Abstract
GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality relaxations of the (generally non-convex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality or integer constraints. It generates a series of lower bounds monotonically converging to the global optimum. Global optimality is detected and isolated optimal solutions are extracted automatically. In this paper we first briefly describe the theoretical background underlying the relaxations. Following a small illustrative example of the use of GloptiPoly, we then evaluate its performance on benchmark test examples from global optimization, combinatorial optimization and polynomial systems of equations.
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References
Anjos, M.: New Convex Relaxations for the Maximum Cut and VLSI Layout Problems. PhD Thesis, Waterloo University, Ontario, Canada, orion.math.uwaterloo.ca/~hwolkowi (2001)
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht (1999), See titan.princeton.edu/TestProblems
The Numerical Algorithms Group Ltd. FRISCO - A Framefork for Integrated Symbolic/Numeric Computation. European Commission Project No. 21-024, Esprit Reactive LTR Scheme (2000), See http://www.nag.co.uk/projects/frisco.html
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)
Henrion, D., Lasserre, J.B.: GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi. ACM Transactions on Mathematical Software 29(2), 165–194 (2003), Available at www.laas.fr/~henrion/software/gloptipoly
Lasserre, J.B.: Global Optimization with Polynomials and the Problem of Moments. SIAM Journal on Optimization 11(3), 796–817 (2001)
Lasserre, J.B.: An Explicit Equivalent Positive Semidefinite Program for 0-1 Nonlinear Programs. SIAM Journal on Optimization 12(3), 756–769 (2002)
Waterloo Maple Software Inc. Maple V release 5 (2001), See http://www.maplesoft.com
The MathWorks Inc. Matlab version 6.1 (2001), See http://www.mathworks.com
Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk, H., Roos, K., Terlaky, T. (eds.) High performance optimization, ch. 17, pp. 405–440. Kluwer Academic Publishers, Dordrecht (2000)
Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD Thesis, California Institute of Technology, Pasadena, California (2000), See http://www.control.ethz.ch/~parrilo
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42, 969–984 (1993)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3, 411–430 (1990)
Sherali, H.D., Adams, W.P.: A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Kluwer Academic Publishers, Dordrecht (1999)
Shor, N.Z.: Quadratic optimization problems. Tekhnicheskaya Kibernetika 1, 128–139 (1987)
Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishers, Dordrecht (1998)
Sturm, J.F.: Using SeDuMi 1.02, a Matlab Toolbox for Optimization over Symmetric Cones. Optimization Methods and Software 11-12, 625–653 (1999), Version 1.05 available at fewcal.kub.nl/sturm/software/sedumi.html
Prajna, S., Papachristodoulou, A., Parrilo, P.A.: SOSTOOLS: Sum of Squares Optimization Toolbox for Matlab. California Institute of Technology, Pasadena, USA (2002), Version 1.00 available at http://www.cds.caltech.edu/sostools
Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38, 49–95 (1996)
Verschelde, J.: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software 25(2), 251–276 (1999), Database of polynomial systems available at www.math.uic.edu/~jan/demo.html
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Henrion, D., Lasserre, JB. (2003). Solving Global Optimization Problems over Polynomials with GloptiPoly 2.1. In: Bliek, C., Jermann, C., Neumaier, A. (eds) Global Optimization and Constraint Satisfaction. COCOS 2002. Lecture Notes in Computer Science, vol 2861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39901-8_4
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DOI: https://doi.org/10.1007/978-3-540-39901-8_4
Publisher Name: Springer, Berlin, Heidelberg
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