Semidefinite programming

SIAM Review

Volumn 38, Issue 1, 1996, Pages 49-95

  Vandenberghe, Lieven ^a   Boyd, Stephen ^b  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b Stanford University   (United States)

Author keywords

Combinatorial optimization; Convex optimization; Eigenvalue optimization; Interior point methods; Semidefinite programming; System and control theory

Indexed keywords

COMBINATORIAL MATHEMATICS; CONTROL THEORY; EIGENVALUES AND EIGENFUNCTIONS; SYSTEM THEORY;

COMBINATORIAL OPTIMIZATION; CONVEX OPTIMIZATION; EIGENVALUE OPTIMIZATION; INTERIOR POINT METHODS; SEMIDEFINITE PROGRAMMING;

MATHEMATICAL PROGRAMMING;

EID: 0030106462     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/1038003     Document Type: Article

References (118)

1. F. ALIZADEH, Combinatorial optimization with interior point methods and semi-definite matrices, Ph.D. thesis, University of Minnesota, Minneapolis, MN, Oct. 1991.
2. _, Combinatorial optimization with semidefinite matrices, in Proc. 2nd Annual Integer Programming and Combinatorial Optimization Conference, Carnegie-Mellon University, Pittsburgh, PA, 1992.
3. _, Optimization over the positive-definite cone: Interior point methods and combinatorial applications, in Advances in Optimization and Parallel Computing, P. Pardalos, ed., North-Holland, the Netherlands, 1992.
4. _, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
5. F. ALIZADEH, J.-P. HAEBERLY, AND M. OVERTON, Primal-dual interior-point methods for semidefinite programming, manuscript, 1994.
6. _, Complementarity and nondegeneracy in semidefinite programming, Math. Programming, Series B, to appear.
7. J. ALLWRIGHT, Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation, SIAM J. Control Optim., 26 (1988), pp. 537-556.
8. _, On maximizing the minimum eigenvalue of a linear combination of symmetric matrices, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 347-382.
9. E. J. ANDERSON AND P. NASH, Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, John Wiley & Sons, New York, 1987.
10. K. M. ANSTREICHER, A combined phase I - phase II scaled potential algorithm for linear programming, Math. Programming, 52 (1991), pp. 429-439.
11. O. AXELSSON, Iterative Solution Methods, Cambridge University Press, London, UK, 1994.
12. M. S. BAZARAA, H. D. SHERALI, AND C. M. SHETTY, Nonlinear Programming. Theory and Algorithms, 2nd ed., Wiley, New York, 1993.
13. R. BELLMAN AND K. FAN, On systems of linear inequalities in Hermitian matrix variables, in Convexity, V. L. Klee, ed., Vol. 7 Proc. Symposia in Pure Mathematics, Amer. Math. Soc., Providence, RI, 1963, pp. 1-11.
14. A. BEN-TAL AND M. P. BENDSØE, A new method for optimal truss topology design, SIAM J. Optim., 3 (1993), pp. 322-358.
15. P. M. BENTLER AND J. A. WOODWARD, Inequalities among lower bounds to reliability: With applications to test construction and factor analysis, Psychometrika, 45 (1980), pp. 249-267.
16. S. BOYD AND L. EL GHAOUI, Method of centers for minimizing generalized eigenvalues, special issue on Numerical Linear Algebra Methods in Control, Signals and Systems, Linear Algebra Appl., 188 (1993), pp. 63-111.
17. S. BOYD, L. EL GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, Vol. 15, Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, June 1994.
18. S. BOYD, L. VANDENBERGHE, AND M. GRANT, Efficient convex optimization for engineering design, in Proc. IFAC Symp. Robust Control Design, Rio de Janeiro, Brazil, Sept. 1994, pp. 14-23.
19. S. BOYD AND S. WU, SDPSOL: A Parser/Solver for Semidefinite Programs with Matrix Structure. User's Guide, Version Alpha, Stanford University, Stanford, CA, March 1995.
