Fenchel decomposition for stochastic mixed-integer programming

This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an FD algorithm for SMIP and establish finite convergence for binary first-stage. Second, we derive FD cuts based on the second-stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first-stage variables. We then devise an alternative algorithm (FD-L algorithm) based on the lifted FD cuts. Finally, we report on computational results based on several test instances from the literature involving the special structure of knapsack problems with nonnegative left-hand side coefficients. The results are promising and show that both algorithms can outperform a standard direct solver and a disjunctive decomposition algorithm on large-scale instances. Furthermore, the FD-L algorithm provides better performance than the FD algorithm in general. Since Fenchel cuts can be computationally expensive in general and are best suited for problems with special structure, both algorithms exploit the special structure of the test instances by reducing the size of the cut generation problems based on the number of nonzero components in the non-integer solution that needs to be cut off.     

Tight bounds on indefinite separable singly-constrained quadratic programs in linear-time

This paper focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer decision variables in both stages. We develop a decomposition algorithm in which the first-stage approximation is solved by a branch-and-bound algorithm with its nodes inheriting Benders’ cuts that are valid for their ancestor nodes. In addition, we develop two closely related convexification schemes which use multi-term disjunctive cuts to obtain approximations of the second-stage mixed-integer programs. We prove that the proposed methods are finitely convergent. One of the main advantages of our decomposition scheme is that we use a Benders-based branch-and-cut approach in which linear programming approximations are strengthened sequentially. Moreover as in many decomposition schemes, these subproblems can be solved in parallel. We also illustrate these algorithms using several variants of an SMIP example from the literature, as well as a new set of test problems, which we refer to as Stochastic Server Location and Sizing. Finally, we present our computational experience with previously known examples as well as the new collection of SMIP instances. Our experiments reveal that our algorithm is able to produce provably optimal solutions (within an hour of CPU time) even in instances for which a highly reliable commercial MIP solver is unable to provide an optimal solution within an hour of CPU time.     

Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming. However, problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem. This research was supported by the NSF awards DMS-0603728 and DMI-0354678.     

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.     

IIS branch-and-cut for joint chance-constrained stochastic programs and application to optimal vaccine allocation

We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a large-scale mixed 0–1 integer program. We derive a new class of optimality cuts called IIS cuts and apply them to our problem. The cuts are based on irreducibly infeasible subsystems (IIS) of an LP defined by requiring that all scenarios be satisfied. We propose a method for improving the upper bound of the problem when no cut can be found. We derive and implement a branch-and-cut algorithm based on IIS cuts, and refer to this algorithm as the IIS branch-and-cut algorithm. We report on computational results with several test instances from optimal vaccine allocation. The computational results are promising as the IIS branch-and-cut algorithm gives better results than a state-of-the-art commercial solver on one class of problems.     

Disjunctive cuts for continuous linear bilevel programming

This work shows how disjunctive cuts can be generated for a bilevel linear programming problem (BLP) with continuous variables. First, a brief summary on disjunctive programming and bilevel programming is presented. Then duality theory is used to reformulate BLP as a disjunctive program and, from there, disjunctive programming results are applied to derive valid cuts. These cuts tighten the domain of the linear relaxation of BLP. An example is given to illustrate this idea, and a discussion follows on how these cuts may be incorporated in an algorithm for solving BLP.     

Solving a multicoloring problem with overlaps using integer programming

This paper presents a new generalization of the graph multicoloring problem. We propose a Branch-and-Cut algorithm based on a new integer programming formulation. The cuts used are valid inequalities that we could identify to the polytope associated with the model. The Branch-and-Cut system includes separation heuristics for the valid inequalities, specific initial and primal heuristics, branching and pruning rules. We report on computational experience with random instances.     

Adaptive multicut aggregation for two-stage stochastic linear programs with recourse

Outer linearization methods for two-stage stochastic linear programs with recourse, such as the L-shaped algorithm, generally apply a single optimality cut on the nonlinear objective at each major iteration, while the multicut version of the algorithm allows for several cuts to be placed at once. In general, the L-shaped algorithm tends to have more major iterations than the multicut algorithm. However, the trade-offs in terms of computational time are problem dependent. This paper investigates the computational trade-offs of adjusting the level of optimality cut aggregation from single cut to pure multicut. Specifically, an adaptive multicut algorithm that dynamically adjusts the aggregation level of the optimality cuts in the master program, is presented and tested on standard large-scale instances from the literature. Computational results reveal that a cut aggregation level that is between the single cut and the multicut can result in substantial computational savings over the single cut method.     

