Abstract
Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space corresponding to the vector of variables. A combinatorial optimization problem is then formulated to increase the number of zeros of the Hessian matrices in the resulting transformed space, and a heuristic greedy algorithm is applied to this formulation. The resulting method can thus be viewed as a preprocessor for converting a problem with hidden sparsity into one in which sparsity is explicit. When it is combined with the sparse semidefinite programming relaxation by Waki et al. for polynomial optimization problems, the proposed method is shown to extend the performance and applicability of this relaxation technique. Preliminary numerical results are presented to illustrate this claim.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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, pp. 1–29. Springer, New York (1993)
Conn, A.R., Gould, N.I.M., Toint, Ph.L: Improving the decomposition of partially separable functions in the context of large-scale optimization: a first approach. In: Hager, W.W., Hearn, D.W., Pardalos, P.M.(eds) Large Scale Optimization: State of the Art, pp. 82–94. Kluwer, Dordrecht (1994)
Conn, A.R., Gould, N.I.M., Toint, Ph.L.: LANCELOT, A Fortran Package for large-Scale Nonlinear Optimization (Release A). Springer, Heidelberg (1992)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Saunders College Publishing, Fort Worth (1976)
Gay, D.M.: Automatically Finding and Exploiting Partially Separable Structure in Nonlinear Programming Problems. Bell Laboratories, Murray Hill (1996)
Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr, a constrained and unconstrained testing environment, revisited. TOMS 29, 373–394 (2003)
Gould, N.I.M., Orban, D., Toint, Ph.L.: GALAHAD, a library of thread-safe Fortran packages for large-scale nonlinear optimization. TOMS 29, 353–372 (2003)
Griewank, A., Toint, Ph.L.: On the unconstrained optimization of partially separable functions. In: Powell, M.J.D.(eds) Nonlinear Optimization 1981, pp. 301–312. Academic Press, New York (1982)
Griewank, A., Toint, Ph.L.: Partitioned variable metric updates for large structured optimization problems. Numerische Mathematik 39, 119–137 (1982)
Griewank, A., Toint, Ph.L.: Local convergence analysis for partitioned quasi-Newton updates. Numerische Mathematik 39, 429–448 (1982)
Griewank, A., Toint, Ph.L.: On the existence of convex decomposition of partially separable functions. Math. Program. 28, 25–49 (1984)
Floudas, C., Pardalos, P., Adjiman, C., Esposito, W., Gümüs, Z., Harding, S., Klepeis, J., Meyer, C., Schweiger, C.: Handbook of Test Problems in Local and Global Optimization. Kluwer, Dordrecht (1999)
GLOBAL Library. http://www.gamsworld.org/global/globallib.htm
Kim, S., Kojima, M., Waki, H.: Generalized Lagrangian duals and sums of squares relaxation of sparse polynomial optimization problems. SIAM J. Optim. 15, 697–719 (2005)
Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103, 45–62 (2005)
Kojima, M., Muramatsu, M.: A note on SOS and SDP relaxations for polynomial optimization problems over symmetric cones. Comput. Optim. Appl. (2008, to appear)
Lasserre, J.B.: Global optimization with polynomials and the problems of moments. SIAM J. Optim. 1, 796–817 (2001)
Lasserre, J.B.: Convergent semidefinite relaxation in polynomial optimization with sparsity. Working paper, LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France (2005)
More, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Soft. 7, 17–41 (1981)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)
Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11 & 12, 625–653 (1999)
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, 218–242 (2006)
Waki, H., Kim, S., Kojima, M., Muramatsu, M.: SparsePOP : a sparse semidefinite programming relaxation of polynomial optimization problems. Research Report B-414, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8552, Japan (2005). ACM Trans Math Softw (to appear)
Yamashita, N.: Sparse quasi-Newton updates with positive definite matrix completion. Technical Report 2005-0008, Applied Mathematics and Physics, Kyoto University 606-8501, Kyoto, Japan (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
S. Kim’s research was supported by Kosef R01-2005-000-10271-0.
M. Kojima’s research was supported by Grant-in-Aid for Scientific Research on Priority Areas 16016234.
Rights and permissions
About this article
Cite this article
Kim, S., Kojima, M. & Toint, P. Recognizing underlying sparsity in optimization. Math. Program. 119, 273–303 (2009). https://doi.org/10.1007/s10107-008-0210-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-008-0210-4