This paper presents a recurrent neural network for solving strict convex quadratic programming problems and related linear piecewise equations. Compared with the existing neural networks for quadratic program, the proposed neural network has a one-layer structure with a low model complexity. Moreover, the proposed neural network is shown to have a finite-time convergence and exponential converg...

In this work we propose an outcome space approach for globally solving generalized concave multiplicative problems, a special class of nonconvex problems which involves the maximization of a finite sum of products of concave functions over a nonempty compact convex set. It is shown that this nonconvex maximization problem can be reformulated as an indefinite quadratic problem with infinitely ma...

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear...

A new method is proposed for solving two-stage problems in iinear and quadratic stochastic programmjng. Such problems are dualized, and the dual, elthouSht itself of high dimension, is approximated by a sequence of quadratic programming subproblems whose djmensionality can be kept low. These subproblems correspond to rnaximizing the dual objective over the convex hull of finitely many dual feas...

Convex optimization has developed a wide variety of useful tools critical to many applications in machine learning. However, unlike linear and quadratic programming, general convex solvers have not yet reached sufficient maturity to fully decouple the convex programming model from the numerical algorithms required for implementation. Especially as datasets grow in size, there is a significant g...

Every function of several variables with the continuous second derivative can be convexified (i.e., made convex) by adding to it a quadratic “convexifier”. In this paper we give simple estimates on the bounds of convexifiers. Using the idea of convexification, many problems in applied mathematics can be reduced to convex mathematical programming models. This is illustrated here for nonlinear pr...

Let F be a subset of the n-dimensional Euclidean spaceR represented in terms of a compact convex subset C0 and a set PF of nitely or in nitely many quadratic functions on R such that F = fx 2 C0 : p(x) 0 (8p( ) 2 PF )g. In this paper, we investigate some fundamental properties related to the nite convergence of the successive SDP (semide nite programming) relaxation method proposed by the autho...

This paper considers state-of-the-art convex relaxations for the AC power flow equations and introduces valid cuts based on convex envelopes and lifted nonlinear constraints. These valid linear inequalities strengthen existing semidefinite and quadratic programming relaxations and dominate existing cuts proposed in the literature. Combined with model intersection and bound tightening, the new l...

Sequential quadratic programming (SQP) methods solve nonlinear optimization problems by finding an approximate solution of a sequence of quadratic programming (QP) subproblems. Each subproblem involves the minimization of a quadratic model of the objective function subject to the linearized constraints. Depending on the definition of the quadratic model, the QP subproblem may be nonconvex, lead...

Support vector machine soft margin classifiers are important learning algorithms for classification problems. They can be stated as convex optimization problems and are suitable for a large data setting. Linear programming SVM classifier is specially efficient for very large size samples. But little is known about its convergence, compared with the well understood quadratic programming SVM clas...