François Margot

An algorithmic framework for convex mixed integer nonlinear programs

This paper is motivated by the fact that mixed integer nonlinear programming is an important and difficult area for which there is a need for developing new methods and software for solving large-scale problems. Moreover, both fundamental building blocks, namely mixed integer linear programming and nonlinear programming, have seen considerable and steady progress in recent years. Wishing to exploit expertise in these areas as well as on previous work in mixed integer nonlinear programming, this work represents the first step in an ongoing and ambitious project within an open-source environment. COIN-OR is our chosen environment for the development of the optimization software. A class of hybrid algorithms, of which branch-and-bound and polyhedral outer approximation are the two extreme cases, are proposed and implemented. Computational results that demonstrate the effectiveness of this framework are reported. Both the library of mixed integer nonlinear problems that exhibit convex continuous relaxations, on which the experiments are carried out, and a version of the software used are publicly available.

Branching and bounds tighteningtechniques for non-convex MINLP

Many industrial problems can be naturally formulated using mixed integer non-linear programming (MINLP) models and can be solved by spatial Branch&Bound (sBB) techniques. We study the impact of two important parts of sBB methods: bounds tightening (BT) and branching strategies. We extend a branching technique originally developed for MILP, reliability branching, to the MINLP case. Motivated by the demand for open-source solvers for real-world MINLP problems, we have developed an sBB software package named couenne (Convex Over- and Under-ENvelopes for Non-linear Estimation) and used it for extensive tests on several combinations of BT and branching techniques on a set of publicly available and real-world MINLP instances. We also compare the performance of couenne with a state-of-the-art MINLP solver.

Symmetry in Integer Linear Programming

An integer linear program (ILP) is symmetric if its variables can be permuted without changing the structure of the problem. Areas where symmetric ILPs arise range from applied settings (scheduling on identical machines), to combinatorics (code construction), and to statistics (statistical designs construction). Relatively small symmetric ILPs are extremely difficult to solve using branch-and-cut codes oblivious to the symmetry in the problem. This paper reviews techniques developed to take advantage of the symmetry in an ILP during its solution. It also surveys related topics, such as symmetry detection, polyhedral studies of symmetric ILPs, and enumeration of all non isomorphic optimal solutions.

An Approximate Dynamic Programming Approach to Benchmark Practice-Based Heuristics for Natural Gas Storage Valuation

The valuation of the real option to store natural gas is a practically important problem that entails dynamic optimization of inventory trading decisions with capacity constraints in the face of uncertain natural gas price dynamics. Stochastic dynamic programming is a natural approach to this valuation problem, but it does not seem to be widely used in practice because it is at odds with the high-dimensional natural gas price evolution models that are widespread among traders. According to the practice-based literature, practitioners typically value natural gas storage heuristically. The effectiveness of the heuristics discussed in this literature is currently unknown because good upper bounds on the value of storage are not available. We develop a novel and tractable approximate dynamic programming method that, coupled with Monte Carlo simulation, computes lower and upper bounds on the value of storage, which we use to benchmark these heuristics on a set of realistic instances. We find that these heuristics are extremely fast to execute but significantly suboptimal compared to our upper bound, which appears to be fairly tight and much tighter than a simpler perfect information upper bound; computing our lower bound takes more time than using these heuristics, but our lower bound substantially outperforms them in terms of valuation. Moreover, with periodic reoptimizations embedded in Monte Carlo simulation, the practice-based heuristics become nearly optimal, with one exception, at the expense of higher computational effort. Our lower bound with reoptimization is also nearly optimal, but exhibits a higher computational requirement than these heuristics. Besides natural gas storage, our results are potentially relevant for the valuation of the real option to store other commodities, such as metals, oil, and petroleum products.

Pruning by Isomorphism in Branch-and-Cut

Abstract. The paper presents a branch-and-cut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non-standard, cutting integer feasible solutions but leaving the optimal value of the problem unchanged. Pruning and cut generation are performed by backtracking procedures using a Schreier-Sims table for representing the group. Applications to hard set covering problems and to the generation of covering designs and error correcting codes are presented.

Exploiting orbits in symmetric ILP

Abstract.This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given.

Reoptimization Approaches for the Vehicle-Routing Problem with Stochastic Demands

We consider the vehicle-routing problem with stochastic demands (VRPSD) under reoptimization. We develop and analyze a finite-horizon Markov decision process (MDP) formulation for the single-vehicle case and establish a partial characterization of the optimal policy. We also propose a heuristic solution methodology for our MDP, named partial reoptimization, based on the idea of restricting attention to a subset of all the possible states and computing an optimal policy on this restricted set of states. We discuss two families of computationally efficient partial reoptimization heuristics and illustrate their performance on a set of instances with up to and including 100 customers. Comparisons with an existing heuristic from the literature and a lower bound computed with complete knowledge of customer demands show that our best partial reoptimization heuristics outperform this heuristic and are on average no more than 10%--13% away from this lower bound, depending on the type of instances.

A Feasibility Pump for mixed integer nonlinear programs

We present an algorithm for finding a feasible solution to a convex mixed integer nonlinear program. This algorithm, called Feasibility Pump, alternates between solving nonlinear programs and mixed integer linear programs. We also discuss how the algorithm can be iterated so as to improve the first solution it finds, as well as its integration within an outer approximation scheme. We report computational results.

On the facets of mixed integer programs with two integer variables and two constraints

In this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables, nonnegative continuous variables and two equality constraints, and we give a complete characterization of its facets. We also derive an analogous characterization of the facets of the underlying finite integer program.

Octane: A New Heuristic for Pure 0-1 Programs

We propose a new heuristic for pure 0--1 programs, which finds feasible integer points by enumerating extended facets of the octahedron, the outer polar of the unit hypercube. We give efficient algorithms to carry out the enumeration, and we explain how our heuristic can be embedded in a branch-and-cut framework. Finally, we present computational results on a set of pure 0--1 programs taken from MIPLIB and other sources.