The Linear Nonconvex Generalized Gradient and Lagrange Multipliers

Abstract

A Lagrange multiplier rule that uses small generalized gradients is introduced. It includes both inequality and set constraints. The generalized gradient is the linear generalized gradient. It is smaller than the generalized gradients of Clarke and Mordukhovich but retains much of their nice calculus. Its convex hull is the generalized gradient of Michel and Penot if a function is Lipschitz.The tools used in the proof of this Lagrange multiplier result are a coderivative, a chain rule, and a scalarization formula for this coderivative. Many smooth and nonsmooth Lagrange multiplier results are corollaries of this result.It is shown that the technique in this paper can be used for cases of equality, inequality, and set constraints if one considers the generalized gradient of Mordukhovich. An open question is: Does a Lagrange multiplier result hold when one has equality constraints and uses the linear generalized gradient?