Accelerating Benders stochastic decomposition for the optimization under uncertainty of the petroleum product supply chain

This paper addresses the solution of a two-stage stochastic programming model for an investment planning problem applied to the petroleum products supply chain. In this context, we present the development of acceleration techniques for the stochastic Benders decomposition that aim to strengthen the cuts generated, as well as to improve the quality of the solutions obtained during the execution of the algorithm. Computational experiments are presented for assessing the efficiency of the proposed framework. We compare the performance of the proposed algorithm with two other acceleration techniques. Results suggest that the proposed approach is able to efficiently solve the problem under consideration, achieving better performance in terms of computational times when compared to other two techniques.     

The parallel QZ decomposition method for solving linear equations

The QR orthogonal decomposition method is well established for solving linear systems and both sequential and parallel implementations have been proposed. In this paper, the QZ orthogonal decomposition method is proposed in which 2 elements are eliminated simultaneously. Consequently, the theoretical analyses presented show that the QZ method is faster than the QR method which is confirmed by the numerical results.     

Reduced gradient decomposition in multistage linear programming

It is shown that multistage linear programming problems without final constraints allow an ultimate decomposition based on reduced gradient computation. The decomposition results in one-stage linear programming problems of small dimensionality without the necessity of coordination involving master problems and subproblems.     

Benders decomposition with integer subproblem

The application of Benders decomposition method to a problem might result in a subproblem including integer variables. In this case, it is not able to apply the classical Benders algorithm. In this study we present a Branch-and-Cut algorithm, which introduces the notion of “Local Cuts” as well as “Global Cuts”. The integrality constraints of the subproblem are relaxed and the relaxed problem is solved in a branch-and-bound framework, where in each node, the Benders algorithm is applied between the master problem and the relaxed subproblem. Benders cuts generated in a node of the branch-and bound tree are proved to be valid for all its descendants, but they are not necessarily valid for the non-descendant nodes. These cuts, referred to as local cuts, can be used to warm start the master problem of each descendant node, thus leading to better initial bounds. Furthermore, a novel way is presented for defining the local cuts in a general form. This general form is in fact a function of the subproblems’ variables and enables us to reuse the generated (local) cuts in the whole tree by updating some values of the function. The performance of the proposed algorithm is tested on the classical Capacitated Fixed Charge Multiple Knapsack Problem (CFCMKP).     

Bilevel stochastic linear programming problems with quantile criterion

We propose a setting for a bilevel stochastic linear programming problem with quantile criterion. We study continuity properties of the criterial function and prove the existence theorem for a solution. We propose a deterministic equivalent of the problem for the case of a scalar random parameter. We show an equivalent problem in the form of a two-stage stochastic programming problem with equilibrium constraints and quantile criterion. For the case of a discrete distribution of random parameters, the problem reduces to a mixed linear programming problem. We show results of numerical experiments.     

A Generate-Test-Aggregate parallel programming library for systematic parallel programming

The Generate-Test-Aggregate (GTA for short) algorithm is modeled following a simple and straightforward programming pattern, for combinatorial problems. First, generate all candidates; second, test and filter out invalid ones; finally, aggregate valid ones to make the final result. These three processing steps can be specified by three building blocks namely, generator, tester, and aggregator. Despite the simplicity of algorithm design, implementing the GTA algorithm naively following the three processing steps, i.e., brute-force, will result in an exponential-cost computation, and thus it is impractical for processing large data. The theory of GTA illustrates that if the definitions of generator, tester, and aggregator satisfy certain conditions, an efficient (usually near-linear cost) MapReduce program can be automatically derived from the GTA algorithm.     

On stochastic linear programming problems with the quantile criterion

We study qualitative properties of a stochastic linear programming problem with quantile criterion for a wide class of distributions. We show convexity conditions for the criterion function with respect to the strategy, and continuity conditions with respect to the strategy and reliability level. We give sufficient conditions for the existence of a solution. We present a new algorithm for finding a guaranteeing solution of the problem, i.e., an admissible solution for which the quantile criterion function's value turns out to be close to optimal.     

Solving dynamic stochastic economic models by mathematical programming decomposition methods

Discrete-time optimal control problems arise naturally in many economic problems. Despite the rapid growth in computing power and new developments in the literature, many economic problems are still quite challenging to solve. Economists are aware of the limitations of some of these approaches for solving these problems due to memory and computational requirements. However, many of the economic models present some special structure that can be exploited in an efficient manner. This paper introduces a decomposition methodology, based on a mathematical programming framework, to compute the equilibrium path in dynamic models by breaking the problem into a set of smaller independent subproblems. We study the performance of the method solving a set of dynamic stochastic economic models. The numerical results reveal that the proposed methodology is efficient in terms of computing time and accuracy.     

Scalable image decomposition

Neural Computing and Applications - Image decomposition, which separates a given input image into structure and texture images, has been used for various applications in the fields of computer...     

An effective heuristic for multistage linear programming with a stochastic right-hand side

The multistage Stochastic Linear Programming (SLP) problem may become numerically intractable for huge instances, in which case one can solve an approximation for example the well known multistage Expected Value (EV) problem. We introduce a new approximation to the SLP problem that we call the multistage Event Linear Programming (ELP) problem. To obtain this approximation the SLP constraints are aggregated by means of the conditional expectation operator. Based on this new problem we derive the ELP heuristic that produces a lower and an upper bound for the SLP problem. We have assessed the validity of the ELP heuristic by solving large scale instances of the network revenue management problem, where the new approach has clearly outperformed the EV approach. One limitation of this paper is that it only considers randomness on the right-hand side, which is assumed to be discrete and stagewise independent.     

A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system

This paper describes a Benders decomposition-based framework for solving the large scale energy management problem that was posed for the ROADEF 2010 challenge. The problem was taken from the power industry and entailed scheduling the outage dates for a set of nuclear power plants, which need to be regularly taken down for refueling and maintenance, in such a way that the expected cost of meeting the power demand in a number of potential scenarios is minimized. We show that the problem structure naturally lends itself to Benders decomposition; however, not all constraints can be included in the mixed integer programming model. We present a two phase approach that first uses Benders decomposition to solve the linear programming relaxation of a relaxed version of the problem. In the second phase, integer solutions are enumerated and a procedure is applied to make them satisfy constraints not included in the relaxed problem. To cope with the size of the formulations arising in our approach we describe efficient preprocessing techniques to reduce the problem size and show how aggregation can be applied to each of the subproblems. Computational results on the test instances show that the procedure competes well on small instances of the problem, but runs into difficulty on larger ones. Unlike heuristic approaches, however, this methodology can be used to provide lower bounds on solution quality.     

Benders decomposition for the uncapacitated multiple allocation hub location problem

In telecommunication and transportation systems, the uncapacitated multiple allocation hub location problem (UMAHLP) arises when we must flow commodities or information between several origin–destination pairs. Instead of establishing a direct node to node connection from an origin to its destination, the flows are concentrated with others at facilities called hubs. These flows are transported on links established between hubs, being then splitted and delivered to its final destination. Systems with this sort of topology are named hub-and-spoke (HS) systems or hub-and-spoke networks. They are designed to exploit the scale economies attainable through the shared use of high capacity links between hubs. Therefore, the problem is to find the least expensive HS network, selecting hubs and assigning traffic to them, given the demands between each origin–destination pair and the respective transportation costs. In the present paper, we present efficient Benders decomposition algorithms based on a well known formulation to tackle the UMAHLP. We have been able to solve some large instances, considered ‘out of reach’ of other exact methods in reasonable time.     

A two-stage stochastic programming model for the parallel machine scheduling problem with machine capacity

This paper proposes a two-stage stochastic programming model for the parallel machine scheduling problem where the objective is to determine the machines' capacities that maximize the expected net profit of on-time jobs when the due dates are uncertain. The stochastic model decomposes the problem into two stages: The first (FS) determines the optimal capacities of the machines whereas the second (SS) computes an estimate of the expected profit of the on-time jobs for given machines' capacities. For a given sample of due dates, SS reduces to the deterministic parallel weighted number of on-time jobs problem which can be solved using the efficient branch and bound of M’Hallah and Bulfin [16]. FS is tackled using a sample average approximation (SAA) sampling approach which iteratively solves the problem for a number of random samples of due dates. SAA converges to the optimum in the expected sense as the sample size increases. In this implementation, SAA applies a ranking and selection procedure to obtain a good estimate of the expected profit with a reduced number of random samples. Extensive computational experiments show the efficacy of the stochastic model.     

A Benders decomposition based framework for solving cable trench problems

In this work, we present an algorithmic framework based on Benders decomposition for the Capacitated p-Cable Trench Problem with Covering. We show that our approach can be applied to most variants of the Cable Trench Problem (CTP) that have been considered in the literature. The proposed algorithm is augmented with a stabilization procedure to accelerate the convergence of the cut loop and with a primal heuristic to derive high-quality primal solutions. Three different variants of the CTP are considered in a computational study which compares the Benders approach with two compact integer linear programming formulations that are solved with CPLEX. The obtained results show that the proposed algorithm significantly outperforms the two compact models and that it can be used to tackle significantly larger instances than previously considered algorithms based on Lagrangean relaxation.