Abstract
We present a hierarchy of semidefinite programming (SDP) relaxations for solving the concave cost transportation problem (CCTP), which is known to be NP-hard, with p suppliers and q demanders. In particular, we study cases in which the cost function is quadratic or square-root concave. The key idea of our relaxation methods is in the change of variables to CCTPs, and due to this, we can construct SDP relaxations whose matrix variables are of size O((min {p, q}) ω) in the relaxation order ω. The sequence of optimal values of SDP relaxations converges to the global minimum of the CCTP as the relaxation order ω goes to infinity. Furthermore, the size of the matrix variables can be reduced to O((min {p, q})ω-1), ω ≥ 2 by using Reznick’s theorem. Numerical experiments were conducted to assess the performance of the relaxation methods.
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References
Altiparmak F., Karaoglan I.: An adaptive tabu-simulated annealing for concave cost transportation problems. J. Oper. Res. Soc. 59(3), 331–341 (2008)
Blair J.R.S., Peyton B.: An introduction to chordal graphs and clique trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation, volume 56 of IMA Volumes in Mathematics and its Applications, pp. 1–29. Springer, Berlin (1993)
Floudas C.A., Pardalos P.M., Adjiman C.S., Esposito W.R., Gumus Z.H., Harding S.T., Klepeis J.L., Meyer C.A., Schweiger C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer, Dordrecht (1999)
Fukuda M., Kojima M., Murota K., Nakata K.: Exploiting sparsity in semidefinite programming via matrix completion. I: General framework. SIAM J. Optim. 11(3), 647–674 (2000)
Fukunaga A.S.: A branch-and-bound algorithm for hard multiple knapsack problems. Ann. Oper. Res. 184(1), 97–119 (2011)
Gallo G., Sandi C., Sodini C.: An algorithm for the min concave cost flow problem. Eur. J. Oper. Res. 4(4), 248–255 (1980)
George A., Liu J.W.H.: Computer Solutions of Large Sparse Positive Definite Systems. Prentice Hall, Englewood Cliffs (1981)
Grimm D., Netzer T., Schweighofer M.: A note on the representation of positive polynomials with structured sparsity. Arch. Math. 89(5), 399–403 (2007)
Guisewite G.M., Pardalos P.M.: Minimum concave cost network flow problems: applications, complexity, and algorithms. Ann. Oper. Res. 25(1), 75–100 (1990)
Guisewite G.M., Pardalos P.M.: Global search algorithms for minimum concave-cost network flow problems. J. Global. Optim. 1(4), 309–330 (1991)
Horst R., Thoai N.V.: An integer concave minimization approach for the minimum concave cost capacitated flow problem on networks. OR Spectrum 20(1), 47–53 (1998)
Kim S., Kojima M., Toint P.: Recognizing underlying sparsity in optimization. Math. Program. Ser. A 119, 273–303 (2009)
Kojima M., Kim S., Waki H.: Sparsity in sums of squares of polynomials. Math. Program. Ser. A 103, 45–62 (2005)
Kojima M., Muramatsu M.: A note on sparse SOS and SDP relaxations for polynomial optimization problems over symmetric cones. Comput. Optim. Appl. 42(1), 31–41 (2009)
Kuno T., Utsunomiya T.: A Lagrangian based branch-and-bound algorithm for production-transportation problems. J. Global. Optim. 18(1), 59–73 (2000)
Lamar B.W.: An improved branch and bound algorithm for minimum concave cost network flow problems. J. Global. Optim. 3(3), 261–287 (1993)
Larsson T., Migdalas A., Ronnqvist M.: A Lagrangean heuristic for the capacitated concave minimum cost network flow problem. Eur. J. Oper. Res. 78(1), 116–129 (1994)
Lasserre J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)
Lasserre J.B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)
Laurent M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, volume 149 of IMA Volumes in Mathematics and its Applications, pp. 157–270. Springer, Berlin (2009)
Mittelmann H.D.: An independent benchmarking of SDP and SOCP solvers. Math. Program. Ser. B 95(2), 407–430 (2003)
Nakata M., Braams B.J., Fujisawa K., Fukuda M., Percus J.K., Yamashita M., Zhao Z.: Variational calculation of second-order reduced density matrices by strong N-representability conditions and an accurate semidefinite programming solver. J. Chem. Phys. 128, 164113 (2008)
Parrilo P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. Ser. B 96, 293–320 (2003)
Pisinger D.: An exact algorithm for large multiple knapsack problems. Eur. J. Oper. Res. 114(3), 528–541 (1999)
Reznick B.: Extremal PSD forms with few terms. Duke Math. J. 45, 363–374 (1978)
Strum J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)
Tuy H., Ghannadan S., Migdalas A., Värbrand P.: A strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables. Math. Program. 72(3), 229–258 (1996)
Waki H., Kim S., Kojima M., Muramatsu M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006)
Waki H., Kim S., Kojima M., Muramatsu M., Sugimoto H.: Algorithm 883: SparsePOP: a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Softw. 35(2), 1–13 (2008)
Yamashita, M., Fujisawa, K., Nakata, K., Nakata, M., Fukuda, M., Kobayashi, K., Goto, K.: A high-performance software package for semidefinite programs: SDPA 7. Research Report B-460, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, (2009)
Yan S., Luo S.-C.: Probabilistic local search algorithms for concave cost transportation network problems. Eur. J. Oper. Res. 117(3), 511–521 (1999)
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Mizutani, T., Yamashita, M. Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables. J Glob Optim 56, 1073–1100 (2013). https://doi.org/10.1007/s10898-012-9924-1
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DOI: https://doi.org/10.1007/s10898-012-9924-1