Journal ArticleDOI
A Polyhedral Approach for Nonconvex Quadratic Programming Problemswith Box Constraints
Yasutoshi Yajima,Tetsuya Fujie +1 more
TLDR
It is shown that cutting plane algorithms can be designed to solve the equivalent problems which minimize a linear function over a convex region and several classes of valid inequalities of the conveX region are proposed.Abstract:
We apply a linearization technique for nonconvex quadratic problems with box constraints. We show that cutting plane algorithms can be designed to solve the equivalent problems which minimize a linear function over a convex region. We propose several classes of valid inequalities of the convex region which are closely related to the Boolean quadric polytope. We also describe heuristic procedures for generating cutting planes. Results of preliminary computational experiments show that our inequalities generate a polytope which is a fairly tight approximation of the convex region.read more
Citations
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Journal ArticleDOI
Mixed-integer nonlinear optimization
TL;DR: An emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP is described and a range of approaches for tackling this challenging class of problems are discussed, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non- Convex structures to obtain improved convex Relaxations.
Journal ArticleDOI
Non-convex mixed-integer nonlinear programming: A survey
Samuel Burer,Adam N. Letchford +1 more
TL;DR: In this paper, the authors survey the literature on non-convex mixed-integer nonlinear programs, discussing applications, algorithms, and software, and special attention is paid to the case in which the objective and constraint functions are quadratic.
Journal ArticleDOI
A Supervised Learning and Control Method to Improve Particle Swarm Optimization Algorithms
Wenyong Dong,MengChu Zhou +1 more
TL;DR: An adaptive particle swarm optimization with supervised learning and control (APSO-SLC) for the parameter settings and diversity maintenance of particle Swarm optimization to adaptively choose parameters, while improving its exploration competence.
Journal ArticleDOI
On convex relaxations for quadratically constrained quadratic programming
TL;DR: This work considers convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and ( possibly nonconvergent) quadRatic constraints.
Journal ArticleDOI
On Nonconvex Quadratic Programming with Box Constraints
Samuel Burer,Adam N. Letchford +1 more
TL;DR: This paper proves several fundamental results concerned with a certain family of convex sets, determining their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets.
References
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Book
Nonlinear Programming: Theory and Algorithms
TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.
Journal ArticleDOI
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
TL;DR: It is argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion.
Journal ArticleDOI
An Interior-Point Method for Semidefinite Programming
TL;DR: A new interior-point-based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices is proposed.
Journal ArticleDOI
Quadratic programming with one negative eigenvalue is NP-hard
TL;DR: It is shown that the problem of minimizing a concave quadratic function with one concave direction is NP-hard, and this result can be interpreted as an attempt to understand exactly what makes nonconvex quadRatic programming problems hard.
Journal ArticleDOI
On the cut polytope
TL;DR: It is shown that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented.
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