Jointly Constrained Biconvex Programming

  title={Jointly Constrained Biconvex Programming},
  author={Faiz A. Al-Khayyal and James E. Falk},
  journal={Math. Oper. Res.},
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… 

Figures from this paper

Sequential Convexification of a Bilinear Set

A sequential convexification procedure is presented to derive a set arbitrary close to the convex hull of -feasible solutions to a general nonconvex continuous bilinear set, which can be used in a cutting plane algorithm to solve a linear optimization problem over the bilinears set to -optimality.

A finite algorithm for concave minimization over a polyhedron

A new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron of the branch-and-bound type finds a globally optimal extreme point solution in a finite number of steps.

The Convex Hull of a Quadratic Constraint over a Polytope

This paper shows that the exact convex hull of a general quadratic equation intersected with any bounded polyhedron is second-order cone representable.

αBB: A global optimization method for general constrained nonconvex problems

The proposed branch and bound type algorithm attains finiteε-convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems.

A reformulation-convexification approach for solving nonconvex quadratic programming problems

This paper considers the class of linearly constrained nonconvex quadratic programming problems, and presents a new approach based on a novel Reformulation-Linearization/Convexification Technique, showing that for many problems, the initial relaxation itself produces an optimal solution.

Global optimization of a quadratic function subject to a bounded mixed integer consraint set

In this paper we consider the optimization of a quadratic function subject to a linearly bounded mixed integer constraint set. We develop two types of piecewise affine convex underestimating

Obtaining an Approximate Solution for Quadratic Maximization Problems

In this paper, we consider indefinite quadratic maximization problems over inequality constraints. Through the Reformulation and Linearization Technique (RLT), we reformulate the problem as a linear

Towards the Biconjugate of Bivariate Piecewise Quadratic Functions

This work focuses on computing the conjugate of a bivariate piecewise quadratic function defined over a polytope, and obtains explicit formulas for the convex envelope of piecewise rational functions.

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

This paper proposes different methods for finding the global minimum of a concave function subject to quadratic separable constraints and shows how this constraint can be replaced by an equivalent system of convex and separablequadratic constraints.

Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques

This work develops the convex envelope and concave envelope of z=x/y over a hypercube and proposes a new relaxation technique for fractional programs which includes the derived envelopes.



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


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.