Abstract
We consider the problem of approximating the global maximum of a quadratic program (QP) subject to convex non-homogeneous quadratic constraints. We prove an approximation quality bound that is related to a condition number of the convex feasible set; and it is the currently best for approximating certain problems, such as quadratic optimization over the assignment polytope, according to the best of our knowledge.
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References
Alizadeh, F. (1991), Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices, Ph.D. thesis, University of Minnesota, Minneapolis, MN.
Anstreicher, K. and Wolkowicz, H. (1998), On Lagrangian relaxation of quadratic matrix constraints, Research report, corr 98–24, University of Waterloo, Waterloo, Ontario.
Bellare, M. and Rogaway, P. (1995), The complexity of approximating a nonlinear program, Mathematical Programming 69: 429–442.
Frieze, A. and Jerrum, M. (1995), Improved approximation algorithms for max k-cut and max bisection, Proc. 4th IPCO Conference pp. 1–13.
Fu, M., Luo, Z.-Q. and Ye, Y. (1998), Approximation algorithms for quadratic programming, Journal of Combinatorial Optimization 2(1): 29–50.
Fujie, T. and Kojima, M. (1997), Semidefinite programming relaxation for nonconvex quadratic programs, J. Global Optimization 10(4): 367–380.
Gibbons, L.E., Hearn, D.W. and Pardalos, P.M. (1996), A continuous based heuristic for the maximum clique problem, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26: 103–124.
Goemans, M. X. and Williamson, D.P. (1995), Improved approximation algorithms for Maximum Cut and Satisfiability problems using semidefinite programming, Journal of ACM 42: 1115–1145.
Lovász, L. and Shrijver, A. (1990), Cones of matrices and setfunctions, and 01 optimization, SIAM Journal on Optimization 1: 166–190.
Nemirovskii, A., Roos, C. and Terlaky, T. (1998), On maximization of quadratic form over intersection of ellipsoids with common center. Technical report, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands.
Nesterov, Yu.E. (1997), Quality of semidefinite relaxation for nonconvex quadratic optimization, CORE Discussion Paper #9719, Belgium.
Nesterov, Yu.E. (1998), Semidefinite relaxation and nonconvex quadratic optimization, Special Issue Celebrating the 60th Birthday of Professor Naum Shor. Optimization Methods and Software, 9: 141–160.
Nesterov, Yu.E. (1998), Global quadratic optimization via conic relaxation, CORE Working Paper, Catholic University of Louvaine, Belgium.
Nesterov, Yu.E. and Nemirovskii, A.S. (1993), Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms SIAM Publications, SIAM, Philadelphia.
Pardalos, P.M. and Rosen, J.B. (1987), Constrained Global Optimization: Algorithms and Applications Springer-Verlag, Lecture Notes in Computer Sciences 268.
Polijak, S., Rendl, F. and Wolkowicz, H. (1995), A recipe for semidefinite relaxation for 0–1 quadratic programming, Journal of Global Optimization 7: 51–73.
Vavasis, S.A. (1991), Nonlinear Optimization: Complexity Issues, Oxford Science, New York.
Ye, Y. (1997), Approximating quadratic programming with bound constraints, Working Paper, Department of Management Science, The University of Iowa, Iowa City, IA 52242.
Ye, Y. (1999), Approximating quadratic programming with bound and quadratic constraints, Mathematical Programming 84: 219–228.
Ye, Y. (1992), On affine scaling algorithms for nonconvex quadratic programming, Mathematical Programming 56: 285–300.
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Ye, Y. Approximating Global Quadratic Optimization with Convex Quadratic Constraints. Journal of Global Optimization 15, 1–17 (1999). https://doi.org/10.1023/A:1008370723217
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DOI: https://doi.org/10.1023/A:1008370723217