## Journal of Applied Mathematics

### A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming

#### Abstract

A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP). We firstly convert the problem (LMP) into an equivalent programming one by introducing $p$ auxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 984168, 10 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807935

Digital Object Identifier
doi:10.1155/2013/984168

Mathematical Reviews number (MathSciNet)
MR3039751

Zentralblatt MATH identifier
1266.90126

#### Citation

Zhou, Xue-Gang; Cao, Bing-Yuan. A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming. J. Appl. Math. 2013 (2013), Article ID 984168, 10 pages. doi:10.1155/2013/984168. https://projecteuclid.org/euclid.jam/1394807935

#### References

• T. Matsui, “NP-hardness of linear multiplicative programming and related problems,” Journal of Global Optimization, vol. 9, no. 2, pp. 113–119, 1996.
• H. P. Benson, “Global maximization of a generalized concave multiplicative function,” Journal of Optimization Theory and Applications, vol. 137, no. 1, pp. 105–120, 2008.
• H. Tuy, Analysis and Global Optimization, vol. 22, Kluwer Academic, Dordrecht, The Netherlands, 1998.
• P. M. Pardalos and J. B. Rosen, Constrained Global Optimization: Algorithms and Applications, vol. 268, Springer, Berlin, Germany, 1987.
• C. A. Floudas and V. Visweswaran, “Quadratic optimization,” in Handbook of Global Optimization, R. Horst and P. M. Pardalos, Eds., pp. 217–269, Kluwer Academic, Dordrecht, The Netherlands, 1995.
• R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, vol. 3, Kluwer Academic, Dordrecht, The Netherlands, 1995.
• H. Watanabe, IC layout generation and compaction using mathematical programming [Ph.D. thesis], University of Rochester, Rochester, NY, USA, 1984.
• H. Konno and Y. Yajima, “Solving rank two bilinear programs by parametric simplex algorithms,” Instituteof Human and Social Sciences Working Paper IHSS 90-17, Tokyo Institute of Technology, Tokyo, Japan, 1990.
• H. Konno, P. T. Thach, and H. Tuy, Optimization on Low Rank Nonconvex Structures, vol. 15, Kluwer Academic, Dordrecht, The Netherlands, 1997.
• O. L. Mangasarian, “Equilibrium points of bimatrix games,” Journal of the Society for Industrial and Applied Mathematics, vol. 12, pp. 778–780, 1964.
• A. M. Frieze, “A bilinear programming formulation of the 3-dimensional assignment problem,” Mathematical Programming, vol. 7, no. 1, pp. 376–379, 1979.
• J. E. Falk, “A linear max-min problem,” Mathematical Programming, vol. 5, pp. 169–188, 1973.
• M. Raghavachari, “On connections between zero-one integer programming and concave programming under linear constraints,” Operations Research, vol. 17, pp. 680–684, 1969.
• G. L. Nemhauser and L. A. Wolsey, Combinatorial Optimization, Wiley, New York, NY, USA, 1998.
• H. Konno, T. Kuno, and Y. Yajima, “Global minimization of a generalized convex multiplicative function,” Journal of Global Optimization, vol. 4, no. 1, pp. 47–62, 1994.
• H. Konno, Y. Yajima, and T. Matsui, “Parametric simplex algorithms for solving a special class of nonconvex minimization problems,” Journal of Global Optimization, vol. 1, no. 1, pp. 65–81, 1991.
• N. V. Thoai, “A global optimization approach for solving the convex multiplicative programming problem,” Journal of Global Optimization, vol. 1, no. 4, pp. 341–357, 1991.
• H. Konno and T. Kuno, “Generalized linear multiplicative and fractional programming,” Annals of Operations Research, vol. 25, no. 1–4, pp. 147–161, 1990.
• T. Kuno, Y. Yajima, and H. Konno, “An outer approximation method for minimizing the product of several convex functions on a convex set,” Journal of Global Optimization, vol. 3, no. 3, pp. 325–335, 1993.
• H.-S. Ryoo and N. V. Sahinidis, “Global optimization of multiplicative programs,” Journal of Global Optimization, vol. 26, no. 4, pp. 387–418, 2003.
• T. Kuno, “Solving a class of multiplicative programs with 0-1 knapsack constraints,” Journal of Optimization Theory and Applications, vol. 103, no. 1, pp. 121–135, 1999.
• H. P. Benson, “An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming,” Journal of Global Optimization, vol. 15, no. 4, pp. 315–342, 1999.
• J. M. Mulvey, R. J. Vanderbei, and S. A. Zenios, “Robust optimization of large-scale systems,” Operations Research, vol. 43, no. 2, pp. 264–281, 1995.
• P. Shen and H. Jiao, “Linearization method for a class of multiplicative programming with exponent,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 328–336, 2006.
• X.-G. Zhou and K. Wu, “A method of acceleration for a class of multiplicative programming problems with exponent,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 975–982, 2009.
• K. P. Bennett, “Global tree optimization: anon-greedy decision tree algorithm,” Computing Sciences AndStatistics, vol. 26, pp. 156–160, 1994.
• P. M. Pardalos, “Polynomial time algorithms for some classes of constrained nonconvex quadratic problems,” Optimization, vol. 21, no. 6, pp. 843–853, 1990.
• J. E. Falk and S. W. Palocsay, “Image space analysis of generalized fractional programs,” Journal of Global Optimization, vol. 4, no. 1, pp. 63–88, 1994.
• S. Schaible and C. Sodini, “Finite algorithm for generalized linear multiplicative programming,” Journal of Optimization Theory and Applications, vol. 87, no. 2, pp. 441–455, 1995.
• H. P. Benson and G. M. Boger, “Outcome-space cutting-plane algorithm for linear multiplicative programming,” Journal of Optimization Theory and Applications, vol. 104, no. 2, pp. 301–322, 2000.
• H. P. Benson and G. M. Boger, “Multiplicative programming problems: analysis and efficient point search heuristic,” Journal of Optimization Theory and Applications, vol. 94, no. 2, pp. 487–510, 1997.
• X. J. Liu, T. Umegaki, and Y. Yamamoto, “Heuristic methods for linear multiplicative programming,” Journal of Global Optimization, vol. 15, no. 4, pp. 433–447, 1999.
• H. P. Benson, “Decomposition branch-and-bound based algorithm for linear programs with additional multiplicative constraints,” Journal of Optimization Theory and Applications, vol. 126, no. 1, pp. 41–61, 2005.
• H. P. Benson, “A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem,” European Journal of Operational Research, vol. 182, no. 2, pp. 597–611, 2007.
• R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, Berlin, Germany, 1993.