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.

. We provide a highly--refined sequential description of the generalized gradients of Clarke and approximate G--subdifferential of a lower semicontinuous extended--real--valued function defined on a Banach space with a fi--smooth equivalent renorm. In the case of a Frech`et differentiable renorm we give a corresponding result for the corresponding singular objects. Keywords: Lipschitz functions, lower semi--continuous functions, sub-derivatives, variational principles, distance functions, tangent cones, normals, smooth renorms, Clarke--subdifferentials, G--subdifferentials. AMS (1991) subject classification: Primary 49J52, Secondary 49J50, 58C20. 2 1 Introduction. There are two natural ways to define a (non--convex) generalized derivative @f of a non--differentiable real--valued function. As discussed in [1, 3] one can do so topologically as for example in [16] or sequentially as in [17, 18, 19, 21, 22]. Topological constructions have an intrinsically complicated structure usually ...

. We discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis. Keywords: Smooth variational principle, fuzzy sum rules, mean value inequalities and partially smooth spaces. Short Title: Partially smooth variational principles. AMS (1991) subject classification: 49J50, 49J52. 1 Introduction Smooth variational analysis [7] has been highly successful in providing tools for the study of non-- smooth analysis and optimization problems: especially when married to viscosity concepts [10, 17]. Outside of smoothable Banach spaces (thus, notably in / L 1 spaces) general constructions such as those of Ioffe [25, 28, 29] require a largely non--constructive intersection over smooth or finite-- dimensional subspaces. Equally, outside of Asplund or Fr'echet spaces the most puissant results [41, 42] fail. Nonetheless, many problems inevitably lie in large (nonsmooth or non--Fr'echet) spaces, X. In such settings the ...

. 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 a Mayer problem of optimal control, whose dynamic constraint is given by a convex-valued differential inclusion. Both state and endpoint constraints are involved. We prove necessary conditions incorporating the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition. Our results weaken the hypotheses and strengthen the conclusions of earlier works. Their main focus is to allow the admissible velocity sets to be unbounded, provided they satisfy a certain continuity hypothesis. They also sharpen the assertion of the Euler-Lagrange inclusion by replacing Clarke’s subgradient of the essential Lagrangian with a subset formed by partial convexification of limiting subgradients. In cases where the velocity sets are compact, the traditional Lipschitz condition implies the continuity hypothesis mentioned above, the assumption of “integrable boundedness” is shown to be superfluous, and our refinement of the Euler-Lagrange inclusion remains a strict improvement on previous forms of this condition.