20. B. CRAVEN AND B. MONO, Linear programming with matrix variables, Linear Algebra Appl., 38 (1981), pp. 73-80.
21. J. CULLUM, W. DONATH, AND P. WOLFE, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Math. Programming Study, 3 (1975), pp. 35-55.
22. D. DEN HERTOG, Interior Point Approach to Linear, Quadratic and Convex Programming, Kluwer, Norwell, MA, 1993.
23. I. DIKIN, Iterative solution of problems of linear and quadratic programming, Soviet Math. Dokl., 8 (1967), pp. 674-675.
24. W. E. DONATH AND A. J. HOFFMAN, Lower bounds for the partitioning of graphs, IBM J. Research and Development, 17 (1973), pp. 420-425.
25. K. FAN, On a theorem of Weyl concerning eigenvalues of linear transformations. I, Proc. Natl. Acad. Sci. U.S.A., 35 (1949), pp. 652-655.
26. M. K. H. FAN, A quadratically convergent algorithm on minimizing the largest eigenvalue of a symmetric matrix, Linear Algebra Appl., 188-189 (1993), pp. 231-253.
27. M. K. H. FAN AND B. NEKOOIE, On minimizing the largest eigenvalue of a symmetric matrix, in Proc. IEEE Conf. Decision and Control, Dec. 1992, pp. 134-139.
28. L. FAYBUSOVICH, On a matrix generalization of affine-scaling vector fields, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 886-897.
29. A. FIACCO AND G. MCCORMICK, Nonlinear programming: Sequential unconstrained minimization techniques, Wiley, New York, 1968; SIAM Classics in Applied Mathematics Series, 1990, reprint.
30. A. FIACCO AND G. MCCORMICK, Nonlinear programming: Sequential unconstrained minimization techniques, Wiley, New York, 1968; SIAM Classics in Applied Mathematics Series, 1990, reprint.
31. R. FLETCHER, A nonlinear programming problem in statistics (educational testing), SIAM J. Sci. Statist. Comput., 2 (1981), pp. 257-267.
32. _, Semidefinite matrix constraints in optimization, SIAM J. Control Optim., 23 (1985), pp. 493-513.
33. R. FOURER AND S. MEHROTRA, Performance of an augmented system approach for solving least-squares problems in an interior-point method for linear programming, Mathematical Programming Society, Committee on Algorithms Newsletter, 1991, pp. 26-31.
34. A. L. FRADKOV AND V. A. YAKUBOVICH, The S-procedure and duality relations in nonconvex problems of quadratic programming, Vestnik Leningrad Univ. Math., 6 (1979), pp. 101-109, 1973. (In Russian.)
35. R. FREUND, Complexity of an algorithm for finding an approximate solution of a semi-definite program with no regularity assumption, Tech. report OR 302-94, Operations Research Center, MIT, Cambridge, MA, 1994.
36. R. FREUND AND N. NACHTIGAL, A new Krylov-subspace method for symmetric indefinite linear systems, Numerical Analysis Manuscript 94-07, AT&T Bell Laboratories, Murray Hill, NJ, 1994.
37. P. GAHINET AND A. NEMIROVSKY, LMI Lab: A Package for Manipulating and Solving LMIs, INRIA, Le Chesnay, France, 1993.
38. P. GAHINET, A. NEMIROVSKY, A. J. LAUB, AND M. CHILALI, The LMI Control Toolbox, in Proc. IEEE Conf. Decision and Control, Dec. 1994, pp. 2083-2041.
39. P. E. GILL, W. MURRAY, D. B. PONCELEON, AND M. A. SAUNDERS, Preconditioned for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292-311.
40. M. GOEMANS AND D. WILLIAMSON, .878-approximation algorithm for max-cut and max-2sat, Tech. report, MIT, Cambridge, MA, 1994.
41. _, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Tech. report, Dept. of Mathematics, MIT, Cambridge, MA, 1995.
42. C. GOH AND D. TEO, On minimax eigenvalue problems via constrained optimization, J. Optim. Theory Appl., 57 (1988), pp. 59-68.
43. C. C. GONZAGA, Path-following methods for linear programming, SIAM Rev., 34 (1992), pp. 167-224.
