Abstract
Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optinmization, and the development of efficient primal-dual interior-point rnethods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NP-hard combinatorial optimnization problenis to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to non-convex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include non-convex potential reduction methods, interior point cutting plane methods, primal-dual IPMs and first-order algorithms for solving SDPs, branch and out approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alizadeh, F. and Goldfarb, D. (2003). Second-order cone programming. Mathematical Programming, 95:3–51.
Alizadeh, F., Haeberly, J.P.A., and Overton, M.L. (1998). Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results. SIAM Journal on Optimization, 8:746–751.
Andersen, E.D., Gondzio, J., Mészáros, Cs., and Xu, X. (1996). Implementation of interior point methods for large scale linear programming. In: T. Terlaky (ed.), Interior Point Methods for Linear Programming, pp. 189–252. Kluwer Academic Publishers, The Netherlands.
Anjos, M.F. (2004). An improved semidefinite programming relaxation for the satisfiability problem. Mathematical Programming, to appear.
Anjos, M.F. and Wolkowicz, H. (2002a). Geometry of semidefinite maxcut relaxations via matrix ranks. Journal of Combinatorial Optimization, 6:237–270.
Anjos, M.F. and Wolkowicz, H. (2002b). Strengthened semidefinite relaxations via a second lifting for the maxcut problem. Discrete Applied Mathematics, 119:79–106.
Anstreicher, K.M. (1997). On Vaidya's volumetric cutting plane method for convex programming. Mathematics of Operations Research, 22:63–89.
Anstreicher, K.M. (1999). Towards a practical volumetric cutting plane method for convex prograniming. SIAM Journal on Optimization, 9:190–206.
Anstreicher, K.M. (2000). The volumetric barrier for semidefinite programming. Mathematics of Operations Research, 25:365–380.
Arora, S. and Lund, C. (1996). Hardness of approximations. In: D. Hochbaum (ed.), Approximation Algorithms for NP-hard Problems, Chapter 10. PWS Publishing.
Atkinson, D.S. and Vaidya, P.M. (1995). A cutting plane algorithm for convex programming that uses analytic centers. Mathematical Programming, 69:1–43.
Avis, D. (2003). On the complexity of testing hypermetric, negative type, k-gonal and gap inequalities. In: J. Akiyama, M. Kano (eds.), Discrete and Computational Geometry, pp. 51–59. Lecture Notes in Computer Science, vol. 2866, Springer.
Avis, D. and Grishukhin, V.P. (1993). A bound on the k-gonality of facets of the hypermetric cone and related complexity problems. Computational Geometry: Theory & Applications, 2:241–254.
Bahn, O., Du Merle, O., Goffin, J.L., and Vial, J.P. (1997). Solving nonlinear multicommodity network flow problems by the analytic center cutting plane method. Mathematical Programming, 76:45–73.
Balas, E., Ceria, S., and Cornuejols, G. (1993). A lift and project cutting plane algorithm for mixed 0–1 programs. Mathematical Prograrnming, 58:295–324.
Barahona, F. (1983). The maxcut problem in graphs not contractible to K 5. Operations Research Letters, 2:107–111.
Barahona, F. and Mahjoub, A.R. (1986). On the cut polytope. Mathematical Programming, 44:157–173.
Benson, H.Y. and Vanderbei, R.J. (2003). Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming. Mathematical Programming, 95:279–302.
Benson, S.J., Ye, Y., and Zhang, X. (2000). Solving large-scale semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10:443–461.
Bomze, I., Dur, M., De Klerk, E., Roos, C., Quist, A.J., and Terlaky, T. (2000). On copositive programming and standard quadratic optimization problems. Journal of Global Optimization, 18:301–320.
Bomze, I. and De Klerk, E. (2002). Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. Journal of Global Optimization, 24:163–185.
Borchers, B. (1999). CSDP: A C library for semidefinite programming. Optimization Methods and Software, 11:613–623.
