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.

The eigenvalues of a symmetric matrix depend on the matrix nonsmoothly. This paper describes the nonsmooth analysis of these eigenvalues. In particular, I present a simple formula for the approximate (limiting Frechet) subdifferential of an arbitrary function of the eigenvalues, subsuming earlier results on convex and Clarke subgradients. As an example I compute the subdifferential of the k'th largest eigenvalue.

We consider spectral functions f , where f is any permutationinvariant mapping from C n to R, and is the eigenvalue map from C nn to C n , ordering the eigenvalues lexicographically. For example, if f is the function \maximum real part", then f is the spectral abscissa, while if f is \maximum modulus", then f is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998), which is particularly well suited to the variational analysis of non-Lipschitz spectral functions. We derive a number of necessary conditions for subgradients of spectral functions. For the spectral abscissa, we give both necessary and sucient conditions for subgradients, and precisely identify the case where subdierential regularity holds. We conclude by introducing the notion of semistable programmin...

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.

In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is the Lipschitzian stability of multi-valued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multi-valued and nonsmooth operators. This approach allows us to obtain effectice sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multi-valued and single-valued implicit functions.

In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results for viscosity solutions of Hamilton-Jacobi equations in fi-smooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustrates the flexibility of viscosity subderivatives as a tool for analysis.

. A very general optimization problem with a variational inequality constraint, inequality constraints and an abstract constraint is studied. Fritz John type and Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are derived. Several constraint qualifications for the Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are introduced and their relationships are studied. Applications to bilevel programming problems are also given. Key words. optimization problems, variational inequality constraints, necessary optimality conditions, constraint qualifications, coderivatives of set-valued maps, nonsmooth analysis. AMS subject classifications. 49K99, 90C, 90D65 1. Introduction. An optimization problem with variational inequality constraints (OPVIC) is a special class of an optimization problem over variables x and y in which some or all of its constraints are defined by a parametric variational inequality with y as its...

. A Lagrange multiplier rule for finite dimensional Lipschitz problems is proven that uses a nonconvex generalized gradient. This result uses either both the linear generalized gradient and the generalized gradient of Mordukhovich or the linear generalized gradient and a qualification condition involving the pseudo-Lipschitz behavior of the feasible set under perturbations. The optimization problem includes equality constraints, inequality constraints and a set constraint. This result extends known nonsmooth results for the Lipschitz case. Abbreviated Title: Nonconvex gradients and Lagrange Multipliers. 1991 Mathematics Subject Classification. 90C30, 49J52. Key words and phrases. Lagrange multipliers, nonsmooth analysis, generalized gradients, optimality conditions. 1. Introduction In this paper we derive necessary conditions for a finite dimensional constrained optimization problem. The main differences between this and other work is that a small nonconvex generalized gradient is u...

. We provide several characterizations of compact epi-Lipschitzness for closed convex sets in normed vector spaces. In particular, we show that a closed convex set is compactly epi-Lipschitzian if and only if it has nonempty relative interior, finite codimension, and spans a closed subspace. Next, we establish that all boundary points of compactly epi-Lipschitzian sets are proper support points. We provide the corresponding results for functions by using inf-convolutions and the Legendre--Fenchel transform. We also give an application to constrained optimization with compactly epi-Lipschitzian data via a generalized Slater condition involving relative interiors. R' esum' e. Plusieurs caract'erisations de la propri'et'ee d"epi-Lipschitz compactit 'e sont prouv'ees pour les ensembles convexes ferm'es dans un espace vectoriel norm'e. En particulier, nous prouvons qu'un ensemble convexe ferm'e est compactement 'epi-Lipschitz si, et seulement si, il a un int'erieur relatif non vide, une co...

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz-John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and Farkas' Lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.