We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz-John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new pr...

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz-John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new pr...

In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are Lipschitz near the optimal solution. Necessary optimality conditions and constraint qualifications in terms of Michel-Penot subdifferential are given.

In this paper we study the nonsmooth semi-infinite programming problem with inequality constraints. First, we consider the notions of local cone approximation $Lambda$ and $Lambda$-subdifferential. Then, we derive the Karush-Kuhn-Tucker optimality conditions under the Abadie and the Guignard constraint qualifications.

In this paper we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints (MPECs). We first show that MPEC-LICQ is not a constraint qualification for the strong (S-) stationary condition when the objective function is nonsmooth. Enhanced Fritz John conditions provide stronger necessary optimality conditions under weaker constraint qualifications. In...

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constrai...

For an inequality system defined by a possibly infinite family of proper functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we obtain characterizations of those reverse-convex inequalities which are consequence of the constrained syst...