Small and strong formulations for unions of convex sets from the Cayley embedding

There is often a significant trade-off between formulation strength and size in mixed integer programming (MIP). When modelling convex disjunctive constraints (e.g. unions of convex sets) this trade-off can be resolved by adding auxiliary continuous variables. However, adding these variables can result in a deterioration of the computational effectiveness of the formulation. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural nonpolyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.National Science Foundation (U.S.) (grant CMMI-1351619

Embedding Formulations and Complexity for Unions of Polyhedra

It is well known that selecting a good Mixed Integer Programming (MIP) formulation is crucial for an effective solution with state-of-the art solvers. While best practices and guidelines for constructing good formulations abound, there is rarely a systematic construction leading to the best possible formulation. We introduce embedding formulations and complexity as a new MIP formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size. More specifically, they yield the smallest possible ideal formulation (i.e. one whose LP relaxation has integral extreme points) among all formulations that only use 0-1 auxiliary variables. We use the paradigm to characterize optimal formulations for SOS2 constraints and certain piecewise linear functions of two variables. We also show that the resulting formulations can provide a significant computational advantage over all known formulations for piecewise linear functions.United States. National Science Foundation. (Grant CMMI-13516

Incremental and encoding formulations for Mixed Integer Programming

The standard way to represent a choice between n alternatives in Mixed Integer Programming is through n binary variables that add up to one. Unfortunately, this approach commonly leads to unbalanced branch-and-bound trees and diminished solver performance. In this paper, we present an encoding formulation framework that encompasses and expands existing approaches to mitigate this behavior. Through this framework, we generalize the incremental formulation for piecewise linear functions to any finite union of polyhedra with identical recession cones

On packing and covering polyhedra in infinite dimensions

We consider the natural generalizations of packing and covering polyhedra in infinite dimensions, and study issues related to duality and integrality of extreme points for these sets. Using appropriate finite truncations we give conditions under which complementary slackness holds for primal/dual pairs of the infinite linear programming problems associated with infinite packing and covering polyhedra. We also give conditions under which the extreme points are integral. We illustrate an application of our results on an infinite-horizon lot-sizing problem. Keywords: Covering polyhedron; Packing polyhedron; Infinite linear program; Complementary slackness; Integral extreme poin

Strong mixed-integer formulations for the floor layout problem

The floor layout problem (FLP) tasks a designer with positioning a collection of rectangular boxes on a fixed floor in such a way that minimizes total communication costs between the components. While several mixed integer programming (MIP) formulations for this problem have been developed, it remains extremely challenging from a computational perspective. This work takes a systematic approach to constructing MIP formulations and valid inequalities for the FLP that unifies and recovers all known formulations for it. In addition, the approach yields new formulations that can provide a significant computational advantage and can solve previously unsolved instances. While the construction approach focuses on the FLP, it also exemplifies generic formulation techniques that should prove useful for broader classes of problems.United States. National Science Foundation. Graduate Research Fellowship Program (Grant 1122374)United States. National Science Foundation. Graduate Research Fellowship Program (Grant CMMI-1351619

Solution strategies to the stochastic design of mineral flotation plants

The aim of this study is two-fold: first, to analyze the effect of stochastic uncertainty in the design of flotation circuits and second, to analyze different strategies for the solution of a two-stage stochastic problem applied to a copper flotation circuit. The paper begins by introducing a stochastic optimization problem whose aim is to find the best configuration of superstructure, equipment design and operational conditions, such as residence time and stream flows. Variability is considered in the copper price and ore grade. This variability is represented by scenarios with their respective probability of occurrence. The resulting optimization problem is a two-stage stochastic mixed integer nonlinear program (TS-MINLP), which can be extremely challenging to solve. For this reason, several solvers for this problem are compared and two stochastic programming methodologies are applied. The combination of these techniques allows the production of high quality solutions and an analysis of their sensitivity to epistemic uncertainty. The results show that the stochastic problem gives better designs because it allows operational parameters to adapt to the uncertainty of the parameters. The results also show that the flotation circuit structure can vary with the feed grade and copper price. The sensitivity analysis shows small to moderate variability with epistemically uncertain parameters

Beating the SDP bound for the floor layout problem: A simple combinatorial idea

For many mixed-integer programming (MIP) problems, high-quality dual bounds can be obtained either through advanced formulation techniques coupled with a state-of-the-art MIP solver, or through semi-definite programming (SDP) relaxation hierarchies. In this paper, we introduce an alternative bounding approach that exploits the ‘combinatorial implosion’ effect by solving portions of the original problem and aggregating this information to obtain a global dual bound. We apply this technique to the one-dimensional and two-dimensional floor layout problems and compare it with the bounds generated by both state-of-the-art MIP solvers and by SDP relaxations. Specifically, we prove that the bounds obtained through the proposed technique are at least as good as those obtained through SDP relaxations, and present computational results that these bounds can be significantly stronger and easier to compute than these alternative strategies, particularly for very difficult problem instances.United States. National Science Foundation. Graduate Research Fellowship Program (Grant 1122374)United States. National Science Foundation. Graduate Research Fellowship Program (Grant CMMI-1351619

Mixed-Integer Convex Representability

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.United States. National Science Foundation. (Grant CMMI-1351619

On the Chvátal–Gomory closure of a compact convex set

In this paper, we show that the Chvátal–Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver (Ann Discret Math 9:291–296, 1980) for irrational polytopes, and generalizes the same result for the case of rational polytopes (Schrijver in Ann Discret Math 9:291–296, 1980), rational ellipsoids (Dey and Vielma in IPCO XIV, Lecture Notes in Computer Science, vol 6080. Springer, Berlin, pp 327–340, 2010) and strictly convex bodies (Dadush et al. in Math Oper Res 36:227–239, 2011). An extended abstract of this paper appeared in [6]. After the completion of this work, it has been brought to our notice that the polyhedrality of the Chvátal–Gomory closure for irrational polytopes has recently been shown independently by Dunkel and Schulz [9]. The proof presented in this paper has been obtained independently.United States. National Science Foundation. (Grant CMMI-1030662)United States. National Science Foundation. (Grant CMMI-1030422

Restricted risk measures and robust optimization

In this paper we consider characterizations of the robust uncertainty sets associated with coherent and distortion risk measures. In this context we show that if we are willing to enforce the coherent or distortion axioms only on random variables that are affine or linear functions of the vector of random parameters, we may consider some new variants of the uncertainty sets determined by the classical characterizations. We also show that in the finite probability case these variants are simple transformations of the classical sets. Finally we present results of computational experiments that suggest that the risk measures associated with these new uncertainty sets can help mitigate estimation errors of the Conditional Value-at-Risk. Keywords: Risk management; Stochastic programming; Uncertainty modelin