44. C. C. GONZAGA AND M. J. TODD, An O(√nL)-iteration large-step primal-dual affine algorithm for linear programming, SIAM J. Optim., 2 (1992), pp. 349-359.
45. M. GRÖTSCHEL, L. LOVÁSZ, AND A. SCHRIJVER, Geometric Algorithms and Combinatorial Optimization, Vol. 2, Algorithms and Combinatorics, Springer-Verlag, New York, Berlin, 1988.
46. J.-P. A. HAEBERLY AND M. OVERTON, Optimizing eigenvalues of symmetric definite pencils, in Proc. 1994 American Control Conf., Baltimore, MD, 1994.
47. C. HELMBERG, F. RENDL, R. J. VANDERBEI, AND H. WOLKOWICZ, An interior-point method for semidefinite programming, SIAM J. Optim., to appear.
48. J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods, Vol. 306, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, Berlin, 1993.
49. J.-B. HIRIART-URRUTY AND D. YE, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numer. Math., 70 (1995), pp. 45-72.
50. M. IRI AND H. IMAI, A multiplicative barrier function method for linear programming, Algorithmica, 1 (1986), pp. 455-482.
51. F. JARRE, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control Optim., 31 (1993), pp. 1360-1377.
52. F. JOHN, On symmetric matrices whose eigenvalues satisfy linear inequalities, in Proc. American Mathematical Society, E. Dyer, W. H. J. Fuchs, A. P. Mattuck, R. C. Buck, F. John, and I. Reiner, eds., Amer. Math. Soc., Providence, RI, 1966, pp. 1140-1145.
53. A. KAMATH AND N. KARMARKAR, A continuous approach to compute upper bounds in quadratic maximization problems with integer constraints, in Recent Advances in Global Optimization, C. Floudas and P. Pardalos, eds., Princeton University Press, Oxford, UK, 1992, pp. 125-140.
54. _, An O(nL) iteration algorithm for computing bounds in quadratic optimization problems, in Complexity in Numerical Optimization, P. M. Pardalos, ed., World Scientific Publishing Co., Singapore, 1993, pp. 254-268.
55. D. KARGER, R. MOTWANI, AND M. SUDAN, Approximate graph coloring by semidefinite programming, Tech. report, Dept. Computer Science, Stanford University, Stanford, CA, 1994.
56. N. KARMARKAR, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), pp. 373-395.
57. L. G. KHACHIYAN AND M. J. TODD, On the complexity of approximating the maximal inscribed ellipsoid for a polytope, Math. Programming, 61 (1993), pp. 137-159.
58. K. C. KIWIEL, Methods of Descent for Nondifferentiable Optimization, Vol. 1133, Lecture Notes in Mathematics, Springer-Verlag, New York, Berlin, 1985.
59. M. KOJIMA, S. KOJIMA, AND S. HARA, Linear algebra for semidefinite programming, Tech. report, Dept. Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan, Oct. 1994.
60. M. KOJIMA, N. MEGIDDO, T. NOMA, AND A. YOSHISE, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, Springer-Verlag, New York, Berlin, 1991.
61. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, An O(√nL) iteration potential reduction algorithm for linear complementarity problems, Math. Programming, 50 (1991), pp. 331-342.
62. M. KOJIMA, S. SHINDOH, AND S. HARA, Interior-point methods for the monotone linear complementarity problem in symmetric matrices, Tech. report, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1994.
63. J. B. LASSERRE, Linear programming with positive semi-definite matrices, Tech. report LAAS-94099, Laboratoire d'Analyse et d'Architecture des Systèmes du CNRS, 1995.
64. B. LIEU AND P. HUARD, La méthode des centres dans un espace topologique, Numer. Math., 8 (1966), pp. 56-67.
65. L. LOVÁsz AND A. SCHRIJVER, Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim., 1 (1991), pp. 166-190.
66. A. I. LUR'E AND V. N. POSTNIKOV, On the theory of stability of control systems, Appl. Math. Mech., 8, 1944. (In Russian.)
