A Lagrangean based branch-and-cut algorithm for global optimization of nonconvex mixed-integer nonlinear programs with decomposable structures

In this work we present a global optimization algorithm for solving a class of large-scale nonconvex optimization models that have a decomposable structure. Such models, which are very expensive to solve to global optimality, are frequently encountered in two-stage stochastic programming problems, engineering design, and also in planning and scheduling. A generic formulation and reformulation of the decomposable models is given. We propose a specialized deterministic branch-and-cut algorithm to solve these models to global optimality, wherein bounds on the global optimum are obtained by solving convex relaxations of these models with certain cuts added to them in order to tighten the relaxations. These cuts are based on the solutions of the sub-problems obtained by applying Lagrangean decomposition to the original nonconvex model. Numerical examples are presented to illustrate the effectiveness of the proposed method compared to available commercial global optimization solvers that are based on branch and bound methods.     

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.     

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.     

Inequalities in completely convex stochastic programming

A two-stage stochastic programming problem in which the random variable enters in a convex manner is called completely convex. For such problems we give a sequence of inequalities and equalities showing the equivalence of optimality over plans and optimality of a two-stage procedure related to dynamic programming and giving upper bounds on the expected value of perfect information. Our assumptions are the weakest possible to guarantee the results in the completely convex case and supersede previous related results which have received erroneous proofs or have been established under highly restrictive conditions. In the course of our argument we exhibit a new measurable selection theorem and a rather general form of Jensen's inequality. We also present a multistage generalization of our central theorem.     

Solving two-stage stochastic programming problems with level decomposition

We propose a new variant of the two-stage recourse model. It can be used e.g., in managing resources in whose supply random interruptions may occur. Oil and natural gas are examples for such resources. Constraints in the resulting stochastic programming problems can be regarded as generalizations of integrated chance constraints. For the solution of such problems, we propose a new decomposition method that integrates a bundle-type convex programming method with the classic distribution approximation schemes. Feasibility and optimality issues are taken into consideration simultaneously, since we use a convex programming method suited for constrained optimization. This approach can also be applied to traditional two-stage problems whose recourse functions can be extended to the whole space in a computationally efficient way. Network recourse problems are an example for such problems. We report encouraging test results with the new method.      

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.     

Generalized mirror averaging and <Emphasis Type="Italic">D</Emphasis>-convex aggregation

We study the problem of aggregation of estimators. Given a collection of M different estimators, we construct a new estimator, called aggregate, which is nearly as good as the best linear combination over an l ₁-ball of ℝ^M of the initial estimators. The aggregate is obtained by a particular version of the mirror averaging algorithm. We show that our aggregation procedure statisfies sharp oracle inequalities under general assumptions. Then we apply these results to a new aggregation problem: D-convex aggregation. Finally we implement our procedure in a Gaussian regression model with random design and we prove its optimality in a minimax sense up to a logarithmic factor.      

Estimation of potential gains from bank mergers: A novel two-stage cost efficiency DEA model

This paper develops a novel two-stage cost efficiency model to estimate and decompose the potential gains from Mergers and Acquisitions (M&As). In this model, a hypothetical DMU is defined as a combination of two or more candidate DMUs. The hypothetical DMU would surpass the traditional Production Possibility Set (PPS). In order to solve the problem, a Merger Production Possibility Set (PPS_M) is constructed. The model minimizes the total cost of the hypothetical DMU while maintaining its outputs at the current level, and estimates the overall merger efficiency by comparing its minimal total cost with its actual cost. Moreover, the overall merger efficiency could be decomposed into technical efficiency, harmony efficiency, and scale efficiency. We show that the model can be extended to a two-stage structure and these efficiencies can be decomposed to both sub-systems. To show the usefulness of the proposed approach, we applied it to a real dataset of top 20 most competitive Chinese City Commercial Banks (CCBs). We concluded that (1) there exist considerably potential gains for the proposed merged banks. (2) It is also shown that the main impact on potential merger gains are from technical and harmony efficiency. (3) As an interesting result we found that the scale effect works against the merger, indicating that it is not favorable for a full-scale merger.     

A regularized stochastic decomposition algorithm for two-stage stochastic linear programs

In this paper a regularized stochastic decomposition algorithm with master programs of finite size is described for solving two-stage stochastic linear programming problems with recourse. In a deterministic setting cut dropping schemes in decomposition based algorithms have been used routinely. However, when only estimates of the objective function are available such schemes can only be properly justified if convergence results are not sacrificed. It is shown that almost surely every accumulation point in an identified subsequence of iterates produced by the algorithm, which includes a cut dropping scheme, is an optimal solution. The results are obtained by including a quadratic proximal term in the master program. In addition to the cut dropping scheme, other enhancements to the existing methodology are described. These include (i) a new updating rule for the retained cuts and (ii) an adaptive rule to determine when additional reestimation of the cut associated with the current solution is needed. The algorithm is tested on problems from the literature assuming both descrete and continuous random variables.A majority of this work is part of the author's Ph.D. dissertation prepared at the University of Arizona in 1990.     

Polyhedral study of the connected subgraph problem

In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.     

A simulation-based approach to two-stage stochastic programming with recourse

In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for two-stage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution. We think that the novelty of the numerical approach developed in this paper is twofold. First, various variance reduction techniques are applied in order to enhance the rate of convergence. Successful application of those techniques is what makes the whole approach numerically feasible. Second, a statistical inference is developed and applied to estimation of the error, validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brasília, Brazil, through a Doctoral Fellowship under grant 200595/93-8.     

