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Branch-and-reduce algorithm for convex programs with additional multiplicative constraints

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  • Benson, Harold P.
  • Sun, Erjiang

Abstract

This article presents a branch-and-reduce algorithm for globally solving for the first time a convex minimization problem (P) with p[greater-or-equal, slanted]1 additional multiplicative constraints. In each of these p additional constraints, the product of two convex functions is constrained to be less than or equal to a positive number. The algorithm works by globally solving a 2p-dimensional master problem (MP) equivalent to problem (P). During a typical stage k of the algorithm, a point is found that minimizes the objective function of problem (MP) over a nonconvex set Fk that contains the portion of the boundary of the feasible region of the problem where a global optimal solution lies. If this point is feasible in problem (MP), the algorithm terminates. Otherwise, the algorithm continues by branching and creating a new, reduced nonconvex set Fk+1 that is a strict subset of Fk. To implement the algorithm, all that is required is the ability to solve standard convex programming problems and to implement simple algebraic steps. Convergence properties of the algorithm are given, and results of some computational experiments are reported.

Suggested Citation

  • Benson, Harold P. & Sun, Erjiang, 2009. "Branch-and-reduce algorithm for convex programs with additional multiplicative constraints," European Journal of Operational Research, Elsevier, vol. 199(1), pages 1-8, November.
  • Handle: RePEc:eee:ejores:v:199:y:2009:i:1:p:1-8
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    References listed on IDEAS

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    1. Kuno, Takahito & Yajima, Yasutoshi & Yamamoto, Yoshitsugu & Konno, Hiroshi, 1994. "Convex programs with an additional constraint on the product of several convex functions," European Journal of Operational Research, Elsevier, vol. 77(2), pages 314-324, September.
    2. H. P. Benson, 2005. "Decomposition Branch-and-Bound Based Algorithm for Linear Programs with Additional Multiplicative Constraints," Journal of Optimization Theory and Applications, Springer, vol. 126(1), pages 41-61, July.
    3. Arthur M. Geoffrion, 1967. "Solving Bicriterion Mathematical Programs," Operations Research, INFORMS, vol. 15(1), pages 39-54, February.
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    Cited by:

    1. Erjiang Sun, 2017. "On Optimization Over the Efficient Set of a Multiple Objective Linear Programming Problem," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 236-246, January.

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