Abstract
Many global optimization problems can be formulated in the form min{c(x, y): x εX, y εY, (x, y) εZ, y εG} where X, Y are polytopes in ℝp, ℝn, respectively, Z is a closed convex set in ℝp+n, while G is the complement of an open convex set in ℝn. The function c:ℝp+n → ℝ is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ℝn. Computational experiments show that the resulting algorithms work well for problems with smalln.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, P. C., Hansen, P., and Jaumard B. (1991), On-Line and Off-Line Vertex Enumeration by Adjacency Lists,Operations Research Letters 10, 403–409.
Horst, R. and Thoai N. V. (1989), Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems,Computing 42, 271–289.
Horst, R. and Thoai, N. V. (1992), Conical Algorithm for the Global Minimisation of Linearly Constrained Decomposable Concave Minimization Problems,J. Opt. Theory a. Appl. 74, 469–486.
Horst, R., Tuy, H. (1993),Global Optimization: Deterministic Approaches, Springer-Verlag.
Horst, R., Phong, T. Q. and Thaoi, N. V. (1990), On Solving General Reserve Convex Programming Problems by a Sequence of Linear Programs and Line Searches,Annals of Oper. Res. 25, 1–18.
Horst, R., Thoai, N. V., Benson, H. P. (1991) Concave Minimization via Conical Partitions and Polyhedral Outer Approximation,Mathematical Programming 50, 259–274.
Horst, R., Thoai, N. V. and Tuy, H. (1987), Outer Approximation by Polyhedral Convex Sets,Oper. Res. Spektrum 9, 153–159.
Horst, R., Thoai, N. V., de Vries, J. (1992), A New Simplical Cover Technique in Constrained Global Optimization,Journal Global Optimization 2, 1–19.
Horst, R., Thoai, N. V., de Vries, J. (1992a), On Geometry and Convergence of a Class of Simplicial Covers,Optimization 25, 53–64.
Muu, L. D. (1993), An Algorithm for Solving Convex Programs with an Additional Convex-Concave Constraint,Mathematical Programming 61, 75–87.
Thach, P. T., Burkard, R. E. and Oettli, W. (1991), Mathematical Programs with a Two-dimensional Reverse Convex Constraint,J. Global Optimization 1, 145–154.
Thoai, N. V. (1991), A Global Optimization Approach for Solving the Convex Multiplicative Programming Problem,J. Global Optimization 1, 341–357.
Thoai, N. V. (1993), Canonical D.C. Programming Techniques for Solving a Convex Program with an Additional Constraint of Multiplication Type,Computing 50, 241–253.
Thoai, N. V., Tuy, H. (1980), Convergent Algorithms for Minimizing a Concave Function,Mathematics of Oper. Res. 5, 556–566.
Tuy, H. (1985), Concave Minimization under Linear Constraints with Special Structure,Optimization 16, 335–352.
Tuy, H. (1986), A General Deterministic Approach to Global Optimization via D.C. Programming. In: Hiriart-Urruty, J. B. (Ed.),Fermat Days 1985: Mathematics for Optimization, Elsevier, Amsterdam, 137–162.
Tuy, H., Khachaturov, V., Utkin, S. (1987), A Class of Exhaustive Cone Splitting Procedures in Conical Algorithms for Concave Minimization,Optimization 18, 791–807.
Tuy, H. (1992), The Complementary Convex Structure in Global Optimization.J. Global Optimization 2, 21–40.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Horst, R., van Thoai, N. Constraint decomposition algorithms in global optimization. J Glob Optim 5, 333–348 (1994). https://doi.org/10.1007/BF01096683
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01096683