Burer, S. (2003). Semidefinite programming in the space of partial positive semidefinite matrices. SIAM Journal on Optimization, 14:139–172.
Burer, S. and Monteiro, R.D.C. (2003a). A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Mathematical Programming, 95:329–357.
Burer, S. and Monteiro, R.D.C. (2003b). Local minima and convergence in low-rank semidefinite programming. Technical Report, Department of Management Sciences, University of Iowa.
Burer, S., Monteiro, R.D.C., and Zhang, Y. (2002a). Solving a class of semidefinite programs via nonlinear programming. Mathematical Programming, 93:97–102.
Burer, S., Monteiro, R.D.C., and Zhang, Y. (2002b). Rank-two relaxation heuristics for max-cut and other binary quadratic programs. SIAM Journal on Optimization, 12:503–521.
Burer, S., Monteiro, R.D.C., and Zhang, Y. (2002c). Maximum stable set formulations and heuristics based on continuous optimization. Mathematical Programming, 94:137–166.
Burer, S., Monteiro, R.D.C., and Zhang, Y. (2003). A computational study of a gradient based log-barrier algorithm for a class of largescale SDPs. Mathematical Programming, 95:359–379.
Chvátal, V. (1983). Linear Programming. W.H. Freeman and Company.
Conn, A.R., Gould, N.I.M., and Toint, P.L. (2000). Trust-Region Methods. MPS-SIAM Series on Optimization, SIAM, Philadelphia, PA.
De Klerk, E. (2002). Aspects of Semidefinite Progranmming: Interior Point Algorithms and Selected Applications. Applied Optimization Series, vol. 65, Kluwer Academic Publishers.
De Klerk, E. (2003). Personal communication.
De Klerk, E., Laurent, M., and Parrilo, P. (2004a). On the equivalence of algebraic approaches to the minimization of forms on the simplex. In: D. Henrion and A. Garulli (eds.), Positive Polynomials in Control, LNCIS, Springer, to appear.
De Klerk, E. and Pasechnik, D. (2002). Approximating the stability number of graph via copositive programming. SIAM Journal on Opmization, 12:875–892.
De Klerk, E., Pasechnik, D.V., and Warners, J.P. (2004b). Approximate graph coloring and max-k-cut algorithms based on the theta function. Journal of Combinatorial Optimization, to appear.
De Klerk, E., Roos, C., and Terlaky, T. (1998). Infeasible-start semidefinite programming algorithms via self-dual embeddings. Fields Institute Communications, 18:215–236.
De Klerk, E. and Van Maaren, H. (2003). On semidefinite programming relaxations of 2+p-SAT. Annals of Mathematics of Artificial Intelligence, 37:285–305.
De Klerk, E., Warners, J., and Van Maaren, H. (2000). Relaxations of the satisfiability problein using semidefinite programming. Journal of Automated Reasoning, 24:37–65.
Deza, M.M. and Laurent, M. (1997). Geornetry of Cuts and Metrics. Springer-Verlag, Berlin.
Elhedhli, S. and Goffin, J.L. (2004). The integration of an interior-point cutting plane method within a branch-and-price algorithm. Mathematical Programming, to appear.
Feige, U. and Goemans, M. (1995). Approximating the value of two prover proof systems withe applications to MAX-2SAT and MAX-DICUT. Proceedings of the 3rd Isreal Symposium on Theory of Computing and Systems, pp. 182–189. Association for Computing Machinery, New York.
Frieze, A. and Jerrum, M.R. (1997). Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18:61–77.
Fukuda, M., Kojima, M., Murota, K., and Nakata, K. (2000). Exploiting sparsity in semidefinite programming via matrix comrpletion I: General framework. SIAM Journal on Optimization, 11:647–674.
Fukuda, M., Kojima, M., and Shida, M. (2002). Lagrangian dual interiorpoint methods for semidefinite programs. SIAM Journal on Optimization, 12:1007–1031.
