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 paper we deal with sensitivity analysis in convex quadratic programming, without making assumptions on nondegeneracy, strict convexity of the objective function, and the existence of a strictly complementary solution. We show that the optimal value as a function of a right--hand side element (or an element of the linear part of the objective) is piecewise quadratic, where the pieces can be characterized by maximal complementary solutions and tripartitions. Further, we investigate differentiability of this function. A new algorithm to compute the optimal value function is proposed. Finally, we discuss the advantages of this approach when applied to mean--variance portfolio models.

. Two-stage stochastic programs with random right-hand side are considered. Verifiable sufficient conditions for the existence of second-order directional derivatives of marginal values are presented. The central role of the strong convexity of the expected recourse function as well as of a Lipschitz stability result for optimal sets is emphasized. Keywords. Two-stage stochastic programs, directional derivatives of marginal values, strong convexity, sensitivity analysis 1991 Mathematics Subject Classification: 90C15, 90C31 1 Introduction Consider the following two-stage stochastic program minfg(x) +Q¯ (Ax) : x 2 Cg (1.1) Q¯ (Ø) = Z IR s ~ Q(z \Gamma Ø)¯(dz); (1.2) ~ Q(t) = minfq ? y : Wy = t; y 0g (1.3) where g : IR m ! IR is a convex function, C ae IR m is a non-empty closed convex set and ¯ is a Borel probability measure on IR s . Furthermore, q 2 IR m and A 2 L(IR m ; IR s ); W 2 L(IR m ; IR s ). To have (1.1.)-(1.3) well-defined we assume This researc...