Pierre Bonami

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.

ALGORITHMS AND SOFTWARE FOR CONVEX MIXED INTEGER NONLINEAR PROGRAMS

This paper provides a survey of recent progress and software for solving convex Mixed Integer Nonlinear Programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in recent years. By exploiting analogies to well-known techniques for solving Mixed Integer Linear Programs and incorporating these techniques into software, significant improvements have been made in the ability to solve these problems.

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 relative strength of split, triangle and quadrilateral cuts

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.

Projected Chvátal–Gomory cuts for mixed integer linear programs

Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure integer linear program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal–Gomory (CG) separation problem as a mixed integer linear program (MILP) which is then solved by a general- purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory mixed integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal–Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, which can typically be solved in a reasonable amount of computing time.

Multiphase Mixed-Integer Optimal Control Approach to Aircraft Trajectory Optimization

In this paper, an approach to aircraft trajectory optimization is presented in which integer and continuous variables are considered. Integer variables model decision-making processes, and continuous variables describe the state of the aircraft, which evolves according to differential-algebraic equations. The problem is formulated as a multiphase mixed-integer optimal control problem. It is transcribed into a mixed-integer nonlinear programming problem by applying a fifth degree Gauss–Lobatto direct collocation method and is then solved using a nonlinear-programming-based branch-and-bound algorithm. The approach is applied to the following en route flight planning problem: Given an aircraft point mass model, a wind forecast, an airspace structure, and the relevant flying information regions with their associated overflying costs, find the control inputs that steer the aircraft from the initial fix to the final fix, following a route of waypoints while minimizing the fuel consumption and overflying costs d...

Heuristics for convex mixed integer nonlinear programs

In this paper, we describe the implementation of some heuristics for convex mixed integer nonlinear programs. The work focuses on three families of heuristics that have been successfully used for mixed integer linear programs: diving heuristics, the Feasibility Pump, and Relaxation Induced Neighborhood Search (RINS). We show how these heuristics can be adapted in the context of mixed integer nonlinear programming. We present results from computational experiments on a set of instances that show how the heuristics implemented help finding feasible solutions faster than the traditional branch-and-bound algorithm and how they help in reducing the total solution time of the branch-and-bound algorithm.

Experiments with Two-Row Cuts from Degenerate Tableaux

There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau. One can construct examples of integer programs for which a single cutting plane generated from two rows dominates the entire split closure. Motivated by these theoretical results, we study the effect of adding a family of cutting planes generated from two rows on a set of instances from the MIPLIB library. The conclusion of whether these cuts are competitive with Gomory mixed-integer cuts is very sensitive to the experimental setup. In particular, we consider the issue of reliability versus aggressiveness of the cut generators, an issue that is usually not addressed in the literature.

Disjunctive cuts for non-convex mixed integer quadratically constrained programs

This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non-convexities: integer variables and nonconvex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities by using the equation Y = xxT. We use the concave constraint 0 ≥ Y - xxT to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y - xxT with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint Y - xxT ≥ 0 to derive convex quadratic cuts and combine both approaches in a cutting plane algorithm. We present preliminary computational results to illustrate our findings.

Lift-and-project cuts for mixed integer convex programs

This paper addresses the problem of generating cuts for mixed integer nonlinear programs where the objective is linear and the relations between the decision variables are described by convex functions defining a convex feasible region. We propose a new method for strengthening the continuous relaxations of such problems using cutting planes. Our method can be seen as a practical implementation of the lift-and-project technique in the nonlinear case. To derive each cut we use a combination of a nonlinear programming subproblem and a linear outer approximation. One of the main features of the approach is that the subproblems solved to generate cuts are typically not more complicated than the original continuous relaxation. In particular they do not require the introduction of additional variables or nonlinearities. We propose several strategies for using the technique and present preliminary computational evidence of its practical interest. In particular, the cuts allow us to improve over the state of the art branch-and-bound of the solver Bonmin, solving more problems in faster computing times on average.