Jointly Constrained Biconvex Programming

Abstract

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 andy 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 over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point. The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented. Introduction. One of the most persistently difficult and recurring nonconvex problems in mathematical programming is the bilinear program, whose general form is minimize c x + x Ay + d y (X,y) (1) subject to x EX,y E Y where c and d are given vectors, A is a given p X q matrix, and X and Y are given polyhedra in _SP and _q, respectively. Mills [15], Mangasarian [12], Mangasarian and Stone [13], and Altman [1] studied the problem as formulated in the bimatrix game context. Solution procedures were either locally convergent (e.g., [1], [2]) or completely enumerative (e.g., [13]). Cabot and Francis [2] proposed an extreme point ranking procedure for the solution. Konno, in a series of papers [9]-[11], develops a cutting plane approach designed to converge locally, and in a finite number of steps to an e-optimal solution. By taking the partial dual of (1) with respect to y, the problem becomes a min-max problem wherein the constraint region available to the "inside optimizer" (i.e., the maximizer) is determined by the selection of the "outside optimizer's" (i.e., the minimizer over X) move. Falk [4] addressed this formulation and proposed a branch-andbound solution procedure. Vaish and Shetty developed two finite procedures: one involving the extension of Tui's method of "polyhedral annexation" [20] and the other being a cutting plane procedure employing polar cuts [21]. Subsequently, Sherali and Shetty [18] improved the latter procedure by incorporating disjunctive face cuts. Finally, Gallo and Ulkiicii [7] addressed the min-max formulation by developing a cutting plane approach in connection with Tui's method. Applications of the bilinear programming problem include constrained bimatrix games, dynamic Markovian assignment problems, multicommodity network flow *

Globally Optimal Algorithms for Stratified Autocalibration

“…Condition (L1) is evidently true by construction. That the Cauchy continuity condition (L2) holds for various bilinear relaxations follows from (McCormick 1976;Al-Khayyal and Falk 1983;Horst and Tuy 2006). We refer the reader to Sahinidis 2001, 2002) to prove the same for Case I in Sect.…”

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

“…Convex hull Al- Khayyal and Falk (1983);McCormick (1976) Outer approximation Geoffrion (1970); Duran and Grossmann (1986a,b);Floudas (1995) Eigenvector projections Rosen and Pardalos (1986);Pardalos (1991) (2006); Karuppiah and Grossmann (2006); Wicaksono and Karimi (2008) continued on the next page (2010); Teles et al (2012); Castro and Teles (2013); Kolodziej et al (2013a); Gupte et al (2013) Trilinear…”

Solving a mixed integer linear program approximation of the toll design problem using constraint generation within a branch-and-cut algorithm

“…For each link flow segment, the linear function which underestimates −t a (v a ), and the convex envelope (Al-Khayyal and Falk, 1983) which underestimates τ a v a are specified. Let P be the number of linear segments defined by P + 1 link flow break points.…”