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.

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.

Abstract. Much effort in recent years has gone into generalizing the classical Hamiltonian and Euler-Lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints. These generalizations, in which nonsmoothness abounds and gradients are systematically replaced by subgradients, have succeeded in furnishing necessary conditions for optimality which reduce to the classical ones in the classical setting, but important issues have remained unsettled, especially concerning the exact relationship of the subgradient versions of the Hamiltonian equations versus those of the Euler-Lagrange equations. Here it is shown that new, tighter subgradient versions of these equations are actually equivalent to each other. The theory of epi-convergence of convex functions provides the technical basis for this development. Key words. Euler-Lagrange equations, Hamiltonian equations, variational analysis, nonsmooth analysis, subgradients, optimality.

Abstract. Problems of optimal control are considered in the neoclassical Bolza format, which centers on states and velocities and relies on nonsmooth analysis. Subgradient versions of the Euler-Lagrange equation and the Hamiltonian equation are shown to be necessary for the optimality of a trajectory, moreover in a newly sharpened form that makes these conditions equivalent to each other. At the same time, the assumptions on the Lagrangian integrand are weakened substantially over what has been required previously in obtaining such conditions. Key words. Optimal control, calculus of variations, nonsmooth analysis, problem of Bolza, Euler-Lagrange condition, Hamiltonian condition, transversality condition AMS subject classifications. 49K15, 49K05, 49K24 1. Introduction. Among

. 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...

. 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 ...

In this paper we consider a mathematical program with equilibrium con-straints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover we propose new constraint qualifi-cations for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse con-vex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualifica-tion, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification and MPEC Abadie constraint qualification. Key words: mathematical program with equilibrium constraints, necessary optimality conditions, sufficient optimality conditions, constraint qualifications AMS subject classification:49K10, 90C30, 91A65

A notion of approximate Jacobian matrix is introduced for a continuous vector-valued map. It is shown for instance that the Clarke generalized Jacobian is an approximate Jacobian for a locally Lipschitz map. The approach is based on the idea of convexificators of real-valued functions. Mean value conditions for continuous vector-valued maps and Taylor's expansions for continuously Gateaux differentiable functions ( i.e. C¹-functions) are presented in terms of approximate Jacobians and approximate Hessians respectively. Second-order necessary, and sufficient conditions for optimality and convexity of C¹-functions are also given.

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.