A comparative study of decomposition algorithms for stochastic combinatorial optimization

This paper presents comparative computational results using three decomposition algorithms on a battery of instances drawn from two different applications. In order to preserve the commonalities among the algorithms in our experiments, we have designed a testbed which is used to study instances arising in server location under uncertainty and strategic supply chain planning under uncertainty. Insights related to alternative implementation issues leading to more efficient implementations, benchmarks for serial processing, and scalability of the methods are also presented. The computational experience demonstrates the promising potential of the disjunctive decomposition (D ²) approach towards solving several large-scale problem instances from the two application areas. Furthermore, the study shows that convergence of the D ² methods for stochastic combinatorial optimization (SCO) is in fact attainable since the methods scale well with the number of scenarios.     

A so-called Cluster Benders Decomposition approach for solving two-stage stochastic linear problems

The optimization of stochastic linear problems, via scenario analysis, based on Benders decomposition requires appending feasibility and/or optimality cuts to the master problem until the iterative procedure reaches the optimal solution. The cuts are identified by solving the auxiliary submodels attached to the scenarios. In this work, we propose the algorithm named scenario Cluster Benders Decomposition (CBD) for dealing with the feasibility cut identification in the Benders method for solving large-scale two-stage stochastic linear problems. The scenario tree is decomposed into a set of scenario clusters and tighter feasibility cuts are obtained by solving the auxiliary submodel for each cluster instead of each individual scenario. Then, the scenario cluster based scheme allows to identify tighter feasibility cuts that yield feasible second stage decisions in reasonable computing time. Some computational experience is reported by using CPLEX as the solver of choice for the auxiliary LP submodels at each iteration of the algorithm CBD. The results that are reported show the favorable performance of the new approach over the traditional single scenario based Benders decomposition; it also outperforms the plain use of CPLEX for medium-large and large size instances.     

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.     

Computing with multi-row Gomory cuts

Recent advances on the understanding of valid inequalities from the infinite group relaxation has opened the possibility of finding a computationally effective extension to GMI cuts. In this paper, we investigate the computational impact of using a subclass of minimally valid inequalities from this relaxation on a wide set of instances.     

An interior-point Benders based branch-and-cut algorithm for mixed integer programs

We present an interior-point branch-and-cut algorithm for structured integer programs based on Benders decomposition and the analytic center cutting plane method (ACCPM). We show that the ACCPM based Benders cuts are both pareto-optimal and valid for any node of the branch-and-bound tree. The valid cuts are added to a pool of cuts that is used to warm-start the solution of the nodes after branching. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances.     

Sequential pairing of mixed integer inequalities

We investigate a scheme, called pairing, for generating new valid inequalities for mixed integer programs by taking pairwise combinations of existing valid inequalities. The pairing scheme essentially produces a split cut corresponding to a specific disjunction, and can also be derived through the mixed integer rounding procedure. The scheme is in general sequence-dependent and therefore leads to an exponential number of inequalities. For some important cases, we identify combination sequences that lead to a manageable set of non-dominated inequalities. We illustrate the framework for some deterministic and stochastic integer programs and we present computational results showing the efficiency of adding the new generated inequalities as cuts.     

Integrated design and operation of remnant inventory supply chains under uncertainty

We consider the simultaneous design and operation of remnant inventory supply chains. Remnant inventory is generated when demand for various lengths of a product may be satisfied by existing inventory, or by cutting a large piece into smaller pieces. We formulate our problem as a two-stage stochastic mixed-integer program. In solving our stochastic program, we enhance the standard L-shaped method in two ways. Our computational experiments demonstrate that these enhancements are effective, dramatically reducing the solution time for large instances.     

Fix and Relax Heuristic for a Stochastic Lot-Sizing Problem

This paper addresses a particular stochastic lot-sizing and scheduling problem. The evolution of the uncertain parameters is modelled by means of a scenario tree and the resulting model is a multistage stochastic mixed-integer program. We develop a heuristic approach that exploits the specific structure of the problem. The computational experiments carried out on a large set of instances have shown that the approach provides good quality solutions in a reasonable amount of time.     

B&B Frameworks for the Capacity Expansion of High Speed Telecommunication Networks Under Uncertainty

The purpose of this paper is to investigate branch and bound strategies and the comparison of branch and cut with pure branch and bound approaches on high speed telecommunication network design under uncertainty. We model the problem as a two-stage stochastic program with discrete first-stage (investment) variables. Two formulations of the problem are used. The first one with general integer investment variables and the second one, a variant of the first model, with 0-1 investment variables. We present computational results for three solution approaches: the integer L-shaped (Benders) decomposition, a branch and bound framework and a disjunctive cutting plane method. This work was supported by France Telecom.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

Threshold Boolean form for joint probabilistic constraints with random technology matrix

We develop a new modeling and solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations for the studied stochastic problem, and we derive a set of strengthening valid inequalities. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the mixed-integer linear programming formulations are orders of magnitude faster to solve than the mixed-integer nonlinear programming formulation. The method integrating the valid inequalities in a branch-and-bound algorithm has the best performance.