Existing global optimization techniques for nonconvex quadratic programming (QP) branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch-and-bound nodes. An open question of theoretical interest is how to develop a finite branch-and-bound algorithm for nonconvex QP. One idea, which guarantees a finite number of branching decisions, is to enforce...

In this paper we propose a primal-dual algorithm for the solution of general nonlinear programming problems. The core of the method is a local algorithm which relies on a truncated procedure for the computation of a search direction, thus resulting suitable for large scale problems. The truncated direction produces a sequence of points which locally converges to a KKT pair with superlinear conv...

This paper deals with optimality conditions to solve nonlinear programming problems. The classical Karush-Kuhn-Tucker (KKT) optimality 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 equalit...

In this paper we study nonlinear programming problems with equality, inequality, and abstract constraints where some of the functions are Fréchet differentiable at the optimal solution, some of the functions are Lipschitz near the optimal solution, and the abstract constraint set may be nonconvex. We derive Fritz John type and Karush–Kuhn–Tucker (KKT) type first order necessary optimality condi...

Abstract We introduce a flexible, open source implementation that provides the optimal sensitivity of solutions of nonlinear programming (NLP) problems, and is adapted to a fast solver based on a barrier NLP method. The program, called sIPOPT evaluates the sensitivity of the KKT system with respect to model parameters. It is paired with the open-source IPOPT NLP solver and reusesmatrix factoriz...

This paper introduces the research status of convex quadratic bilevel programming at present firstly. Secondly, it analyzes the problem of convex quadratic bilevel programming models, concepts and properties. On this basis, using the optimality conditions of KKT, the problem will be transformed into a single complementary slackness relaxation problem. To solve this problem, we propose an orthog...

In this paper we consider the problem of minimizing a quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the rst part of the paper, we show that (i) the number of values of the objective function at KKT points is bounded by 3n + 1 where n is the dimension of the problem; (ii) given a KKT point that is not a global minimizer, it is ...

The paper rst investigates Newton-type methods for generalized equations with compact solution sets. The analysis of their local convergence behavior is based, besides other conditions, on the upper Lipschitz-continuity of the local solution set mapping of a simply perturbed generalized equation. This approach is then applied to the KKT conditions of a nonlinear program with inequality constrai...

A modiication of the (infeasible) primal{dual interior point method is developed. The method uses multiple corrections to improve the centrality of the current iterate. The maximum number of corrections the algorithm is encouraged to make depends on the ratio of the eeorts to solve and to factorize the KKT systems. For any LP problem, this ratio is determined right after preprocessing the KKT s...