Convex composite non-Lipschitz programming

In this paper necessary and sufficient optimality conditions are established without Lipschitz continuity for convex composite continuous optimization model problems subject to inequality constraints. Necessary conditions for the special case of the optimization model involving max–min constraints, which frequently arise in many engineering applications, are also given. Optimality conditions in the presence of Lipschitz continuity are routinely obtained using chain rule formulas of the Clarke generalized Jacobian which is a bounded set of matrices. However, the lack of derivative of a continuous map in the absence of Lipschitz continuity is often replaced by a locally unbounded generalized Jacobian map for which the standard form of the chain rule formulas fails to hold. In this paper we overcome this situation by constructing approximate Jacobians for the convex composite function involved in the model problem using ε-perturbations of the subdifferential of the convex function and the flexible generalized calculus of unbounded approximate Jacobians. Examples are discussed to illustrate the nature of the optimality conditions.