Abstract
We show the importance of exploiting the complementary convex structure for efficiently solving a wide class of specially structured nonconvex global optimization problems. Roughly speaking, a specific feature of these problems is that their nonconvex nucleus can be transformed into a complementary convex structure which can then be shifted to a subspace of much lower dimension than the original underlying space. This approach leads to quite efficient algorithms for many problems of practical interest, including linear and convex multiplicative programming problems, concave minimization problems with few nonlinear variables, bilevel linear optimization problems, etc...
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Thach, P. T. (1987), D. C. Sets, D. C. Functions and Systems of Equations, Preprint, Institute of Mathematics, Hanoi. To appear in Mathematical Programming.
Tuy, H. and R., Horst (1991), The Geometric Complementarity Problem and Transcending Stationarity in Global Optimization in “Applied Geometry and Discrete Mathematics”, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 4, 341–354.
Henderson, J. M. and R. E. Quant (1971), Microeconomic Theory, McGraw-Hill.
Maling, K., S. H. Mueller, and W. R. Heller (1982), On Finding Most Optimal Rectangular Package Plans, Proceedings of the 19th Design Automation Conference, 663–670.
Konno, H. and M., Inori (1988), Bon Portfolio Optimization by Bilinear Fractional Programming, J. of Oper. Res. of Japan 32, 143–158.
Paradalos, P. M. (1988), Polynomial Time Algorithms for Some Classes of Constrained Non-Convex Quadratic Problems, Preprint, Computer Science Department, the Pennsylvania State University.
Forgo, F. (1975), The Solution of a Special Quadratic Problem, Szigma, 53–59 (in Hungarian).
Gabasov, R. and F. M. Kirillova (1980), Linear Programming Methods, Part 3 (Special Problems), Minsk (in Russian).
Pardalos, P. M. (1988), On the Global Minimization of the Product of Two Linear Functions over a Polytope, Preprint, Computer Science Department, The Pennsylvania State University.
Konno H. and T. Kuno (1989), Linear Multiplicative Programming, IHSS Report 89-13, Institute of Human and Social Sciences, Tokyo Institute of Technology.
Tuy, H. and B. T. Tam (1990), An Efficient Solution Method for Rank Two Quasiconcave Minimization Problems, to appear in Optimization.
Idrissi, H., P., Loridan, and C., Michelot (1988), Approximation of Solutions for Location Problems, Journal of Optimization Theory and Applications 56, 127–143.
Tuy, H. and Faiz A. Al-Khayyal (1991), A Class of Global Optimization Problems Solvable by Sequential Unconstrained Convex Minimization, to appear in C. Floudas and P. Pardalos (eds), Recent Advances in Global Optimization, Princeton University Press.
Ben-Ayed, O. and C. E., Blair (1990), Computational Difficulties of Bilevel Linear Programming, Operations Research 38, 556–560.
Tuy, H., A. Migdalas, and P. Värbrand (1990), A Global Optimization Approach for the Linear Two Level Program, Preprint, Department of Mathematics, Linköping University (submitted).
Tuy, H. (1990), On Polyhedral Annexation Method for Concave Minimization, in Lev, J. Leifman (ed.), Functional Analysis, Optimization, and Mathematical Economics (volume dedicated to the memory of Kantorovich), Oxford University Press, New York, 248–260.
Konno, H. (1976), A Cutting Plane Algorithm for Solving Bilinear Programs, Mathematical Programming 11, 14–27.
Tuy, H. (1987), Convex Programs with an Additional Reverse Convex Constraint, Journal of Optimization Theory and Applications 52, 463–485.
Tuy, H. (1991), Polyhedral Annexation, Dualization, and Dimension Reduction Technique in Global Optimization, Journal of Global Optimization, 1, 229–244.
Thach, P. T. (1991), Quasiconjugates of Functions and Duality Correspondence between Quasiconcave Minimization under a Reverse Convex Constraint and Quasi Convex Maximization under a Convex Constraint, Journal of Mathematical Analysis and Applications.
Thach, P. T., and R. E. Burkard (1990), Reverse Convex Programs Dealing with the Product of Two Linear Functions, Preprint, Institute of Mathematics, Graz Technical University.
Thach, P. T. and H., Tuy (1990), Dual Solution Methods for Concave Programs and Reverse Convex Programs, Preprint, IHSS, Tokyo, Institute of Technology.
Horst, R. and H. Tuy (1990), Global Optimization (Deterministic Approaches), Springer-Verlag.
Kuno, T. and H. Konno (1990), Parametric Successive Underestimation Method for Convex Multiplicative Programming Problems, Preprint IHSS, Tokyo Institute of Technology.
Konno H. and T. Kuno (1991), Generalized Linear Multiplicative and Fractional Programming, to appear in Annals of Operations Research.
Rockafella, R. T. (1970), Convex Analysis, Princeton University Press.
Horst, R., N. V., Thoai, and J.de, Vries (1988), On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, Operations Research Letters 7, 85–90.
Tuy, H. (1991), Normal Conical Algorithm for Concave Minimization over Polytopes, Mathematical Programming, 51, 229–245.
Falk, J. E. (1973), A Linear Max-Min Problem, Mathematical Programming 5, 169–188.
Tuy, H. and N. V., Thuong (1988), On the Global Minimization of a Convex Function under General Nonconvex Constraints, Applied Mathematics and Optimization 18, 119–142.
Kuno T. and H. Konno (1990), A Parametric Successive Underestimation Method for Convex Programming Problems with an Additional Convex Multiplicative Constraint, Preprint, IHSS, Tokyo Institute of Technology.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tuy, H. The complementary convex structure in global optimization. J Glob Optim 2, 21–40 (1992). https://doi.org/10.1007/BF00121300
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00121300