Abstract.
The purpose of this paper is threefold. First we propose splitting schemes for reformulating non-separable problems as block-separable problems. Second we show that the Lagrangian dual of a block-separable mixed-integer all-quadratic program (MIQQP) can be formulated as an eigenvalue optimization problem keeping the block-separable structure. Finally we report numerical results on solving the eigenvalue optimization problem by a proximal bundle algorithm applying Lagrangian decomposition. The results indicate that appropriate block-separable reformulations of MIQQPs could accelerate the running time of dual solution algorithms considerably.
Similar content being viewed by others
References
Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, 1999
Al-Khayyal, F.A., Larsen, C., van Voorhis, T.: A relaxation method for nonconvex quadratically constrained quadratic programs. J. Glob. Opt. 6, 215–230 (1995)
Al-Khayyal, F.A., van Voorhis, T.: Accelerating convergence of branch-and-bound algorithms for quadratically constrained optimization problems. In: State of the Art in Global Optimization: Computational Methods and Applications, C.A. Floudas, (ed.), Kluwer Academic Publishers, 1996
Alperin, H., Nowak, I.: Lagrangian Smoothing Heuristics for MaxCut. Technical report, HU–Berlin NR–2002–6, 2002
Balas, E., Ceria, S., Cornuejols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Progr. 58, 295–324 (1993)
Bixby, R.E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: MIP: theory and practice - closing the gap. In: Powell, M.J.D., Scholtes, S. (eds.), System Modelling and Optimization: Methods, Theory and Applications, pp. 19–49. Kluwer Academic Publishers, 2000.
Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib- A Collection of Test Models for Mixed-Iinteger Nonlinear Programming. INFORMS J. Comput. 15 (1), (2003)
Chardaire, P., Sutter, A.: A decomposition method for quadratic zero-one programming. Manage. Sci. 41, 704–712 (1995)
Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. 2nd edn, Springer, New York, 1993
Dantzig, G.B., Wolfe, P.: Decomposition principle for linear programs. Oper. Res. 8, 101–111 (1960)
Demands, E.V., Tang, C.S.: Linear control of a Markov production system. Oper. Res. 40, 259–278 (1992)
Douglas, J., Rachford, H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
Dür, M.: Dual bounding procedures lead to convergent Branch-and-Bound algorithms. Math. Progr. 91, 117–125 (2001)
Ferris, M.C., Horn, J.D.: Partitioning mathematical programs for parallel solution. Math. Progr. 80, 35–61 (1998)
Filar, J.A., Schultz, T.A.: Bilinear programming and structured stochastic games. J. Opt. Theor. Appl. 53, 85–104 (1999)
Fujisawa, K., Kojima, M., Nakata, K.: Exploiting sparsity in primal-dual interior-point methods for semidefinite programming. Math. Progr. 79, 235–254 (1997)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York, 1979
Gomes, F., Sorensen, D.: ARPACK++: a C++ Implementation of ARPACK eigenvalue package, 1997: http://www.crpc.rice.edu/software/ARPACK/
Guignard, M., Kim, S.: Lagrangian decomposition: a model yielding stronger Lagrangean bounds. Math. Progr. 39 (2), 215–228 (1987)
Helmberg, C.: Semidefinite Programming for Combinatorial Optimization. Technical report, ZIB–Report 00–34, 2000
Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Opt 10 (3), 673–695 (2000)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I and II. Springer, Berlin, 1993
Horst, R., Pardalos, P., Thoai, N.: Introduction to Global Optimization. Kluwer Academic Publishers, 1995
Phan-huy-Hao, E.: Quadratically constrained quadratic programming: Some applications and a method for solution. ZOR 26, 105–119 (1982)
Kim, S., Kojima, M.: Second Order Cone Programming Relaxation of Nonconvex Quadratic Optimization Problems. Technical report, Research Reports on Mathematical and Comuting Sciences, Series B: Operations Research, Tokyo Institute of Technology, 2000
Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Progr. 46, 105–122 (1990)
Kiwiel, K.C.: User’s Guide for NOA 2.0/3.0: A FORTRAN Package for Convex Nondifferentiable Optimization. Polish Academy of Science, System Research Institute, Warsaw, 1993/1994
Kojima, M., Tunçel, L.: Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems. Math. Program. 89, 97–111 (2000)
Lemaréchal, C., Renaud, A.: A geometric study of duality gaps, with applications. Math. Progr. 90, 399–427 (2001)
Martin, A.: Integer programs with block structure. Technical report, ZIB–Report 99–03, Habilitationsschrift, 1999
Nesterov, Y., Wolkowicz, H., Ye, Y.: Semidefinite programming relaxaxations of nonconvex quadratic optimization. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.), Handbook of Semidefinite Programming, Kluwer Academic Publishers, 2000, pp. 361–419
NETLIB: EISPACK: http://www.netlib.org/eispack/ 1972-1973
Neumaier, A.: Constrained Global Optimization. In: COCONUT Deliverable D1, Algorithms for Solving Nonlinear Constrained and Optimization Problems: The State of The Art, pp. 55–111. http://www.mat.univie.ac.at/∼neum/glopt/coconut/StArt.html, 2001
Nowak, I.: Dual bounds and optimality cuts for all-quadratic programs with convex constraints. J. Glob. Opt. 18, 337–356 (2000)
Nowak, I., Alperin, H., Vigerske, S.: LaGO - An object oriented library for solving MINLPs. In: Bliek, C., Jermann, C., Neumaier, A. (eds.), Global Optimization and Constraint Satisfaction, pp. 32–42. Springer, Berlin, 2003. http://www.mathematik.hu-berlin.de/∼eopt/papers/LaGO.pdf
Parrilo, P., Sturmfels, B.: Minimizing polynomial functions. To appear in DIMACS volume of the Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, 2001
Phing, T.Q., Tao, P.D., Hoai An, L.T.: A method for solving D. C. programming problems, application to fuel mixture nonconvex optimization problems. J. Global Opt. 6, 87–105 (1994)
Poljak, S., Rendl, F., Wolkowicz, H.: A recipe for semidefinite relaxation for (0,1)-quadratic programming. J. Global Opt. 7 (1), 51–73 (1995)
Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Progr. 77 (2), 273–299 (1997)
Ruszczyński, A.: Decomposition methods in stochastic programming. Math. Progr. 79, 333–353 (1997)
Rutenberg, D.P., Shaftel, T.L.: Product design: sub-assemblies for multiple markets. Manage. Sci. 18, B220–B231 (1971)
Stern, R., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Opt. 5 (2), 286–313 (1995)
Thoai, N.: Duality bound method for the general quadratic programming problem with quadratic constraints. J. Optim. Theor. Appl. 107 (2), (2000)
Visweswaran, V., Floudas, C.A.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs : II. Application of theory and test problems. Comp. Chem. Eng. 1990
Weintraub, A., Vera, J.: A cutting plane approach for chance-constrained linear programs. Oper. Res. 39, 776–785 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work was supported by the German Research Foundation (DFG) under grant NO 421/2-1
Mathematics Subject Classification (2000): 90C22, 90C20, 90C27, 90C26, 90C59
Rights and permissions
About this article
Cite this article
Nowak, I. Lagrangian decomposition of block-separable mixed-integer all-quadratic programs. Math. Program. 102, 295–312 (2005). https://doi.org/10.1007/s10107-003-0500-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-003-0500-9