CHARACTERIZING THE CONTINGENT CONE'S CONVEX KERNEL (original) (raw)

Convex subcones of the contingent cone in nonsmooth calculus and optimization

Transactions of the American Mathematical Society, 1987

ABSTRACT. The tangential appraximants most useful in nonsmooth analysis and optimization are those which lie" between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that ...

A Different Approach to Cone-Convex Optimization

2013

In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently, Lassere [1] considered a scalar programming problem and showed that if the convexity of the constraint functions is replaced by the convexity of the feasible set, this crucial feature of convex programming can still be preserved. In this paper, we generalize his results by making them applicable to vector optimization problems (VOP) over cones. We consider the minimization of a cone-convex function over a convex feasible set described by cone constraints that are not necessarily cone-convex. We show that if a Slater-type cone constraint qualification holds, then every weak minimizer of (VOP) is a KKT point and conversely every KKT point is a weak minimizer. Further a Mond-Weir type dual is formulated in the modified situation and various duality results are established.

On the intersection of contingent cones

Journal of Optimization Theory and Applications, 1991

In this paper, we give a new condition that ensures the equality TK(x)c~ TL(x)= TK~L(X) for convex closed sets K, L. This condition, which is given in terms of support functions of the sets K, L, generalizes, in a Hilbert space, the usual condition 0eint(K-L).

Generalized (Phi, Rho)-convexity in nonsmooth vector optimization over cones

An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 2016

In this paper, new classes of cone-generalized (Phi,Rho)-convex functions are introduced for a nonsmooth vector optimization problem over cones, which subsume several known studied classes. Using these generalized functions, various sufficient Karush-Kuhn-Tucker (KKT) type nonsmooth optimality conditions are established wherein Clarke's generalized gradient is used. Further, we prove duality results for both Wolfe and Mond-Weir type duals under various types of cone-generalized (Phi,Rho)-convexity assumptions.Phi,Rho

Karush-Kuhn-Tucker Optimality Conditions and Constraint Qualifications trhough a Cone Approach

Synergismus scyentifica …, 2009

This paper deals with optimality conditions to solve nonlinear programming problems. The classical Karush-Kuhn-Tucker (KKT) conditions are demonstrated through a cone approach, using the well known Farkas' Lemma. These conditions are valid at a minimizer of a nonlinear programming problem if a constraint qualification is satisfied. First we prove the KKT theorem supposing the equality between the polar of the tangent cone and the polar of the first order feasible variations cone. Although this condition is the weakest assumption, it is extremely difficult to be verified. Therefore, other constraints qualifications, which are easier to be verified, are discussed, as: Slater's, linear independence of gradients, Mangasarian-Fromovitz's and quasiregularity.