Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.

This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools.

For a general category of variational inclusions in finite dimensions, a class of parameterizations, called “ample” parameterizations, is identified that is rich enough to provide a full theory of Lipschitz-type properties of solution mappings without the need for resorting to the auxiliary introduction of canonical parameters. Ample parameterizations also support a detailed description of the graphical geometry that underlies generalized differentiation of solutions mappings. A theorem on proto-derivatives is thereby obtained. The case of a variational inequality over a polyhedral convex set is given special treatment along with an application to minimizing a parameterized function over such a set.

S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the first-order optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the classical implicit function theorem, if not quite all, but went far beyond that in ideas and format. Here, Robinson’s theorem is viewed from the perspective of more recent developments in variational analysis as well as some lesser-known results in the implicit function literature on equations, prior to the advent of generalized equations. Extensions are presented which fully cover such results, translating them at the same time to generalized equations broader than variational inequalities. Robinson’s notion of first-order approximations in the absence of differentiability is utilized in part, but even looser forms of approximation are shown to furnish significant information about solutions.

Abstract. A basic relationship is derived between generalized subgradients of a given function, possibly nonsmooth and nonconvex, and those of a second function obtained from it by partial conjugation. Applications are made to the study of multiplier rules in finite-dimensional optimization and to the theory of the Euler-Lagrange condition and Hamiltonian condition in nonsmooth optimal control. Keywords. Subgradients, nonsmooth analysis, Lagrange multiplier rules, Euler-Lagrange

In this paper we study nonlinear programming problems with equality, inequality, and abstract constraints where some of the functions are Fr'echet differentiable at the optimal solution, some of the functions are Lipschitz near the optimal solution, and the abstract constraint set may be nonconvex. We derive Fritz John type and Karush--Kuhn--Tucker (KKT) type first order necessary optimality conditions for the above problem where Fr'echet derivatives are used for the differentiable functions and subdifferentials are used for the Lipschitz continuous functions. Constraint qualifications for the KKT type first order necessary optimality conditions to hold include the generalized Mangasarian--Fromovitz constraint qualification, the no nonzero abnormal multiplier constraint qualification, the metric regularity of the constraint region, and the calmness constraint qualification.

. We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson-Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson-Ursescu theorem we also establish a refined version of the multidirectional mean value inequality which is of independent interest. Key Words. Nonsmooth analysis, subdifferentials, coderivatives, implicit function theorem, solvability, stability, open mapping theorem, metric regularity, multidirectional mean value inequality. AMS (1991) subject classification: 26B05. 1 Research...

In this work we are concerned with state-constrained optimal control problems. Our aim is to derive the optimality conditions and to prove the convergence of the numerical approximations. To deal with these questions, whose difficulty is motivated by the presence of the state constraints, we consider some concepts of stability of the optimal cost functional with respect to small perturbations of the set of feasible states. While weak and strong stability on the right allow us to derive optimality conditions in a non-qualified and qualified form respectively, the weak stability on the left is the key to prove the convergence of the numerical approximations. 1 Introduction In this paper we consider an optimal control problem with pointwise state constraints governed by a semilinear elliptic partial differential equation. We study the influence of some stability properties in the derivation of the optimality conditions satisfied by the optimal control, obtained through the penalization o...

Concepts of conditioning have long been important in numerical work on solving systems of equations, but in recent years attempts have been made to extend them to feasibility conditions, optimality conditions, complementarity conditions and variational inequalities, all of which can be posed as solving “generalized equations” for set-valued mappings. Here, the conditioning of such generalized equations is systematically organized around four key notions: metric regularity, subregularity, strong regularity and strong subregularity. Various properties and characterizations already known for metric regularity itself are extended to strong regularity and strong subregularity, but metric subregularity, although widely considered, is shown to be too fragile to support stability results such as a radius of good behavior modeled on the Eckart-Young theorem.

ABSTRACT. To minimize or upper-bound the value of a function “robustly”, we might instead minimize or upper-bound the “ǫ-robust regularization”, 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 computations. 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 functions have graphs that are finite unions of sets defined by finitely-many polynomial inequalities, and are commonly encountered in applications. CONTENTS