Deviation Measures in Linear Two-Stage Stochastic Programming

We consider a linear two-stage stochastic program. Whereas optimization in the traditional setting is based solely on expectation, we include risk measures reflecting dispersions of the random objective. Presenting the mean-risk models, we aim to extend existing results for the expectation-based model. In particular, we discuss structural properties such as continuity, differentiability and convexity and address stability issues. Furthermore, we propose algorithmic treatment with a slight variation of the L-shaped method     

Two-stage stochastic variational inequalities: an ERM-solution procedure

We propose a two-stage stochastic variational inequality model to deal with random variables in variational inequalities, and formulate this model as a two-stage stochastic programming with recourse by using an expected residual minimization solution procedure. The solvability, differentiability and convexity of the two-stage stochastic programming and the convergence of its sample average approximation are established. Examples of this model are given, including the optimality conditions for stochastic programs, a Walras equilibrium problem and Wardrop flow equilibrium. We also formulate stochastic traffic assignments on arcs flow as a two-stage stochastic variational inequality based on Wardrop flow equilibrium and present numerical results of the Douglas–Rachford splitting method for the corresponding two-stage stochastic programming with recourse.     

A New Condition and Approach for Zero-Sum Stochastic Games with Average Payoffs

Abstract

This article deals with discrete-time two-person zero-sum stochastic games with Borel state and action spaces. The optimality criterion to be studied is the long-run expected average payoff criterion, and the (immediate) payoff function may have neither upper nor lower bounds. We first replace the optimality equation widely used in the previous literature with two so-called optimality inequalities, and give a new set of conditions for the existence of solutions to the optimality inequalities. Then, from the optimality inequalities we ensure the existence of a pair of average optimal stationary strategies. Our new condition is slightly weaker than those in the previous literature, and as a byproduct some interesting results such as the convergence of a value iteration scheme to the value of the discounted payoff game is obtained. Finally, we first apply the main results in this article to generalized inventory systems, and then further provide an example of controlled population processes for which all of our conditions are satisfied, while some of conditions in some of previous literature fail to hold.     

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.     

A probability metrics approach for reducing the bias of optimality gap estimators in two-stage stochastic linear programming

Monte Carlo sampling-based estimators of optimality gaps for stochastic programs are known to be biased. When bias is a prominent factor, estimates of optimality gaps tend to be large on average even for high-quality solutions. This diminishes our ability to recognize high-quality solutions. In this paper, we present a method for reducing the bias of the optimality gap estimators for two-stage stochastic linear programs with recourse via a probability metrics approach, motivated by stability results in stochastic programming. We apply this method to the Averaged Two-Replication Procedure (A2RP) by partitioning the observations in an effort to reduce bias, which can be done in polynomial time in sample size. We call the resulting procedure the Averaged Two-Replication Procedure with Bias Reduction (A2RP-B). We provide conditions under which A2RP-B produces strongly consistent point estimators and an asymptotically valid confidence interval. We illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small-sample behavior of A2RP-B on a number of two-stage stochastic linear programs from the literature. Our computational results indicate that the procedure effectively reduces bias. We also observe variance reduction in certain circumstances.     

Left-invariant Riemannian problems on the groups of proper motions of hyperbolic plane and sphere

On the Lie groups PSL₂(?) and SO₃ we consider left-invariant Riemannian metrics with two equal eigenvalues. The global optimality of geodesics is investigated. We find the parametrization of geodesics, the cut locus and the equations for the cut time. When the third eigenvalue of a metric tends to the infinity the cut locus and the cut time converge to the cut locus and the cut time of the sub-Riemannian problem.     

交通网络下的多厂商两阶段随机非合作博弈问题——基于随机变分不等式

研究集生产、运输和销售为一体的多个制造商在随机市场环境下的两阶段随机非合作博弈问题.首先,建立了该两阶段随机非合作博弈问题的模型,然后将其转化为两阶段随机变分不等式(Stochastic Variational Inequality,简称SVI).在温和的假设条件下,证明了该问题存在均衡解,并通过Progressive Hedging Method(简称PHM)进行求解.最后,通过改变模型中随机变量的分布和成本参数,分析与研究厂商的市场行为.     

Plant location with minimum inventory

We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality.     

The ring-star problem: A new integer programming formulation and a branch-and-cut algorithm

A ring star in a graph is a subgraph that can be decomposed into a cycle (or ring) and a set of edges with exactly one vertex in the cycle. In the minimum ring-star problem (mrsp) the cost of a ring star is given by the sum of the costs of its edges, which vary, depending on whether the edge is part of the ring or not. The goal is to find a ring-star spanning subgraph minimizing the sum of all ring and assignment costs. In this paper we show that the mrsp can be reduced to a minimum (constrained) Steiner arborescence problem on a layered graph. This reduction is used to introduce a new integer programming formulation for the mrsp. We prove that the dual bound generated by the linear relaxation of this formulation always dominates the one provided by an early model from the literature. Based on our new formulation, we developed a branch-and-cut algorithm for the mrsp. On the primal side, we devised a grasp heuristic to generate good upper bounds for the problem. Computational tests with these algorithms were conducted on a benchmark of public domain. In these experiments both our exact and heuristics algorithms had excellent performances, noticeably in dealing with instances whose optimal solution has few vertices in the ring. In addition, we also investigate the minimum spanning caterpillar problem (mscp) which has the same input as the mrsp and admits feasible solutions that can be viewed as ring stars with paths in the place of rings. We present an easy reduction of the mscp to the mrsp, which makes it possible to solve to optimality instances of the former problem too. Experiments carried out with the mscp revealed that our branch-and-cut algorithm is capable to solve to optimality instances with up to 200 vertices in reasonable time.