Garey, M.R., and Johnson, D.S. (1979). Computers and Intractability: A Guid to the Theory of NP-Completeness. W.H. Freeman & Company, San Francisco, CA.
Goemans, M. and Williamson, D.P. (1995). Improved approximation algorithms for max cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42:1115–1145.
Goemans, M. and Williamson, D.P. (2001). Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming. In: Proceedings of the 33rd Annual ACM Symposiurn on Theory of Computing, pp. 443–452. Association for Computing Machinery, New York.
Goffin, J.L., Gondzio, J., Sarkissian, R., and Vial, J.P. (1997). Solving nonlinear multicommodity flow problems by the analytic center cutting plane method. Mathematical Programming, 76:131–154.
Goffin, J.L., Luo, Z.Q., and Ye, Y. (1996). Complexity analysis of an interior point cutting plane method for convex feasibility problems. SIAM Journal on Optimization, 6:638–652.
Goffin, J.L. and Vial, J.P. (2000). Multiple cuts in the analytic center cutting plane method. SIAM Journal on Optimization, 11:266–288.
Goffin, J.L. and Vial, J.P. (2002). Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method. Optimization Methods and Software, 17:805–867.
Gondzio, J., du Merle, O., Sarkissian, R., and Vial, J.P. (1996). ACCPM–A library for convex optimization based on an analytic center cutting plane method. European Journal of Operations Research, 94:206–211.
Grone, B., Johnson, C.R., Marques de Sa, E., and Wolkowicz, H. (1984). Positive definite completions of partial Hermitian matrices. Linear Algebra and its Applications, 58:109–124.
Grötschel, M., Lovász, L., and Schrijver, A. (1993). Geometric Algorithms and Combinatorial Optimization. Springer Verlag.
Håstad, J. (1997). Some optimal inapproximability results. Proceedings of the 29th ACM Symposium on Theory and Computing, pp. 1–10.
Helmberg, C. (2000a). Semidefinite Programming for Combinatorial Optimization. Habilitation Thesis, ZIB-Report ZR-00-34, Konrad-Zuse-Zentrum Berlin.
Helmberg, C. (2000b). Fixing variables in semidefinite relaxations. SIAM Journal on Matrix Analysis and Applications, 21:952–969.
Helmberg, C. (2003). Numerical evaluation of SBmethod. Mathematical Programming, 95:381–406.
Helmberg, C. and Oustry, F. (2000). Bundle methods to minimize the maximum eigenvalue function. In: H. Wolkowicz, R. Saigal, and L. Vandenberghe (eds.), Handbook of Semidefinite Programming, pp. 307–337. Kluwer Academic Publishers.
Helmberg, C. and Rendl, F. (1998). Solving quadratic (0,1) problems by semidefinite programs and cutting planes. Mathematical Programming, 82:291–315.
Helmnberg, C. and Rendl, F. (2000). A spectral bundle method for semidefinite programming. SIAM Journal on Optimization, 10:673–696.
Helmberg, C., Rendi, F., Vanderbei, R., and Wolkowicz, H. (1996). An interior point method for semidefinite programming. SIAM Journal on Optimization, 6:673–696.
Helmberg, C. Semidefinite programming webpage. http://www-user.tu-chemnitz.de/~helmberg/semidef.html
Henrion, D. and Lasserre, J. (2003a). Gloptipoly: Global optimization over polynomials with MATLAB and SeDuMi. Transactions on Mathematical Software, 29:165–194.
Henrion, D. and Lasserre, J. (2003b). Detecting global optimality and extracting solutions in Globtipoly. Technical Report LAAS-CNRS. ??au]Interior-Point Methods Online. http://www-unix.mcs.anl.gov/otc/InteriorPoint
Kamath, A.P., Karmarkar, N., Ramakrishnan, K.G., and Resende M.G.C. (1990). Computational experience with an interior point algorithm on the satisfiability problem. Annals of Operations Research, 25:43–58.
Kamath, A.P., Karmarkar, N., Ramakrishnan, K.G., and Resende, M.G.C. (1992). A continuous approach to inductive inference. Mathematical Programming, 57:215–238.
Karger, D., Motwani, R., and Sudan, M. (1998). Approximate graph coloring by semidefinite programming. Journal of the ACM, 45:246–265.
Karloff, H. (1999). How good is the Goemans-Williamson MAX CUT algorithm? SIAM Journal on Computing, 29:336–350.
Karloff, H. and Zwick, U. (1997). A 7/8 approximation algorithm for MAX 3SAT? In: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pp. 406–415. IEEE Computer Science Press, Los Alamitos, CA.
Karmarkar, N. (1984). A new polynomial time algorithm for linear programming. Combinatorica, 4:373–395.
Karmarkar, N. (1990). An interior point approach to NP-complete problems. Contemporary Mathematics, 114:297–308.
Karmarkar, N., Resende, M.G.C., and Ramakrishnan, K.G. (1991). An interior point approach to solve computationally difficult set covering problems. Mathematical Programming, 52:597–618.
Kim, S. and Kojima, M. (2001). Second order cone programming relaxations of nonconvex quadratic optimization problems. Optimization Methods & Software, 15:201–224.
Kleinberg, J. and Goemans, M. (1998). The Lovász theta function and a semidefinite programming relaxation of vertex cover. SIAM Journal on Discrete Mathematics, 11:196–204.
Kocvara, M. and Stingl, M. (2003). PENNON: A code for convex nonlinear and semidefinite programming. Optimization Methods & Software, 18:317–333.
Kojima, M., Shindoh, S., and Hara, S. (1997). Interior-point methods for the monotone linear complementarity problem in symmetric matrices. SIAM Journal on Optimization, 7:86–125.
Kojima, M. and Tuncel, L. (2000). Cones of matrices and successive convex relaxations of nonconvex sets. SIAM Journal on Optimization, 10:750–778.
Krishnan, K. and Mitchell, J.E. (2003a). Properties of a cutting plane method for semidefinite programming. Technical Report, Department of Computational & Applied Mathematics, Rice University.
Krishnan, K. and Mitchell, J.E. (2003b). An unifying survey of existing cutting plane methods for semidefinite programming. Optimization Methods & Software, to appear; AdvOL-Report No. 2004/1, Advanced Optimization Laboratory, McMaster University.
Krishnan, K. and Mitchell, J.E. (2004). Semidefinite cut-and-price approaches for the maxcut problem. AdvOL-Report No. 2004/5, Advanced Optimization Laboratory, McMaster University.
Krishnan, K., Pataki, G., and Zhang, Y. (2004). A non-polyhedral primal active set approach to semidefinite programming. Technical Report, Dept. of Computational & Applied Mathematics, Rice University, forthcoming.
Lasserre, J.B. (2001). Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11:796–817.
Lasserre, J.B. (2002). An explicit exact SDP relaxation for nonlinear 0–1 programs. SIAM Journal on Optimization, 12:756–769.
Laurent, M. (1998). A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems. In: Topics in Semidefinite and Interior Point Methods, The Fields Institute for Research in Mathematical Sciences, Communications Series, vol. 18, American Mathematical Society, Providence, RI, AMS.
Laurent, M. (2003). A comparison of the Sherali-Adams, Lovász — Schrijver and Lasserre relaxations for 0–1 programming. Mathematics of Operations Research, 28:470–496.
Laurent, M. (2004). Semidefinite relaxations for max-cut. In: M. Grötschel (ed.), The Sharpest Cut, Festschrift in honor of M. Padberg's 60th birthday, pp. 291–327. MPS-SIAM.
Laurent, M. and Rendl, F. (2003). Semidefinite programming and integer programming. Technical Report PNA-R0210, CWI, Amsterdam.
Lemaréchal, C. and Oustry, F. (1999). Semidefinite relaxations and Lagrangian duality with applications to combinatorial optimization. Technical Report RR-3710, INRIA Rhone-Alpes.
Lovász, L. (1979). On the Shannon capacity of a graph. IEEE Transactions on Information Theory, 25:1–7.
Lovász, L. and Schrijver, A. (1991). Cones of matrices and set functions and 0–1 optimization. SIAM Journal on Optimization, 1:166–190.
Luo, Z.Q. and Sun, J. (1998). An analytic center based column generation method for convex quadratic feasibility problems. SIAM Journal on Optimization, 9:217–235.
Mahajan, S. and Hariharan, R. (1999). Derandomizing semidefinite programming based approximation algorithms. SIAM Journal on Computing, 28:1641–1663.
Mitchell, J.E. (2000). Computational experience with an interior point cutting plane algorithm. SIAM Journal on Optimization, 10:1212–1227.
Mitchell, J.E. (2003). Polynomial interior point cutting plane methods. Optimization Methods & Software, 18:507–534.
Mitchell, J.E. (2001). Restarting after branching in the SDP approach to MAX-CUT and similar combinatorial optimization problems. Journal of Combinatorial Optimization, 5:151–166.
Mitchell, J.E. and Borchers, B. (1996). Solving real-world linear ordering problems using a primal-dual interior point cutting plane method. Annals of Operations Research, 62:253–276.
Mitchell, J.E., Pardalos, P., and Resende, M.G.C. (1998). Interior point methods for combinatorial optimization. Handbook of Combinatorial Optimization, Kluwer Academic Publishers, 1:189–297.
Mitchell, J.E. and Ramaswamy, S. (2000). A long step cutting plane algorithm for linear and convex programming. Annals of Operations Research, 99:95–122.
Mitchell, J.E. and Todd, M.J. (1992). Solving combinatorial optimization problems using Karmarkar's algorithm. Mathematical Programming, 56:245–284.
Mittleman, H.D. (2003). An independent benchmarking of SDP and SOCP software. Mathematical Programming, 95:407–430.
Mokhtarian, F.S. and Goffin, J.L. (1998). A nonlinear analytic center cutting plane method for a class of convex programming problems. SIAM Journal on Optimization, 8:1108–1131.
Monteiro, R.D.C. (1997). Primal-dual path following algorithms for semidefinite programming. SIAM Journal on Optimization, 7:663–678.
Monteiro, R.D.C. (2003). First and second order methods for semidefinite programming. Mathematical Programmrning, 97:209–244.
Motzkin, T.S. and Strauss, E.G. (1965). Maxima for graphs and a new proof of a theorem of Turan. Canadian Journal of Mathematics, 17:533–540.
Muramatsu, M. and Suzuki, T. (2002). A new second-order cone programming relaxation for maxcut problems. Journal of Operations Research of Japan, to appear.
Murthy, K.G. and Kabadi, S.N. (1987). Some NP-complete problems in quadratic and linear programming. Mathematical Programming, 39:117–129.
Nakata, K. Fujisawa, K., Fukuda, M., Kojima, M., and Murota, K. (2003). Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical results. Mathematical Programming, 95:303–327.
Nemirovskii, A.S. and Yudin, D.B. (1983). Problem Complexity and Method Efficiency in Optimization. John Wiley.
Nesterov, Y.E. (1998). Semidefinite relaxation and nonconvex quadratic optimization. Optimization Methods and Software, 9:141–160.
Nesterov, Y.E. (2000). Squared functional systems and optimization problems. In: J.B.G. Frenk, C. Roos, T. Terlaky, and S. Zhang (eds.). High Performance Optimization, pp. 405–440. Kluwer Academic Publishers.
Nesterov Y.E. and Nemirovskii, A. (1994). Interior-Point Polynormial Algorithms in Con'vex Programming. SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA.
Nesterov, Y.E. and Todd, M.J. (1998). Primal dual interior point methods for self-scaled cones. SIAM Journal on Optimization, 8:324–364.
Optimization Online. http://www.optimization-online.org
Oskoorouchi, M. and Goffin, J.L. (2003a). The analytic center cutting plane method with semidefinite cuts. SIAM Journal on Optimization, 13:1029–1053.
Oskoorouchi, M. and Goffin, J.L. (2003b). An interior point cutting plane method for convex feasibility problems with second order cone inequalities. Mathematics of Operations Research, to appear; Technical Report, College of Business Administration, California State University, San Marcos.
Oustry, F. (2000). A second order bundle method to minimize the maximum eigenvalue function. Mathematical Programming, 89:1–33.
Papadimitriou, C.H. and Steiglitz, K. (1982). Combinatorial Optimization: Algorithms and Complexity. Prentice Hall.
Parrilo, P.A. (2000). Structured Semidefinite Programs and Semialgebraic Methods in Robustness and Optimization. Ph.D. Thesis, California Institute of Technology.
Parrilo, P.A. (2003). Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming, 96:293–320.
Pataki, G. (1996a). Cone-LPs and semidefinite programs: geometry and simplex type method. Proceedings of the 5th IPCO Conference, pp. 162–174. Lecture Notes in Computer Science, vol. 1084, Springer.
Pataki, G. (1996b). Cone Programming and Nonsmooth Optimization: Geometry and Algorithms. Ph.D. Thesis, Graduate School of Industrial Administration, Carnegie Mellon University.
Pataki, G. (1998). On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Mathematics of Operations Research, 23:339–358.
Poljak, S., Rendl, F., and Wolkowicz, H. (1995). A recipe for semidefinite relaxations for {0, 1}-quadratic programming. Journal of Global Optimization, 7:51–73.
Poljak, S. and Tuza, Z. (1994). The expected relative error of the polyhedral approximation of the maxcut problem. Operations Research Letters, 16:191–198.
Porkoláb, L. and Khachiyan, L. (1997). On the complexity of semidefinite programs. Journal of Global Optimization, 10:351–365.
Powers, V. and Reznick, B. (2001). A new bound for Polya's theorem with applications to polynomials positive on polyhedra. Journal of Pure and Applied Algebra, 164:221–229.
Prajna, S., Papachristodoulou, A., and Parrilo, P.A. (2002). SOS-TOOLS: Sum of squares optimization toolbox for MATLAB. http://www.cds.caltech.edu/sostools.
Putinar, M. (1993). Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42:969–984.
Quist, A.J., De Klerk, E., Roos, C., and Terlaky, T. (1998). Copositive relaxations for general quadratic programming. Optirmization Methods and Software, 9: 185–209.
Ramana, M. (1993). An Algorithrnic Analysis of Multiquadratic and Semidefinite Programming Problems. Ph.D. Thesis, The John Hopkins University.
Ramana, M. (1997). An exact duality theory for semidefinite programming and its complexity implications. Mathernatical Programming, 77:129–162.
Ramana, M., Tuncel, L., and Wolkowicz, H. (1997). Strong duality for semidefinite programming. SIAM Journal on Optimization, 7:641–662.
Rendl, F. and Sotirov, R. (2003). Bounds for the quadratic assignment problem using the bundle method. Technical Report, University of Klagenfurt, Universitaetsstrasse 65–67, Austria.
Renegar, J. (2001). A Mathematical View of Interior-Point Methods in Convex Optimization. MPS-SIAM Series on Optimization.
Roos, C., Terlaky, T., and Vial, J.P. (1997). Theory and Algorithms for Linear Optimization: An Interior Point Approach. John Wiley & Sons, Chichester, England.
Schrijver, A. (1979). A comparison of the Delsarte and Lovász bounds. IEEE Transactions on Information Theory, 25:425–429.
Schrijver, A. (1986). Theory of Linear and Integer Programming. Wiley-Interscience, New York.
Seymour, P.D. (1981). Matroids and multicommodity flows. European Journal of Combinatorics, 2:257–290.
Sherali, H.D. and Adams, W.P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics, 3:411–430.
Shor, N. (1998). Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishers.
Sturm, J.F. (1997). Primal-Dual Interior Point Approach to Semidefinite Programming. Tinbergen Institute Research Series, vol. 156, Thesis Publishers, Amsterdam, The Netherlands; Also in: Frenk et al. (eds.), High Performance Optimization, Kluwer Academic Publishers.
Sturm, J.F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11–12:625–653.
Sun, J., Toh, K.C., and Zhao, G.Y. (2002). An analytic center cutting plane method for the semidefinite feasibility problem. Mathematics of Operations Research, 27:332–346.
Terlaky, T. (ed.) (1996). Interior Point Methods of Mathematical Programming. Kluwer Academic Publishers.
Todd, M.J. (1999). A study of search directions in interior point methods for semidefinite programming. Optimization Methods and Software, 12:1–46.
Todd, M.J. (2001). Semidefinite optimization. Acta Numerica, 10:515–560.
Todd, M.J., Toh, K.C., and Tütüncü, R.H. (1998). On the Nesterov — Todd direction in semidefinite programming. SIAM Journal on Optimization, 8:769–796.
Toh, K.C. (2003). Solving large scale semidefinite programs via an iterative solver on the augmented system. SIAM Journal on Optimization, 14:670–698.
Toh, K.C. and Kojima, M. (2002). Solving some large scale semidefinite programs via the conjugate residual method. SIAM Journal on Optimization, 12:669–691.
Toh, K.C., Zhao, G.Y., and Sun, J. (2002). A multiple-cut analytic center cutting plane method for semidefinite feasibility problems. SIAM Journal on Optimization, 12:1026–1046.
Tütüncü, R.H., Toh, K.C., and Todd, M.J. (2003). Solving semidefinite-quadratic-linear programs using SDPT3. Mathematical Programming, 95:189–217.
Vaidya, P.M. A new algorithm for minimizing convex functions over convex sets. Mathematical Programming, 73:291–341, 1996.
Vandenberghe, L. and Boyd, S. (1996). Semidefinite programming. SIAM Review, 38:49–95.
Vavasis, S.A. (1991). Nonlinear Optimization. Oxford Science Publications, New York.
Warners, J.P., Jansen, B., Roos, C., and Terlaky, T. (1997a). A potential reduction approach to the frequency assignment problem. Discrete Applied Mathematics, 78:252–282.
Warners, J.P., Jansen, B., Roos, C., and Terlaky, T. (1997b). Potential reduction approaches for structured combinatorial optimization problems. Operations Research Letters, 21:55–65.
Wright, S.J. (1997). Primal-Dual Interior Poznt Methods. SIAM, Philadelphia, PA.
Wolkowicz, H., Saigal, R., and Vandenberghe, L. (2000). Handbook on Semidefinite Programming, Kluwer Academic Publishers.
Ye, Y. (1997). Interior Point Algorithms: Theory and Analysis. John Wiley & Sons, New York.
Ye, Y. (1999). Approximating quadratic programming with bound and quadratic constraints. Mathematical Programming, 84:219–226.
Ye, Y. (2001). A.699 approximation algorithm for max bisection. Mathematical Programming, 90:101–111.
Zhang, Y. (1998). On extending some primal-dual interior point algorithms from linear programming to semidefinite programming. SIAM Journal on Optimization, 8:365–386.
Zwick, U. (1999). Outward rotations: A tool for rounding solutions of semidefinite programming relaxations, with applications to maxcut and other problems. Proceedings of 31st STOC, pp. 496–505.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Krishnan, K., Terlaky, T. (2005). Interior Point and Semidefinite Approaches in Combinatorial Optimization. In: Avis, D., Hertz, A., Marcotte, O. (eds) Graph Theory and Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25592-3_5
Download citation
DOI: https://doi.org/10.1007/0-387-25592-3_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-25591-0
Online ISBN: 978-0-387-25592-7
eBook Packages: Business and EconomicsBusiness and Management (R0)