Jointly Constrained Biconvex Programming

@article{AlKhayyal1983JointlyCB,
  title={Jointly Constrained Biconvex Programming},
  author={Faiz A. Al-Khayyal and James E. Falk},
  journal={Math. Oper. Res.},
  year={1983},
  volume={8},
  pages={273-286}
}
This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x and y over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a special case of the problem addressed in this paper. We prove that the minimum of a biconcave function… 

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References

SHOWING 1-10 OF 21 REFERENCES

An Algorithm for Separable Nonconvex Programming Problems

An algorithm for solving mathematical programming problems of the form Find x = x1,..., xn to minimize Σφixi subject to x ∈ G and l ≤ x ≤ L, which solves a sequence of problems in each of which the objective function is convex.

An algorithm for nonconvex programming problems

  • R. Horst
  • Computer Science, Mathematics
    Math. Program.
  • 1976
A general algorithm which solves a sequence of problems in each of which the objective function is convex or even linear and proves a convergence theorem under suitable regularity assumptions.

Bilinear Programming: Part I. Algorithm for Solving Bilinear Programs.

An algorithm for determining a global optimum of a structured non-concave quadratic programming problem called the bilinear programming problem (BLP): maximize c supt x + d supt y + x supt Cy subject to Ex or =0 Fy or = 0 is established.

Bilinear Programming: Part II. Application of Bilinear Programming.

  • H. Konno
  • Computer Science, Mathematics
  • 1971
Abstract : In the paper a number of new problems such as constrained bematrix game, multi-stage Markovian assignment problem, complementary (orthogonal) planning problem, the problem of reducing a

AN ALGORITHM FOR NONCONVEX PROGRAMMING

The central aim here is to present a new framework for reaching global optimum, which involves two interconnected mechanisms, a method for structuring the search and a decision rule for selecting the course of the search.

A cutting plane algorithm for solving bilinear programs

A cutting plane algorithm is proposed to solve a special class of nonconvex quadratic program referred to as a bilinear program in the literature and the preliminary results of numerical experiments are encouraging.

A linear max—min problem

We consider a two person max—min problem in which the maximizing player moves first and the minimizing player has perfect information of the outcome of this move. The move of the maximizing player

Two-person nonzero-sum games and quadratic programming

Solving Certain Nonconvex Quadratic Minimization Problems by Ranking the Extreme Points

This paper presents a procedure for solving these problems, it involves determining a related linear program having the same constraints, and the extreme-point-ranking approach of Murty is then applied to this linear program to obtain an optimum solution to the quadratic program.

Bilinear programming: An exact algorithm

This work gives a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution to the Bilinear Programming Problem.