Linear programs with an additional reverse convex constraint

Abstract

Abstract. A constraint g ( x )>1 0 is said to be a reverse convex constraint if the function g is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.

Outer approximation algorithms for DC programs and beyond

“…Tuy [105] extended a property in Hillestad and Jacobsen [33] by proving that, the closure of the convex hull of the feasible set is a polyhedral set. Therefore, this type of global minimization problems can be theoretically reduced to a concave minimization problem.…”

“…Fulop [28] used this property to propose a cutting plane method for solving the reverse convex optimization problem. Another important property of this problem is that the convex hull of bounded feasible region is a polytope and the solution lies at a vertex of this polytope, which has been exploited to propose an algorithm by Hillestad and Jacobsen [33]. Furthermore, Sen and Sherali [84] pointed out that, for certain special reverse convex sets, they construct a type of finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set, which provides the capability of generating any facet cut.…”

“…One of the first Cut and Split algorithms was devised by Hillestad and Jacobsen [33] to solve reverse convex programs, i.e. canonical DC problems where Ω is a polyhedron.…”

A Parallel Algorithm for Linear Programs with an Additional Reverse Convex Constraint

“…Hillestad and Jacobsen [8] gave characterizations of optimal solutions and provided a finite algorithm based on these optimality properties. Subsequently, Thuong and Tuy [28] proposed an algorithm involving a sequence of linear programming steps and concave programming steps.…”

Linear programs with an additional separable concave constraint

“…The first class consists of algorithms based on the edge property of F \ G. As will be shown in Section 2, at least one optimal solution to LPAC lies on the intersection of the edges of F and the boundary of G. Exploiting this property, Hillestad [5] proposed a simplex-type pivoting algorithm for searching an optimal intersection point. 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].…”