This paper presents a branch-and-bound algorithm for nonconvex quadratic programming, which is based on solving semidefinite relaxations at each node of the enumeration tree. The method is motivated by a recent branch-and-cut approach for the box-constrained case that employs linear relaxations of the first-order KKT conditions. We discuss certain limitations of linear relaxations when handling...

It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, and the nonsin...

In this paper, the common centralized DEA models are extended to the bi-level centralized resource allocation (CRA) models based on revenue efficiency. Based on the Karush–Kuhn–Tucker (KKT) conditions, the bi-level CRA model is reduced to a one-level mathematical program subject to complementarity constraints (MPCC). A recurrent neural network is developed for solving this one-level mathematica...

These notes characterize maxima and minima in terms of first derivatives. I focus primarily on maximization. The problem of minimizing a function f has the same solution (or solutions) as the problem of maximizing −f , so all of the results for maximization have easy corollaries for minimization. The main result of these notes is the Karush-Kuhn-Tucker (KKT) Theorem, recorded as Theorem 3 in Se...

This thesis is mainly concerned with the efficient numerical solution of optimization problems subject to linear PDE-constraints, with particular focus on robust preconditioners and diffuse domain methods. Associated with such constrained optimization problems are the famous first-order KarushKuhn-Tucker (KKT) conditions. For certain minimization problems, the functions satisfying the KKT condi...

In this paper, a stochastic solid transportation problem (SSTP) is constructed where the demand of item at destinations are randomly distributed. Such SSTP formulated with profit maximization form containing selling revenue, cost and holding/shortage item. The proposed framed as nonlinear which optimized through Karush–Kuhn–Tucker (KKT) conditions Lagrangian function. primary model bifurcated i...

Investigation of optimality conditions has been one the most interesting topics in theory multiobjective optimisation problems (MOP). To derive necessary MOP, we consider assumptions called constraints qualifications. It is recognised that Guignard Constraint Qualification (GCQ) efficient and general assumption for scalar objective problems; however, GCQ does not ensure Karush-Kuhn Tucker (KKT)...

In this technical note, we suggest a new definition for an approximate KKT point. The concept of approximate KKT point can then be used on iterates (points found by an optimization algorithm) to check whether the iterates lead to a KKT point. We will concentrate on the following simple optimization problem (P): Min f(x) subject to gi(x) ≤ 0, i = 1, 2, . . . ,m. (1) We will assume that f and eac...

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions,...