A complete characterization of disjunctive conic cuts for mixed integer second order cone optimization

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feasible set is bounded. Using a similar technique, we show that one can extend our previous results to the case in which that intersection is unbounded. We provide a complete characterization in closed form of the conic inequalities required to describe the convex hull when the hyperplanes defining the disjunction are parallel.     

Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integrality-preserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Section 6 reviews lift-and-project cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with lift-and-project cuts. This is an updated and extended version of the paper published in LNCS 2241, Springer, 2001 (as given in Balas, 2001). Research was supported by the National Science Foundation through grant #DMI-9802773 and by the Office of Naval Research through contract N00014-97-1-0196.     

A hierarchy of relaxations for nonlinear convex generalized disjunctive programming

We propose a framework to generate alternative mixed-integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations. We extend the concept of “basic steps” defined for disjunctive linear programs to the nonlinear case. A basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts. We show that the strength of the relaxations increases as the number of conjuncts decreases, leading to a hierarchy of relaxations. We prove that the tightest of these relaxations, allows in theory the solution of the disjunctive convex program as a nonlinear programming problem. We present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed-integer program. Finally, we apply the theory developed to improve the computational efficiency of solution methods for nonlinear convex generalized disjunctive programs (GDP). This methodology is validated through a set of numerical examples.     

Sequential convexification in reverse convex and disjunctive programming

This paper is about a property of certain combinatorial structures, called sequential convexifiability, shown by Balas (1974, 1979) to hold for facial disjunctive programs. Sequential convexifiability means that the convex hull of a nonconvex set defined by a collection of constraints can be generated by imposing the constraints one by one, sequentially, and generating each time the convex hull of the resulting set. Here we extend the class of problems considered to disjunctive programs with infinitely many terms, also known as reverse convex programs, and give necessary and sufficient conditions for the solution sets of such problems to be sequentially convexifiable. We point out important classes of problems in addition to facial disjunctive programs (for instance, reverse convex programs with equations only) for which the conditions are always satisfied. Finally, we give examples of disjunctive programs for which the conditions are violated, and so the procedure breaks down.The research underlying this report was supported by Grant ECS-8601660 of The National Science Foundation and Contract N00014-85-K-0198 with the Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.On leave from the University of Aarhus, Denmark.     

On mixed-integer sets with two integer variables

We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alternative characterization of the crooked cross cut closure of mixed-integer sets similar to the one on the equivalence of different definitions of split cuts presented in Cook et al. (1990) [4]. This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard (2008) [8]. Finally, we extend our results to mixed-integer sets that are defined as the set of points (with some components being integral) inside a closed, bounded and convex set.     

Convex programming for disjunctive convex optimization

Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming. Received November 27, 1996 / Revised version received June 11, 1999?Published online November 9, 1999     

Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characterization

Disjunctive Programs can often be transcribed as reverse convex constrained problems with nondifferentiable constraints and unbounded feasible regions. We consider this general class of nonconvex programs, called Reverse Convex Programs (RCP), and show that under quite general conditions, the closure of the convex hull of the feasible region is polyhedral. This development is then pursued from a more constructive standpoint, in that, for certain special reverse convex sets, we specify a finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set. When interpreted in the context of convexity/intersection cuts, this provides the capability of generating any (negative edge extension) facet cut. Although this characterization is more clarifying than computationally oriented, our development shows that if certain bounds are available, then convexity/intersection cuts can be strengthened relatively inexpensively.     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.      

A cutting-plane approach to mixed 0–1 stochastic integer programs

We consider a mixed 0–1 integer programming problem with dual block-angular structure arising in two-stage stochastic programming. A relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable. The convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how Lift-and-Project cuts can be generated for one subproblem and made valid for different outcomes.     

A convex-analysis perspective on disjunctive cuts

We treat with tools from convex analysis the general problem of cutting planes, separating a point from a (closed convex) set P. Crucial for this is the computation of extreme points in the so-called reverse polar set, introduced by E. Balas in 1979. In the polyhedral case, this enables the computation of cuts that define facets of P. We exhibit three (equivalent) optimization problems to compute such extreme points; one of them corresponds to selecting a specific normalization to generate cuts. We apply the above development to the case where P is (the closed convex hull of) a union, and more particularly a union of polyhedra (case of disjunctive cuts). We conclude with some considerations on the design of efficient cut generators. The paper also contains an appendix, reviewing some fundamental concepts of convex analysis. Supported by NSF grant DMII-0352885, ONR grant N00014-03-1-0188, INRIA grant ODW and IBM.     

Max Horn SAT and the minimum cut problem in directed hypergraphs

In this paper we consider the Maximum Horn Satisfiability problem, which is reduced to the problem of finding a minimum cardinality cut on a directed hypergraph. For the latter problem, we propose different IP formulations, related to three different definitions of hyperpath weight. We investigate the properties of their linear relaxations, showing that they define a hierarchy. The weakest relaxation is shown to be equivalent to the relaxation of a well known IP formulation of Max Horn SAT, and to a max-flow problem on hypergraphs. The tightest relaxation, which is a disjunctive programming problem, is shown to have integer optimum. The intermediate relaxation consists in a set covering problem with a possible exponential number of constraints. This latter relaxation provides an approximation of the convex hull of the integer solutions which, as proven by the experimental results given, is much tighter than the one known in the literature. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.     

Minimum Distance to the Complement of a Convex Set: Duality Result

The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs.     

Partial convexification cuts for 0–1 mixed-integer programs

In this research, we propose a new cut generation scheme based on constructing a partial convex hull representation for a given 0–1 mixed-integer programming problem by using the reformulation–linearization technique (RLT). We derive a separation problem that projects the extended space of the RLT formulation into the original space, in order to generate a cut that deletes a current fractional solution. Naturally, the success of such a partial convexification based cutting plane scheme depends on the process used to tradeoff the strength of the cut derived and the effort expended. Accordingly, we investigate several variable selection rules for performing this convexification, along with restricted versions of the accompanying separation problems, so as to be able to derive strong cuts within a reasonable effort. We also develop a strengthening procedure that enhances the generated cut by considering the binariness of the remaining unselected 0–1 variables. Finally, we present some promising computational results that provide insights into implementing the proposed cutting plane methodology.     

Some results on linear delay integral equations

Recently Borwein has proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borwein's definition but improper by Geoffrion's definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borwein's definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrion's definition when S is the non-negative orthant. The proposed definition seems preferable to Borwein's for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone.     

An Exact Reformulation Algorithm for Large Nonconvex NLPs Involving Bilinear Terms

Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not affect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an efficient graph-theoretical algorithm for effecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branch-and Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.     

Convex hull representations of models for computing collisions between multiple bodies

In this paper, we consider a collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of S.Z. Selim and H.A. Almohamad [European Journal of Operational Research 119 (1) (1999) 121–129], and compare the relative effectiveness in solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear segment. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once. We provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios, as well as the times at which the first collision occurs.     

Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation

Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [21] and Ceria and Soares [6], we propose a convex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a Mixed-Integer Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional big-M formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.