Convexity Cuts Research Articles

A class of multi-level balanced Foundation-Penalty cuts for mixed-integer programs

Glover and Sherali (2003) introduced a wide class of Foundation-Penalty (FP) cuts for GUB and ordinary mixed-integer programs. The FP cuts are easy to generate by exploiting routine branch-and-bound penalty calculations, and encompass other classical cuts such as disjunctive cuts, lift-and-project cuts, convexity cuts, Gomory cuts, and mixed-integer rounding cuts. Here we focus on two special classes of FP cuts, called balanced FP cuts and multi-level balanced FP cuts, and exhibit their relationship to special forms of disjunctive cuts. A numerical example illustrates the rich variety of cuts that can be generated.

Depth-Optimized Convexity Cuts

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems.

Finitely convergent cutting planes for concave minimization

In 1964 Tuy introduced a new type of cutting plane, the concavity cut, in the context of concave minimization. These cutting planes, which are also known as convexity cuts, intersection cuts and Tu...

How to Extend the Concept of Convexity Cuts to Derive Deeper Cutting Planes

A new type of cutting plane, termed a decomposition cut, is introduced that can be constructed under the same assumptions as the well-known convexity cut. Therefore it can be applied in algorithms (e.g. cutting plane, branch-and-cut) for various problems of global optimization, such as concave minimization, bilinear programming, reverse-convex programming, and integer programming. In computational tests with cutting plane algorithms for concave minimization, decomposition cuts were shown to be superior to convexity cuts.

Optimization of linear-convex programs

Let and be bounded polyhedral sets. Let . Consider the minimization of f(u.v) subject to u∊U v∊V. For fixed v fis linear in u and for fixed U∊ U. it is convex in v. If f ∗is the global minimum function value, then there is at least one vertex u ∗∊ U such that for some v ∗ We characterize KARUSH-KUHN-TUCKER and local minimum points of the problem. An algorithm is proposed where convexity cuts are developed from extreme points of U. To insure finite convergence. another type of cuts “facial cuts are used. We propose a shallow “facial cuts ” that requires negligible computation. Finally, we give rules for deleting redundant shallow facial cuts

Improved convexity cuts for lattice point problems

The generalized lattice point (GLP) problem provides a formulation that accommodates a variety of discrete alternative problems. In this paper, we show how to substantially strengthen the convexity cuts for the GLP problem. The new cuts are based on the identification ofsynthesized lattice point conditions to replace those that ordinarily define the cut. The synthesized conditions give an alternative set of hyperplanes that enlarge the convex set, thus allowing the cut to be shifted deeper into the solution space. A convenient feature of the strengthened cuts is the evidence of linking relationships by which they may be constructively generated from the original cuts. Geometric examples are given in the last section to show how the new cuts improve upon those previously proposed for the GLP problem.

Polyhedral convexity cuts and negative edge extensions

This note shows that convexity cuts defined relative to polyhedral convex sets can utilize negative as well as positive edge extensions under appropriate circumstances, yielding stronger cuts than customarily available. We also show how to partially collapse the polyhedron to further improve these cuts.

Convexity Cuts and Cut Search

This note focuses on two new and related cut strategies for integer programming: the “convexity-cut” and “cut-search” strategies. The fundamental notions underlying the convexity-cut approach are due to Richard D. Young and Egon Balas, whose “hypercylindrical” and “intersection” cuts provide the conceptual starting points for the slightly more general framework developed here. We indicate the utility of our framework (and hence the importance of the original Young and Balas ideas) by specifying a variety of new cuts that can be obtained from it. The second new strategy, cut search, shares with the convexity-cut strategy the notion of generating a cut by passing a hyper-plane through the terminal endpoints of edges extended from the vertex of a cone. However, whereas the convexity-cut approach determines the extensions of these edges by reference to a convex set that contains the vertex of the cone in its interior, the cut-search approach determines these extensions by reference to associations between certain “proxy” sets of points (e.g., collections of hyperplanes) and “candidate solutions” to the integer program. The cut-search approach typically involves more work than the convexity-cut approach, but offers the chance to identify feasible solutions in the process, and can sometimes also yield somewhat stronger cuts than the convexity cuts.