LaGO: a (heuristic) Branch and Cut algorithm for nonconvex MINLPs

We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixed-integer nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxation cannot be applied, and we are restricted to sampling techniques in case of nonquadratic nonconvex functions. The linear relaxation is further improved by mixed-integer-rounding cuts. Also box reduction techniques are applied to improve efficiency. Numerical results on medium size test problems are presented to show the efficiency of the method.     

Linearization-based algorithms for mixed-integer nonlinear programs with convex continuous relaxation

We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund \((\text{ LP/NLP-BB })\) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance.     

Computations with disjunctive cuts for two-stage stochastic mixed 0-1 integer programs

Two-stage stochastic mixed-integer programming (SMIP) problems with recourse are generally difficult to solve. This paper presents a first computational study of a disjunctive cutting plane method for stochastic mixed 0-1 programs that uses lift-and-project cuts based on the extensive form of the two-stage SMIP problem. An extension of the method based on where the data uncertainty appears in the problem is made, and it is shown how a valid inequality derived for one scenario can be made valid for other scenarios, potentially reducing solution time. Computational results amply demonstrate the effectiveness of disjunctive cuts in solving several large-scale problem instances from the literature. The results are compared to the computational results of disjunctive cuts based on the subproblem space of the formulation and it is shown that the two methods are equivalently effective on the test instances.     

A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming

 We establish a precise correspondence between lift-and-project cuts for mixed 0-1 programs, simple disjunctive cuts (intersection cuts) and mixed-integer Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higher-dimensional cut generating linear program (CGLP) into bases of the linear programming relaxation. It provides new bounds on the number of facets of the elementary closure, and on the rank, of the standard linear programming relaxation of the mixed 0-1 polyhedron with respect to the above families of cutting planes. Based on the above correspondence, we develop an algorithm that solves (CGLP) without explicitly constructing it, by mimicking the pivoting steps of the higher dimensional (CGLP) simplex tableau by certain pivoting steps in the lower dimensional (LP) simplex tableau. In particular, we show how to calculate the reduced costs of the big tableau from the entries of the small tableau and based on this, how to identify a pivot in the small tableau that corresponds to one or several improving pivots in the big tableau. The overall effect is a much improved lift-and-project cut generating procedure, which can also be interpreted as an algorithm for a systematic improvement of mixed integer Gomory cuts from the small tableau. Received: October 5, 2000 / Accepted: March 19, 2002 Published online: September 5, 2002 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.     

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.     

Undercover: a primal MINLP heuristic exploring a largest sub-MIP

We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover, such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP.     

On generalized surrogate duality in mixed-integer nonlinear programming

Mathematical Programming - The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global $$\epsilon $$ -optimality with spatial branch and bound is a tight,...     

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability. This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation.     

A heuristic to generate rank-1 GMI cuts

Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts.     

Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

An efficient strategy for the activation of MIP relaxations in a multicore global MINLP solver

Solving mixed-integer nonlinear programming (MINLP) problems to optimality is a NP-hard problem, for which many deterministic global optimization algorithms and solvers have been recently developed. MINLPs can be relaxed in various ways, including via mixed-integer linear programming (MIP), nonlinear programming, and linear programming. There is a tradeoff between the quality of the bounds and CPU time requirements of these relaxations. Unfortunately, these tradeoffs are problem-dependent and cannot be predicted beforehand. This paper proposes a new dynamic strategy for activating and deactivating MIP relaxations in various stages of a branch-and-bound algorithm. The primary contribution of the proposed strategy is that it does not use meta-parameters, thus avoiding parameter tuning. Additionally, this paper proposes a strategy that capitalizes on the availability of parallel MIP solver technology to exploit multicore computing hardware while solving MINLPs. Computational tests for various benchmark libraries reveal that our MIP activation strategy works efficiently in single-core and multicore environments.     

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.     

Three enhancements for optimization-based bound tightening

Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT. Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs. We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17–19 % on average. Most importantly, more instances can be solved when using OBBT.     

Parametric mixed-integer 0–1 linear programming: The general case for a single parameter

Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. Parametric MILPs are considered in which a single parameter can simultaneously influence the objective function, the right-hand side and the matrix. The first algorithm is based on branch-and-bound on the integer variables, solving a parametric linear program (LP) at each node. The second algorithm is based on the optimality range of a qualitatively invariant solution, decomposing the parametric optimization problem into a series of regular MILPs, parametric LPs and regular mixed-integer nonlinear programs (MINLPs). The number of subproblems required for a particular instance is equal to the number of critical regions. For the parametric LPs an improvement of the well-known rational simplex algorithm is presented, that requires less consecutive operations on rational functions. Also, an alternative based on predictor–corrector continuation is proposed. Numerical results for a test set are discussed.     

A computational study of the cutting plane tree algorithm for general mixed-integer linear programs

The cutting plane tree algorithm provides a finite procedure for solving general mixed-integer linear programs with bounded integer variables. The computational evidence provided in this work illustrates that this algorithm is powerful enough to close a significant fraction of the integrality gap for moderately sized MIPLIB instances.     

Branch-and-price for a class of nonconvex mixed-integer nonlinear programs

This work attempts to combine the strengths of two major technologies that have matured over the last three decades: global mixed-integer nonlinear optimization and branch-and-price. We consider a class of generally nonconvex mixed-integer nonlinear programs (MINLPs) with linear complicating constraints and integer linking variables. If the complicating constraints are removed, the problem becomes easy to solve, e.g. due to decomposable structure. Integrality of the linking variables allows us to apply a discretization approach to derive a Dantzig-Wolfe reformulation and solve the problem to global optimality using branch-andprice. It is a remarkably simple idea; but to our surprise, it has barely found any application in the literature. In this work, we show that many relevant problems directly fall or can be reformulated into this class of MINLPs. We present the branch-and-price algorithm and demonstrate its effectiveness (and sometimes ineffectiveness) in an extensive computational study considering multiple large-scale problems of practical relevance, showing that, in many cases, orders-of-magnitude reductions in solution time can be achieved.

     

On optimizing over lift-and-project closures

The strengthened lift-and-project closure of a mixed integer linear program is the polyhedron obtained by intersecting all strengthened lift-and-project cuts obtained from its initial formulation, or equivalently all mixed integer Gomory cuts read from all tableaux corresponding to feasible and infeasible bases of the LP relaxation. In this paper, we present an algorithm for approximately optimizing over the strengthened lift-and-project closure. The originality of our method is that it relies on a cut generation linear programming problem which is obtained from the original LP relaxation by only modifying the bounds on the variables and constraints. This separation LP can also be seen as dual to the cut generation LP used in disjunctive programming procedures with a particular normalization. We study properties of this separation LP, and discuss how to use it to approximately optimize over the strengthened lift-and-project closure. Finally, we present computational experiments and comparisons with recent related works.     

Lifting inequalities: a framework for generating strong cuts for nonlinear programs

In this paper, we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs. The theory is geometric and provides insights into lifting-based cut generation procedures, yielding short proofs of earlier results in mixed-integer programming. Using convex extensions, we obtain conditions that allow for sequence-independent lifting in nonlinear settings, paving a way for efficient cut-generation procedures for nonlinear programs. This sequence-independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general-purpose, strong cuts for integer programs. We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear knapsack sets. Finally, we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the problem.