property called 'Aubin continuity'. The possible single-valuedness and continuity of multifunctions is at the heart of some of the most fundamental issues in variational analysis and its application to optimization. We investigate the impact of our characterizations on several of these issues; discovering exactly when certain generalized subderivatives can be identified with classical derivatives, and determining precisely when solutions to generalized variational inequalities are locally unique and Lipschitz continuous. As an application of our results involving generalized variational inequalities, we characterize when the Karush-Kuhn-Tucker pairs associated with a parameterized optimization problem are locally unique and Lipschitz continuous. Mathematics Subject Classification (1991). 49J52.

Characterizing the Single-Valuedness of Multifunctions ?

We analyze the perturbations of quasi-solutions to a parameterized nonlinear programming problem, these being feasible solutions accompanied by a Lagrange multiplier vector such that the Karush-Kuhn-Tucker optimality conditions are satisfied. We show under a standard constraint qualification, not requiring uniqueness of the multipliers, that the quasi-solution mapping is differentiable in a generalized sense, and we present a formula for its derivative. The results are distinguished from previous ones in the subject, in that they do not entail having to impose conditions to ensure that dual as well as primal elements behave well with respect to sensitivity.

Adam B. Levy

Papers

Solution Sensitivity from General Principles

Variational conditions and the proto-differentiation of partial subgradient mappings

Sensitivity of Solutions to Variational Inequalities on Banach Spaces

Characterizing the Single-Valuedness of Multifunctions ?

Sensitivity of Solutions in Nonlinear Programming Problems with Nonunique Multipliers