A sharp Lagrange multiplier rule for nonsmooth mathematical programming problems involving equality constraints

摘要

It is shown that a Lagrange multiplier rule that uses approximate Jacobians holds for mathematical programming problems involving Lipschitzian functions, finitely many equality constraints, and convex set constraints. It is sharper than the corresponding Lagrange multiplier rules for the convex-valued subdifferentials such as those of Clarke [ Optimization and Nonsmooth Analysis, 2nd ed., SIAM, 1990] and Michel and Penot [Differential Integral Equations, 5 (1992), pp. 433-454]. The Lagrange multiplier result is obtained by means of a controllability criterion and the theory of fans developed by A. D. Ioffe [Math. Oper. Res., 9 (1984), pp. 159-189, Math. Programming, 58 (1993), pp. 137-145]. As an application, necessary optimality conditions are derived for a class of constrained minimax problems. An example is discussed to illustrate the nature of the multiplier rule. [References: 21]