Abstract
Recently Kojima and Tunçel proposed new successive convex relaxation methods and their localized-discretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically by solving a finite number of linear programs, several important implementation issues remain unsolved. In this paper, we discuss those issues, present practically implementable algorithms and report numerical results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Alizadeh (1995), “Interior point methods in semidefinite programming with applications to combinatorial optimization,” SIAM Journal on Optimization, 5, 13–51.
F. A. AL-Khayyal and J. E. Falk (1983), “Jointly constrained biconvex programming”, Mathematics of Operations Research, 8, 273–286.
E. Balas, S. Ceria and G. Cornuéjols (1993), “A lift-and-project cutting plane algorithm for mixed 0–1 programs,” Mathematical Programming, 58, 295–323.
R. W. Cottle, J. S. Pang and R. E. Stone (1992), The Linear Complementarity Problem, Academic Press, New York.
T. G.W. Epperly and R.E. Swaney (1996), “Branch and bound for global NLP: Iterative LP algorithm and results”, in I. E. Grossmann ed., Global Optimization in Engineering Design, Kluwer, Dordrecht.
J. E. Falk and S. W. Palocsay (1992), “Optimizing the sum of linear fractional functions”, in C. A. Floudas eds., Collection: Recent Advances in Global Optimization, Kluwer, Dordrecht, 221–258.
C. A. Floudas and P. M. Pardalos (1990), A Collection of Test Problems for Constrained Global Optimization Algorithms, Lecture Notes in Computing Science, Springer, New York.
T. Fujie and M. Kojima (1997), “Semidefinite relaxation for nonconvex programs,” Journal of Global Optimization, 10, 367–380.
K. Fujisawa, M. Kojima and K. Nakata (1995), “SDPA (SemiDefinite Programming Algorithm) - User’s manual - version 4.10,” Technical Report B-308, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan, revised May 1998, available at http://www.is.titech.ac.jp/pub/OpRes/software/SDPA.
M. X. Goemans (1997), “Semidefinite programming in combinatorial optimization,” Mathematical Programming, 79, 143–161.
K. C. Goh, M. G. Safonov and G. P. Papavassilopoulos (1995), “Global optimization for the biaffine matrix inequality problem,” Journal of Global Optimization, 7, 365–380.
I. E. Grossmann and A. Kravanja (1997), “Mixed-integer nonlinear programming: A survey of algorithms and applications,” in L. T. Biegler eds., Large-Scale Optimization with Applications, Part II: Design and Control, Springer-Verlag, 73–100.
M. Kojima and L. Tunçel (1998), “Cones of matrices and successive convex relaxations of nonconvex sets,” Technical Report B-338, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan. Also issued as CORR 98–6, Dept. of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, revised June 1999, to appear in SIAM Journal on Optimization.
M. Kojima and L. Tunçel (1998), “Discretization and Localization in Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems,” Technical Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan, revised July 1999.
L. Lovâsz and A. Schrijver (1991), “Cones of matrices and set functions and 0–1 optimization,” SIAM Journal on Optimization, 1, 166–190.
M. Mesbahi and G. P. Papavassilopoulos (1997), “A cone programming approach to the bilinear matrix inequality problem and its geometry,” Mathematical Programming, 77, 247–272.
Yu. E. Nesterov (1997), “Semidefinite relaxation and nonconvex quadratic optimization,” CORE Discussion Paper, #9744.
Yu. E. Nesterov and A. S. Nemirovskii (1994), Interior-Point Polynomial Algorithms in Convex Programming, SIAM, Philadelphia.
G. Pataki and L. Tunçel (1997), “On the generic properties of convex optimization problems in conic form,” Research Report 97–16, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.
S. Poljak, F. Rendl and H. Wolkowicz (1995), “A recipe for semidefinite relaxation for (0,1)-quadratic programming,” Journal of Global Optimization, 7, 51–73.
M. V. Ramana (1993), An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, MD.
N. V. Sahinidis (1998), “BARON: Branch and reduce optimization navigator, User’s manual ver. 3.0”, Department of Chemical Engineering, University of Illinois at Urbana-Champaign.
N. V Sahinidis and I.E. Grossmann (1991), “Reformulation of multiperiod MILP models for planning and scheduling of chemical processes”, Computers e4 Chemical Engineering, 15, 255–272.
S. Schaible (1995), “Fractional programming” in R. Horst and P. Pardalos eds., Handbook of Global Optimization, Kluwer, Dordrecht, 495–608.
H. D. Sherali and A. Alameddine (1992), “A new reformulation-linearization technique for bilinear programming problems,” Journal of Global Optimization, 2, 379–410.
H. D. Sherali and C. H. Tuncbilek (1995), “A reformulation-convexification approach for solving nonconvex quadratic programming problems,” Journal of Global Optimization, 7, 1–31.
N. Z. Shor (1990), “Dual quadratic estimates in polynomial and boolean programming,” Annals of Operations Research, 25, 163–168.
R. E. Swaney (1990), “Global solution of algebraic nonlinear programs,” AIChE Annual Meeting, Chicago.
V. Visweswaran and C. A. Floudas (1990), “A global optimization (GOP) for certain classes of nonconvex NLPs — II. Application of theory and test problems,” Computers and Chemical Engineering, 14, 1419–1434.
V. Visweswaran, C. A. Floudas, M. G. Ierapetritou and E. N. Pistikopoulos (1996), “A decomposition-based global optimization approach for solving bilevel linear and quadratic programs,” in C. A. Floudas eds., State of the Art in Global Optimization, Kluwer, Dordrecht, 139–162.
Y. Ye (1997), “Approximating quadratic programming with quadratic constraints,” Working paper, Dept. of Management Sciences, The University of Iowa.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Takeda, A., Dai, Y., Fukuda, M., Kojima, M. (2000). Towards Implementations of Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems. In: Pardalos, P.M. (eds) Approximation and Complexity in Numerical Optimization. Nonconvex Optimization and Its Applications, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3145-3_28
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3145-3_28
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4829-8
Online ISBN: 978-1-4757-3145-3
eBook Packages: Springer Book Archive