Reverse convex programming

Abstract

Reverse convex programs generally have disconnected feasible regions. Basic solutions are defined and properties of the latter and of the convex hull of the feasible region are derived. Solution procedures are discussed and a cutting plane algorithm is developed. IntroductionA constraint h ( x ) >1 0 is called a reverse convex constraint if h is pseudo-convex. Optimization problems with several such constraints generally have disconnected feasible regions. To our knowledge, problems of this form were first studied by Rosen [13], in a control theoretic setting, and subsequently, in an engineering design setting, by Avriel and Williams [1,2]. These authors developed a procedure for finding a Kuhn-Tucker point and this procedure, in a more general setting, has been proven to converge to such a point by Meyer [12] from whom the term "reverse convex" is taken. As shown by Avriel and Williams, such constraints arise in geometric programming when some coefficients of product terms are negative, a likely occurrence in engineering design problems. Bansal and Jacobsen [4,5] and Hillestad [9] show that such problems arise when there are b u d g e t constraints which reflect economies-of-scale. It is also of interest to note that problems with 0-1 restrictions can be cast into the reverse convex form. For instance, the constraint x i = 0 or 1 can be rewritten as -x i + x~ >t 0, 0 < x i • 1. Ueing [ 16] has developed a combinatorial procedure for reverse convex problems and this procedure is expanded upon in this paper.

Linear programs with an additional reverse convex constraint

“…convex problems of the form max cx A l X < bi g(x) <. < 0 where A I is the matrix formed by one of the k possible selections of n -1 rows of A and similarly for b t. This procedure also applies if g is pseudo-convex [9]. It is also possible to apply Tuy cuts [14] to P; however, the convergence of such a procedure is uncertain.…”

“…It is also possible to apply Tuy cuts [14] to P; however, the convergence of such a procedure is uncertain. In fact, when there are two reverse convex constraints, in addition to A x >~b, reference [9] shows that a cutting plane algorithm based upon Tuy cuts may fail to converge to a feasible point. In this paper we show that the convex hull of the feasible region is a convex polytope and, as a result, an optimal solution for P lies on an edge of F A.…”

Sequential convexification in reverse convex and disjunctive programming

“…For general reverse convex programs, cony G. is often a polytope (this is subject, as already mentioned, to certain differentiability and compactpactness assumptions; see Hillestad and Jacobsen [1980] for details). In this case cony G. is generated by a finite number of so-called quasivertices, and the checking of the Constraint Boundary Condition can be restricted to pairs x, y of such quasivertices.…”

Linear programs with an additional separable concave constraint

“…Hillestad's algorithm has been modified and still developed by Hillestad-Jacobsen [7] and Thuong-Tuy [17]. The second class is outer approximation algorithms, which involves e.g., Hillestad-Jacobsen [6] and Fülöp [4]. Hillestad-Jacobsen [6] developed a procedure for cutting off a portion from F by a valid cut constructed at an infeasible vertex of F for the associated concave minimization.…”

“…The second class is outer approximation algorithms, which involves e.g., Hillestad-Jacobsen [6] and Fülöp [4]. Hillestad-Jacobsen [6] developed a procedure for cutting off a portion from F by a valid cut constructed at an infeasible vertex of F for the associated concave minimization. The convergence of their algorithm is not guaranteed; but Fülöp [4] improved this point later.…”