Journal of Global Optimization

, Volume 12, Issue 3, pp 267–283 | Cite as

Global Optimization of Nonconvex Polynomial Programming Problems Having Rational Exponents

  • Hanif D. Sherali
Article

Abstract

This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.

Polynomial programs Reformulation-Linearization Technique (RLT) Nonconvex programming Global optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-Khayyal, F. A. and J. E. Falk (1983), Jointly constrained biconvex programming, Math. of Oper. Res. 8, 273–283.Google Scholar
  2. Al-Khayyal, F. A., C. Larson and T. Van Voorhis (1994), A relaxation method for nonconvex quadratically constrained quadratic programs.Google Scholar
  3. Brimberg, J. and R. F. Love (1991), Estimating travel distances by the weighted ℓp norm, Naval Research Logistics 38 241–259.Google Scholar
  4. Cole, F., W. Gochet and Y. Smeers (1985), A comparison between a primal and a dual cutting plane algorithm for posynomial geometric programming problems, Journal of Optimization Theory and Applications 47, 159–180.Google Scholar
  5. Cole, F., W. Gochet, F. Van Assche, J. Ecker and Y. Smeers (1980), Reversed geometric programming: A branch-and-bound method involving linear subproblems, European Journal of Operational Research 5, 26–35.Google Scholar
  6. Dembo, R. S. (1976), Aset of geometric programming test problems and their solutions, Mathematical Programming 10, 192–213.Google Scholar
  7. Dembo, R. S. (1978), Current state of the art of algorithms and computer software for geometric programming, Journal of Optimization theory and Applications 26, 149–183.Google Scholar
  8. Duffin, R. J., E. L. Peterson and C. Zener (1967), Geometric Programming. John Wiley & Sons, New York.Google Scholar
  9. Floudas, C. A. and P. M. Pardalos (1990), Acollection of test problems for constrained global optimization algorithms, Lecture Notes in Computer Science, Vol. 455, (eds. G Goos and J. Hartmanis). Springer Verlag, Berlin.Google Scholar
  10. Floudas, C. A. and V. Visweswaran (1995), Quadratic optimization, in Handbook of Global Optimization, Nonconvex Optimization and its Applications (eds. R. Horst and P. M. Pardalos). Kluwer Academic Publishers, 217–270.Google Scholar
  11. Floudas, C. A. and V. Visweswaran (1990), A global optimization algorithm (GOP) for certain classes of nonconvex NLP's I: Theory, Computers and Chemical Engineering 14, 1397–1417.Google Scholar
  12. Hansen, P., and B. Jaumard (1992), Reduction of indefinite quadratic programs to bilinear programs, {tiJournal of Global Optimization} 2(1), 41–60.Google Scholar
  13. Hansen, P., B. Jaumard and J. Xiong (1993), Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions, Journal of Global Optimization 3, 421–437.Google Scholar
  14. Horst, R. and H. Tuy (1993), Global Optimization: Deterministic Approaches, 2nd ed. Springer Verlag, Berlin.Google Scholar
  15. Kortanek, K. L., X. Xu and Y. Ye (1995), An infeasible interior-point algorithm for solving primal and dual geometric programs. Manuscript, Department of Management Science, The University of Iowa, Iowa City, IA 52242.Google Scholar
  16. Kostreva, M. M. and L. A. Kinard (1991), A differentiable homotopy approach for solving polynomial optimization problems and noncooperative games, Computers Math. Applic. 21(6/7), 135–143.Google Scholar
  17. Lasdon, L. S., A. D. Waren, S. Sarkar and F. Palacios, (1979), Solving the pooling problem using generalized reduced gradient and successive linear programming algorithms, SIGMAP Bull. 77, 9–15.Google Scholar
  18. Maranas, C. D. and C. A. Floudas (1994), Global optimization in generalized geometric programming. Working Paper, Department of Chemical Engineering, Princeton University, Princeton, NJ.Google Scholar
  19. Passy, U. (1978), Global solutions of mathematical programs with intrinsically concave functions, {tiJournal of Optimization Theory and Applications} 26, 97–115.Google Scholar
  20. Peterson, E. L. (1976), Geometric programming, SIAM Review 18, 1–15.Google Scholar
  21. Rickaert, M. J. and X. M. Martens (1978), Comparison of generalized geometric programming algorithms, {tiJournal of Optimization Theory and Applications} 26, 205–242.Google Scholar
  22. Shectman, H. P. and N. V. Sahindis (1994), A finite algorithm for global minimization of separable concave programs. Technical Report, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana-Champagne, IL.Google Scholar
  23. Sherali, H. D., A. Alameddine and T. S. Glickman (1994/95), Biconvex models and algorithms for risk management problems, American Journal of Mathematical and Management Sciences 14(2&3), 197–228.Google Scholar
  24. Sherali, H. D. and E. P. Smith (1995), A global optimization approach to a water distribution network design problem. Research Report #HDS95-6, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA (to appear in the Journal of Global Optimization).Google Scholar
  25. Sherali, H. D. and C. H. Tuncbilek (1992), A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique, Journal of Global Optimization 2, 101–112.Google Scholar
  26. Sherali, H. D. and C. H. Tuncbilek (1995), A reformulation-convexification approach for solving nonconvex quadratic programming problems, Journal of Global Optimization 7, 1–31.Google Scholar
  27. Sherali, H. D. and C. H. Tuncbilek (1997), Comparison of two Reformulation-Linearization Technique based linear programming relaxations for polynomial programming problems, Journal of Global Optimization 10, 381–390.Google Scholar
  28. Sherali, H. D. and C. H. Tuncbilek (1996), New reformulation-linearization/convexification relaxations for univariate and multivariate polynomial programming problems, under revision for Operations Research Letters.Google Scholar
  29. Shor, N. Z. (1990), Dual quadratic estimates in polynomial and boolean programming, Annals of Operations Research 25, 163–168.Google Scholar
  30. Watson, L. T., S. C. Billups and A. P. Morgan (1987), Algorithm 652 HOMPACK: A suite of codes for globally convergent homotopy algorithms, ACM Transactions on Mathematical Software 13(3), 281–310.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Hanif D. Sherali
    • 1
  1. 1.Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgU.S.A.

Personalised recommendations