Every local minimizer of a smooth constrained optimization problem satisfies the sequential Approximate Karush-Kuhn-Tucker (AKKT) condition. This optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms. It is natural to ask for conditions on the constraints under which AKKT implies KKT. These conditions will be called Strict Constraint Qua...

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lipschitz and not necessarily convex and need not be smooth. We show that if the Slater’s constraint...

For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others.

In recent years, there has been an increasing interest in studying the develpoment of optimality conditions for nondifferentiable multiobjective programming problems. Many authors established and employed some different Kuhn and Tucker type necessary conditions or other type necessary conditions to research optimal solutions; see [1–27] and references therein. In [7], Lai and Ho used the Pareto...

In this paper we obtain secondand first-order optimality conditions of Kuhn-Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that t...

This paper deals with semi-infifinite programming multiple fuzzy-valued objective functions. Firstly, some types of effificient solutions are proposed and illustrated in examples. Then, necessary suffificient Karush-Kuhn-Tucker optimality conditions for functions established.

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the secondorder sufficient condition for optimality...

The notion of low rank approximations arises from many important applications. When the low rank data are further required to comprise nonnegative values only, the approach by nonnegative matrix factorization is particularly appealing. This paper intends to bring about three points. First, the theoretical Kuhn-Tucker optimality condition is described in explicit form. Secondly, a number of nume...

The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used sensitivity analysis, global convergence of first- and second-order algorithms, computing the directional derivative value function. In this paper we discuss naive extensions rank-type qualifications to cone programming semidefinite which are based on Approximate-Karush–K...