67. I. J. LUSTIG, R. E. MARSTEN, AND D. F. SHANNO, Interior point methods for linear programming: Computational state of the art, ORSA J. Comput., 6 (1994), pp. 1-14.
68. B. MOHAR AND S. POLJAK, Eigenvalues in combinatorial optimization, in Combinatorial and GraphTheoretical Problems in Linear Algebra, R. A. Brualdi, S. Friedland, and V. Klee, eds., Springer-Verlag, New York, Berlin, 1993, pp. 107-151.
69. A. NEMIROVSKY, Long-step method of analytic centers for fractional problems, Tech. report 3/94, Technion, Haifa, Israel, 1994.
70. A. NEMIROVSKY AND P. GAHINET, The projective method for solving linear matrix inequalities, in Proc. American Control Conf., June 1994, pp. 840-844.
71. Y. NESTEROV, Infeasible start interior point primal-dual methods in nonlinear programming, Tech. report, CORE, Universite Catholique de Louvain, Belgium, 1994.
72. Y. NESTEROV AND A. NEMIROVSKY, A general approach to polynomial-time algorithms design for convex programming, Tech. report, Centr. Econ. and Math. Inst., USSR Acad. Sci., Moscow, USSR, 1988.
73. _, Optimization over positive semidefinite matrices: Mathematical background and user's manual, USSR Acad. Sci. Centr. Econ. and Math. Inst., Moscow, USSR, 1990.
74. _, Self-concordant functions and polynomial time methods in convex programming, Tech. report, Centr. Econ. and Math. Inst., USSR Acad. Sci., Moscow, USSR, Apr. 1990.
75. _, Conic formulation of a convex programming problem and duality, Tech. report, Centr. Econ. and Math. Inst., USSR Acad. Sci., Moscow, USSR, 1991.
76. _, An interior point method for generalized linear-fractional programming, Math. Programming, Series B., 69 (1995), pp. 117-204.
77. _, Interior-point polynomial methods in convex programming, Vol. 13, Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994.
78. Y. E. NESTEROV AND M. J. TODD, Self-scaled cones and interior-point methods in nonlinear programming, Tech. report 1091, Cornell University, Ithaca, NY, Apr. 1994.
79. M. OVERTON, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 256-268.
80. _, Large-scale optimization of eigenvalues, SIAM J. Optim., 2 (1992), pp. 88-120.
81. M. OVERTON AND R. WOMERSLEY, On the sum of the largest eigenvalues of a symmetric matrix, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 41-45.
82. _, Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices, Math. Programming, 62 (1993), pp. 321-357.
83. C. C. PAIGE AND M. A. SAUNDERS, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
84. C. C. PAIGE AND M. S. SAUNDERS, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), pp. 43-71.
85. E. PANIER, On the need for special purpose algorithms for minimax eigenvalue problems, J. Optim. Theory Appl., 62 (1989), pp. 279-287.
86. G. PATAKI, On the multiplicity of optimal eigenvalues, Tech. report MSRR-604, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA, 1994.
87. _, Cone-LP's and semi-definite programs: Facial structure, basic solutions, and the simplex method, Tech. report, GSIA, Carnegie-Mellon University, Pittsburgh, PA, 1995.
88. S. POLJAK, F. RENDL, AND H. WOLKOWICZ, A recipe for best semidefinite relaxation for (0, 1)-quadratic programming, Tech. report CORR 94-7, University of Waterloo, Ontario, Canada, 1994.
89. F. PUKELSHEIM, Optimal Design of Experiments, Wiley, New York, 1993.
90. M. RAMANA, An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 1993.
91. _, An exact duality theory for semidefinite programming and its complexity implications, DIMACS Tech. report 95-02, RUTCOR, Rutgers University, New Brunswick, NJ, 1995.
