Nonconvex Quadratic Programming Research Articles

A quadratic programming algorithm

By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed.

A generalized design approach to solution of the non-convex quadratic programming problem

This paper outlines a generalized, systematic design approach to solution of the non-convex quadratic programming problem. It is based on a previous formulation of utility of a general system in terms of efficiency and robustness of the system. The approach is to introduce a robustness term of arbitrary magnitude into the design utility function. Mathematically, this makes the problem convex. From a design approach, it yields a more general solution allowing specialization to proceed by decreasing robustness (on an initially convex utility surface in the feasible design space) until the region of the overall optimum is approached. The approach is mathematically related to the Metropolis technique of simulated annealing but a more systematic (less random) solution process is used. It is analogous also to the heuristic technique of Burkard and Bonniger. These two previous techniques are the most effective so far reported for the quadratic programming problem. The robustness approach provides an underpinning for each and opens up further solution options. Applications include layout of buildings and other constructed facilities and information technology layout problems.

Characterization of Positive Definite and Semide Finite Matrices via Quadratic Programming Duality

Positive definite and semidefinite matrices induce well-known duality results in quadratic programming. The converse is established here. Thus if certain duality results hold for a pair of dual quadratic programs, then the underlying matrix must be positive definite or semidefinite. For example, if a strict local minimum of a quadratic program exceeds or equals a strict global maximum of the dual, then the underlying symmetric matrix Q is positive definite. If a quadratic program has a local minimum, then the underlying matrix Q is positive semidefinite if and only if the primal minimum exceeds or equals the dual global maximum and $x^T Qx = 0$ implies $Qx = 0$. A significant implication of these results is that the Wolfe dual may not be meaningful for nonconvex quadratic programs and for nonlinear programs without locally positive definite or semidefinite Hessians, even if the primal second order sufficient optimality conditions are satisfied.

Closed form solutions to nonserial, nonconvex quadratic programming problems using dynamic programming

This paper presents a method for obtaining closed form solutions to serial and nonserial dynamic programming problems with quadratic stage returns and linear transitions. Global parametric optimum solutions can be obtained regardless of the convexity of the stage returns. The closed form solutions are developed for linear, convex, and nonconvex quadratic returns, as well as the procedure for recursively solving each stage of the problem. Dynamic programming is a mathematical optimization technique which is especially powerful for certain types of problems. This paper presents a procedure for obtaining analytical solutions to a class of dynamic programming problems. In addition, the procedure has been programmed on the computer to facilitate the solution of large problems.

The method of bilinear programming for nonconvex quadratic programming

The method of bilinear programming for nonconvex quadratic programming

Var planning and nonconvex quadratic programming

Var planning and nonconvex quadratic programming

A cutting plane algorithm for solving bilinear programs

This paper addresses itself to a special class of nonconvex quadratic program referred to as a bilinear program in the literature. We will propose here a cutting plane algorithm to solve this class of problems. The algorithm is along the lines of H. Tui and K. Ritter, but it differs in its exploitation of the special structure of the problem. Though the algorithm is not guaranteed at this stage of the research to converge to a global optimum, the preliminary results of numerical experiments are encouraging.

Möglichkeiten zur anwendung der erkenntnisse der parametrischen optimierung für die lösnng indefiniter quadratiseher optimierungspromeme

In this paper we give some posibilities for the application of the results of the convex and nonconvex parametric quadratic programming to the computation of nonconvex quadratic programming problems. In the first case we can solve the problem by solving a strongly convex parametric quadratic problem with in general more than one parameter and sme smaller quadratic problems, in the second case by solving an oneparametric nonconvex quadratic problem.

Nonconvex Quadratic Programming via Generalized Polars

A new approach is proposed to linearly constrained nonconvex quadratic programming. The approach is based on generalized polar sets, and is akin to the convex analysis approach to integer programming. We construct a generalized polar of the Kuhn–Tucker polyhedron associated with a quadratic program. This generalized polar is a convex polyhedral cone whose interior contains no complementary feasible solution better than the best known one. This and other properties are used to generate cutting planes similar to the outer polar cuts used in 0-1 programming. An algorithm is then proposed, which does not use cutting planes, but constructs a polytope containing the feasible set and contained in the polar of the latter. This is done by starting with a simplex defined by a subset of the problem constraints, and successively activating other problem constraints, until the resulting polytope is contained in the polar of the feasible set. When this is achieved, the best complementary solution found in the process is optimal, or none exists.

Local optimization for nonconvex quadratic programming

Naval Research Logistics QuarterlyVolume 21, Issue 3 p. 465-490 Article Local optimization for nonconvex quadratic programming Antal Majthay, Antal Majthay University of FloridaSearch for more papers by this authorAndrew Whinston, Andrew Whinston Purdue UniversitySearch for more papers by this authorJohn Coffman, John Coffman Georgia State UniversitySearch for more papers by this author Antal Majthay, Antal Majthay University of FloridaSearch for more papers by this authorAndrew Whinston, Andrew Whinston Purdue UniversitySearch for more papers by this authorJohn Coffman, John Coffman Georgia State UniversitySearch for more papers by this author First published: September 1974 https://doi.org/10.1002/nav.3800210308Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume21, Issue3September 1974Pages 465-490 RelatedInformation

On factored quadratic functions

Necessary and sufficient conditions are derived, which are fulfilled by quadratic functions that can be written as a product of two affine-linear functions plus an additive constant. This criterium characterizes convex and nonconvex quadratic programming problems, which can be solved by one ofSwarup's [1966a, 1966b, 1966c] algorithm.