Bilinear programming: An exact algorithm

The Bilinear Programming Problem is a structured quadratic programming problem whose objective function is, in general, neither convex nor concave. Making use of the formal linearity of a dual formulation of the problem, we give a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution. The Bilinear Programming Problem, in its general form, is to determine x, an n-vector, and y, an n'-vector, to maximize cTx + x-r QXy + dry, subject to Ax <-a, Bry <~ b, (0.1) x~>0, y~>0, where A is an m by n matrix, B T an rn' by n' matrix, QT an n by n' matrix, c, d, a and b are n, n', m m'-vectors respectively. We will assume that X = {x [ Ax <~ a, x ~> 0} and Y = {y I BTy <-b, y >1 0} are bounded and nonempty. It can be easily verified that the set of all optimal solutions of (0.1) contains at least one element (x*, y*), such that x* is a vertex of X and y* is a vertex of Y. It can be directly derived from the Duality Theory that (0.1) is equivalent to the problem of determining x, an n-vector, and u, an m'-vector, to maximize (crx + min b TU), subject to Ax<~a, B u > i d + Q x , (0.2) x~>0, u~>0. In this paper, we solve (0.2) directly. First, in Sections 1 and 2, some geometric properties of the solution set are determined, and a necessary and