92. M. RAMANA AND A. GOLDMAN, Some geometric results in semidefinite programming, J. Global Opt., 7 (1995), pp. 33-50.
93. F. RENDL, R. VANDERBEI, AND H. WOLKOWICZ, A primal-dual interior-point method for the max-min eigenvalue problem, Tech. report, University of Waterloo, Dept. of Combinatorics and Optimization, Waterloo, Ontario, Canada, 1993.
94. U. T. RINGERTZ, Optimal design of nonlinear shell structures, Tech. report FFA TN 1991-18, The Aeronautical Research Institute of Sweden, Stockholm, Sweden, 1991.
95. R. T. ROCKAFELLAR, Convex Analysis, 2nd ed., Princeton University Press, Princeton, NJ, 1970.
96. J. B. ROSEN, Pattern separation by convex programming, J. Math. Anal. Appl., 10 (1965), pp. 123-134.
97. A. SHAPIRO, Weighted minimum trace factor analysis, Psychometrika, 47 (1982), pp. 243-264.
98. _, Extremal problems on the set of nonnegative definite matrices, Linear Algebra Appl., 67 (1985), pp. 7-18.
99. A. SHAPIRO AND M. K. H. FAN, On eigenvalue optimization, SIAM J. Optim., 5 (1995), pp. 552-569.
100. N. Z. SHOR, Cut-off method with space extension in convex programming problems, Cybernetics, 13 (1977), pp. 94-96.
101. _, Minimization Methods for Non-differentiable Functions, Springer Series in Computational Mathematics, Springer-Verlag, New York, Berlin, 1985.
102. _, Quadratic optimization problems, Soviet J. Circuits Systems Sciences, 25 (1987), pp. 1-11.
103. G. SONNEVEND, An 'analytical centre' for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in Lecture Notes in Control and Information Sciences, Vol. 84, Springer-Verlag, New York, Berlin, 1986, pp. 866-878.
104. _, New algorithms in convex programming based on a notion of 'centre' (for systems of analytic inequalities) and on rational extrapolation, Internat. Ser. Numer. Math., 84 (1988), pp. 311-326.
105. P. TARAZAGA AND M. TROSSET, An optimization problem on subsets of the positive-semidefinite matrices, J. Optim. Theory Appl., 79, (1993), pp. 513-524.
106. F. UHLIG, A recurring theorem about pairs of quadratic forms and extensions: A survey, Linear Algebra Appl., 25 (1979), pp. 219-237.
107. L. VANDENBERGHE AND S. BOYD, A polynomial-time algorithm for determining quadratic Lyapunov functions for nonlinear systems, in Eur. Conf. Circuit Theory Design, Davos, Switzerland, 1993, pp. 1065-1068.
108. _, SP: Software for Semidefinite Programming. User's Guide, Beta Version, K. U. Leuven, Leuven, Belgium, and Stanford University, Stanford, CA, Oct. 1994.
109. _, Primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming, Series B, 69 (1995), pp. 205-236.
110. R. J. VANDERBEI AND T. J. CARPENTER, Symmetric indefinite systems for interior point methods, Math. Programming, 58 (1993), pp. 1-32.
111. G. A. WATSON, Algorithms for minimum trace factor analysis, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1039-1053.
112. _, Characterization of the subdifferential of some matrix norms, Linear Algebra Appl., 170 (1992), pp. 33-45.
113. H. WOLKOWICZ, Some applications of optimization in matrix theory, Linear Algebra Appl., 40 (1981), pp. 101-118.
114. M. WRIGHT, Interior methods for constrained optimization, Acta Numerica, 1 (1992), pp. 341-407.
115. 3L) potential reduction algorithm for linear programming, Math. Programming, 50 (1991), pp. 239-258.
116. D. B. YUDIN AND A. S. NEMIROVSKY, Informational complexity and efficient methods for solving complex extremal problems, Matekon, 13 (1977), pp. 25-45.
117. K. ZIETAK, On the characterization of the extremal points of the unit sphere of matrices, Linear Algebra Appl., 106 (1988), pp. 57-75.
118. _, Subdifferentials, faces, and dual matrices, Linear Algebra Appl., 185 (1993), pp. 125-141.