Abstract
A successful technique for some problems in combinatorial optimization is the so-called SDP relaxation, essentially due to L. Lovász, and much developed by M. Goemans and D.P. Williamson. As observed by S. Poljak, F. Rendl and H. Wolkowicz, this technique can be interpreted from the point of view of Lagrangian duality. A central tool for this is dualization of quadratic constraints, an operation pioneered by N.Z. Shor. We synthesize these various operations, in a language close to that of nonlinear programming. Then we show how the approach can be applied to general combinatorial problems.
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J.P. Aubin and I. Ekeland. Estimates of the duality gap in nonconvex optimization. Mathematics of Operations Research, 1 (3): 225–245, 1976.
I. Ekeland and R. Temam. Convex Analysis and Variational Problems. Classics in Applied Mathematids 28. SIAM Publications, 1999.
J.E. Falk. Lagrange multipliers and nonconvex programs. SIAM Journal on Control, 7 (4): 534–545, 1969.
S. Feltenmark and K. C. Kiwiel. Dual applications of proximal bundle methods, including lagrangian relaxation of nonconvex problems. SIAM Journal on Optimization, 10 (3): 697–721, 2000.
T. Fujie and M. Kojima. Semidefinite protramming relaxation for nonconvex quadratic programs. Journal of Global Optimization, 10: 367–380, 1997.
C. Garrod and J.K. Percus. Reduction of the N-particle variational problem. Journal of Mathematical Physics, 5(12), 1964.
A.M. Geoffrion. Lagrangean relaxation for integer programming. Mathematical Programming Study, 2: 82–114, 1974.
M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 6: 1115–1145, 1995.
C. Helmberg, F. Rendl, and R. Weismantel. Quadratic knapsack relaxations using cutting planes and semidefinite programming In W. H. Cunningham, S. T. McCormick, and M. Queyranne, editors, Integer Programming and Combinatorial Optimization V, volume 1084 of Lecture Notes in Computer Science, pages 175–189. Springer Verlag, Berlin, 1996.
J.-B. Hiriart-Urruty and C. Lemaréchal. Convex Analysis and Minimization Algorithms. Springer-Verlag, 1993.
R.A. Horne and Ch.R. Johnson. Matrix analysis. Cambridge University Press, 1989. (New edition, 1999 ).
D. Karger, R. Motwani, and M. Sudan. Approximate graph coloring by semidefinite programming. Journal of the ACM, 45(2):246–265, 1998.
D.E. Knuth. The sandwich theorem. Electronic Journal of Combinatorics, 1(A1), 1994.
F. Körner. A tight bound for the boolean quadratic optimization problem and its use in a branch and bound algorithm. Optimization, 19 (5): 711–721, 1988.
F. Körner and C. Richter. Zur effektiven lösung von booleschen, quadratischen optimierungsproblemen. Numerische Mathematik, 40: 99–109, 1982.
C. Lemaréchal, Yu. Nesterov, and F. Oustry. Duality gap analysis for problems with quadratic constraints, 2001. In preparation.
C. Lemaréchal and F. Oustry. Semi-definite relaxations and lagrangian duality with application to combinatorial optimization. Rapport de Recherche 3710, INRIA, 1999. Submitted to Mathematical Programming.
C. Lemaréchal and A. Renaud. A geometric study of duality gaps, with applications. Mathematical Programming, 2001. to appear.
L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, IT 25: 1–7, 1979.
L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0–1 optimization. SIAM Journal on Optimization, 1(2):166–190, 1991.
Yu.E. Nesterov, H. Wolkowicz, and Y.Y Ye. Semidefinite programming relaxations of nonconvex quadratic optimization. In Lieven Vandenberghe R. Saigal and H. Wolkovicz, editors, Han-book on Semidefinite Programming. Theory, Algorithms and Applications. Kluwer, 2000.
P. Pardalos and H. Wolkowicz. Topics in Semidefinite and Interior-Point Methods. Fields Institute Communications Series 18. American Mathematical Society, 1998.
S. Poljak, F. Rendi, and H. Wolkowicz. A recipe for semidefinite relaxation for (0,1)-quadratic programming. Journal of Global Optimization, 7: 51–73, 1995.
B.N. Pshenichnyi The Linearization Method for Constrained Optimization. Number 22 in Computational Mathematics. Springer-Verlag, 1994.
N.Z. Shor. Class of global minimum bounds of polynomial functions. Cybernetics, 23 (6): 731–734, 1987.
N.Z. Shor. Quadratic optimization problems. Soviet Journal of Computer and Systems Sciences, 25: 1–11, 1987.
N.Z. Shor. Dual estimates in multiextremal problems. Journal of Global Optimization, 2: 411–418, 1992.
N.Z. Shor and A.S. Davydov. Method of opbtaining estimates in quadratic extremal probems with boolean variables. Cybernetics, (2): 207–211, 1985.
L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38 (1): 49–95, 1996.
H. Wolkowicz and Z. Zhao. Semidefinite relaxations for the graph partitioning problem. Discrete Applied Mathematics, 96–97: 461–479, 1999.
Q. Zhao, S.E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite programming relaxations for the quadratic assignment problem. Journal of Combinatorial Optimization, 2 (1): 71–109, 1998.
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Lemaréchal, C., Oustry, F. (2001). SDP Relaxations in Combinatorial Optimization from a Lagrangian Viewpoint. In: Hadjisavvas, N., Pardalos, P.M. (eds) Advances in Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications, vol 54. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0279-7_6
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DOI: https://doi.org/10.1007/978-1-4613-0279-7_6
Publisher Name: Springer, Boston, MA
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