Abstract In this paper, for a nonsmooth semi-infinite multiobjective programming with locally Lipschitz data, some weak and strong Karush-KuhnTucker type optimality conditions are derived. The necessary conditions are proposed under a constraint qualification, and the sufficient conditions are explored under assumption of generalized invexity. All results are expressed in terms of Clarke subdif...

We prove a continuous version of a relaxation theorem for the nonconvex Darboux problem x,T e F(t, z, x, x,,xz). This result allows us to use Warga's open mapping theorem for deriving necessary conditions in the form of a maximum principle for optimization problems with endpoint constraints. Neither constraint qualification nor regularity assumption is supposed.

We obtain a tight semidefinite relaxation of the MAX CUT problem which improves several previous SDP relaxation in the literature. Not only is it a strict improvement over the SDP relaxation obtained by adding all the triangle inequalities to the well-known SDP relaxation, but also it satisfy Slater constraint qualification (strict feasibility).

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...

We consider a class of constrained optimization problems where the objective function is a sum of a smooth function and a nonconvex non-Lipschitz function. Many problems in sparse portfolio selection, edge preserving image restoration and signal processing can be modelled in this form. First we propose the concept of the Karush-Kuhn-Tucker (KKT) stationary condition for the non-Lipschitz proble...

We consider a class of constrained optimization problems where the objective function is a sum of a smooth function and a nonconvex non-Lipschitz function. Many problems in sparse portfolio selection, edge preserving image restoration and signal processing can be modelled in this form. First we propose the concept of the Karush-Kuhn-Tucker (KKT) stationary condition for the non-Lipschitz proble...

For a locally optimal solution to the nonlinear semidefinite programming,under Robinson’s constraint qualification, we show that the nonsingularity of Clarke’sJacobian of the Fischer-Burmeister (FB) nonsmooth system is equivalent to the strongregularity of the Karush-Kuhn-Tucker point. Consequently, from Sun’s paper (Mathe-matics of Operations Research, vol. 31, pp. 761-776, 200...

We consider optimization problems in Banach spaces involving a complementarity constraint defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Ba...

We derive firstand second-order necessary optimality conditions for set-constrained optimization problems under the constraint qualificationtype conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting constraints are giv...

Variational problems under uniform quasiconvex constraints on the gradient are studied. Our technique consists in approximating the original problem by a one−parameter family of smooth unconstrained optimization problems. Existence of solutions to the problems under consideration is proved as well as existence of lagrange multipliers associated to the uniform constraint; no constraint qualifica...