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.

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r> 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction. 1

. Optimization problems with complementarity constraints are closely related to optimization problems with variational inequality constraints and bilevel programming problems. In this paper, under mild constraint qualifications, we derive some necessary and su#cient optimality conditions involving the proximal coderivatives. As an illustration of applications, the result is applied to the bilevel programming problems where the lower level is a parametric linear quadratic problem. Key words. optimization problems, complementarity constraints, optimality conditions, bilevel programming problems, proximal normal cones AMS subject classifications. 49K99, 90C, 90D65 PII. S1052623497321882 1. Introduction. The main purpose of this paper is to derive necessary and su#cient optimality conditions for the optimization problem with complementarity constraints (OPCC) defined as follows: (OPCC) min f(x, y, u) s.t. #u, #(x, y, u)# = 0, u # 0, #(x, y, u) # 0 (1.1) L(x, y, u) = 0, g(x, y, u) ...

When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and second-order conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem’s solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epiderivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of “amenable” functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions we can assure optimal controls are bounded? This question is related to the one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial for closing the gap between the conditions arising in the existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the later problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover

this paper we establish some useful calculus rules in the general Banach space setting.

We characterize the optimal incentive scheme for a manager who faces costly e¤ort decisions and whose ability to generate pro…ts for the …rm varies stochastically over time. The optimal contract is obtained as the solution to a dynamic mechanism design problem with hidden actions and persistent shocks to the agent’s productivity. When the agent is risk-neutral, the optimal contract can often be implemented with a simple pay package that is linear in the …rm’s pro…ts. Furthermore, the power of the incentive scheme typically increases over time, thus providing a possible justi…cation for the frequent practice of putting more stocks and options in the package of managers with a longer tenure in the …rm. In contrast to other explanations proposed in the literature (e.g., declining disutility of e¤ort or career concerns), the optimality of seniority-based reward schemes is not driven by variations in the agent’s preferences or in his outside option. It results from an optimal allocation of the manager’s informational rents over time. Building on the insights from the risk-neutral case, we then explore the properties of optimal incentive schemes for risk-averse managers. We …nd that, other things equal, risk-aversion reduces the bene…t of inducing higher e¤ort over time. Whether (risk-averse) managers with a longer tenure receive more or less high-powered incentives than younger ones then depends on the interaction between the degree of risk aversion and the dynamics of the impulse responses for the shocks to the manager’s type. JEL classi…cation: D82

. In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton-Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential type (superdifferential type) mixed boundary condition. We also show that if the value function is upper semicontinuous then it is the maximum subsolution of the Hamilton-Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous then it is the minimum solution of the Hamilton-Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satisfied, then the value function is the unique lower semicontinuous solution of the Hamilton-Jacobi equation with a natural boundary condition and a subdifferential type bound...

. The primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad class of Banach spaces admitting smooth renorms with respect to some bornology, we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions, an extremal principle for nonconvex sets, and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks. Further refinements of the equivalence result are obtained in the case of a Fr'echet differentiable renorm. Based on the new enhanced sum rule, we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces. 1991 Mathematical Subject Classification. Primary 49J52; Secondary 58C20, 46B20. Key words and phrases. Nonsmooth analysis, smooth Banach spaces, variational and extremal pri...

We introduce the notion of "subgradient" for arbitrary functions from R n to R. This "subgradient" generalizes the classical gradient concept. We exhibit a close relationship with Clarke's generalized gradient for arbitrary functions from R n to R as defined in [1]. We apply this "subgradient" to the problem of optimizing an arbitrary function f : R n ! R. Next, we show how the "subgradient " leads to information about "descent directions". Finally, we describe a role that this "subgradient" may play in a particular class of optimal control problems. 1 Introduction Many problems in pure and applied mathematics deal with non differentiable data. For instance, non differentiable "objective functions" arise naturally and frequently in optimization problems. See [1, Section 1.1] for some examples. When theory and techniques are to be developed to optimize (e.g. minimize) such functions, a good generalization of the classical gradient concept seems indispensable. Since the early 19...