Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming
 Immanuel M. Bomze,
 Etienne De Klerk
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The problem of minimizing a (nonconvex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomialtime approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.
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 Title
 Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming
 Journal

Journal of Global Optimization
Volume 24, Issue 2 , pp 163185
 Cover Date
 20021001
 DOI
 10.1023/A:1020209017701
 Print ISSN
 09255001
 Online ISSN
 15732916
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Approximation algorithms
 Stability number
 Semidefinite programming
 Copositive cone
 Standard quadratic optimization
 Linear matrix inequalities
 Industry Sectors
 Authors

 Immanuel M. Bomze ^{(1)}
 Etienne De Klerk ^{(2)}
 Author Affiliations

 1. ISDS, Universität Wien, Vienna, Austria
 2. Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands