×

zbMATH — the first resource for mathematics

Inverse conic programming with applications. (English) Zbl 1140.90465
Summary: Linear programming duality yields efficient algorithms for solving inverse linear programs. We show that special classes of conic programs admit a similar duality and, as a consequence, establish that the corresponding inverse programs are efficiently solvable. We discuss applications of inverse conic programming in portfolio optimization and utility function identification.

MSC:
90C05 Linear programming
91B28 Finance etc. (MSC2000)
91B16 Utility theory
Software:
GloptiPoly; SeDuMi
PDF BibTeX Cite
Full Text: DOI
References:
[1] Ahuja, R.K.; Orlin, J.B., Inverse optimization, Oper. res, 49, 771-783, (2001) · Zbl 1163.90764
[2] Alizadeh, F., Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. optim, 5, 1, 13-51, (1995) · Zbl 0833.90087
[3] Alizadeh, F.; Goldfarb, D., Second-order cone programming, Math. prog. ser. B, 95, 3-51, (2003) · Zbl 1153.90522
[4] Ben-Tal, A.; Nemirovski, A., Robust convex optimization, Math. oper. res, 23, 4, 769-805, (1998) · Zbl 0977.90052
[5] Ben-Tal, A.; Nemirovski, A., Robust solutions of uncertain linear programs, Oper. res. lett, 25, 1, 1-13, (1999) · Zbl 0941.90053
[6] A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series in Optimization, SIAM, Philadelphia, PA, 2001. · Zbl 0986.90032
[7] Biel, D.R.; Wein, L.M., An inverse-optimization-based auction mechanism to support a multiattribute RFQ process, Manage. sci, 49, 1529-1545, (2003) · Zbl 1232.91274
[8] Burton, D.; Toint, P.-L., On an instance of inverse shortest path problems, Math. prog, 53, 45-61, (1992) · Zbl 0756.90089
[9] Burton, D.; Toint, P.-L., On the inverse shortest path algorithm for recovering linearly correlated costs, Math. prog, 55, 1-22, (1994) · Zbl 0795.90080
[10] Dial, R., Minimal revenue congestion pricing. part ia fast algorithm for the single-origin case, Transport. res. ser. B, 33, 189-202, (1999)
[11] Dial, R., Minimal revenue congestion pricing. part Iian efficient algorithm for the general case, Transport. res. ser. B, 34, 645-665, (2000)
[12] Eppstein, D., Setting parameters by example, SIAM J. comput, 32, 3, 643-653, (2003) · Zbl 1027.90092
[13] Goldfarb, D.; Iyengar, G., Robust portfolio selection problems, Math. oper. res, 28, 1, 1-37, (2003) · Zbl 1082.90082
[14] D. Henrion, J. Lasserre, Detecting global optimality and extracting solutions using Globtipoly, Technical Report, LAAS-CNRS, 2003.
[15] C. Heuberger, Inverse combinatorial optimization: a survey of problems, methods, and results, J. Combin. Opt., 2003, to appear, available at http://www.opt.math.tu-garz.ac.at/ cheub/publications/inverseopt.pdf. · Zbl 1084.90035
[16] Huang, C.-F.; Litzenberger, R.H., Foundations for financial economics, (1998), Prentice-Hall Englewood Cliffs, NJ
[17] G. Iyengar, W. Kang, Inverse convex optimization, Technical Report TR-2003-02, IEOR Department Columbia University, 2003, available at http://www.corc.ieor.columbia.edu/reports/techreports/tr-2003-02.pdf.
[18] Labbé, M.; Marcotte, P.; Savard, G., A bilevel model of taxation and its application to optimal highway pricing, Manage. sci, 44, 12, 1608-1622, (1998) · Zbl 0989.90014
[19] Lasserre, J.B., Global optimization with polynomial and the problem of moments, SIAM J. opt, 11, 796-817, (2001) · Zbl 1010.90061
[20] Michaud, R.O., Efficient asset management: a practical guide to stock portfolio management and asset allocation, (1998), Oxford University Press Oxford
[21] Y. Nesterov, Structure of non-negative polynomials and optimization problems, Technical Report 9749, CORE, Universite Catholique de Louvain, 1997.
[22] Nesterov, Y.; Nemirovskii, A., Interior-point polynomial algorithms in convex programming, (1993), SIAM Philadelphia · Zbl 0824.90112
[23] Nolet, G., Seismic tomography, (1987), Reidel Dordrecht
[24] Parrillo, P.A., Semidefinite programming relaxations for semialgebraic problems, Math. prog, 96, 293-320, (2003) · Zbl 1043.14018
[25] P.T. Sokkalingam, The minimum cost flow problem: primal algorithms and cost perturbations, Ph.D. Thesis, Mathematics Department, Indian Institute of Technology, Kanpur, India, 1995.
[26] Sokkalingam, P.T.; Ahuja, R.K.; Orlin, J.B., Solving inverse spanning tree problem through network flow techniques, Oper. res, 47, 2, 291-298, (1999) · Zbl 0979.90119
[27] Stürm, J., Using sedumi 1.02, a Matlab toolbox for optimization over symmetric cones, Optim. methods software, 11-12, 625-653, (1999) · Zbl 0973.90526
[28] Tarantola, A., Inverse problem theory: methods for data Fitting and model parameter estimation, (1987), Elsevier Amsterdam · Zbl 0875.65001
[29] Zhang, J.; Cai, M., Inverse problem of minimum cuts, Math. methods oper. res, 48, 51-58, (1998) · Zbl 0941.90013
[30] Zhang, J.; Liu, Z., Calculating some inverse programming problems, J. comput. appl. math, 72, 261-273, (1996) · Zbl 0856.65069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.