Block-angular Linear Programming Research Articles

Shipping optimization systems (SOS) for tramp: stochastic cargo soft time windows

Cargo time windows have been discussed in the shipping literature in several ways and within several contexts. One way considers the hard time windows, where a fixed date is assigned to both the open and close laycan for the cargo loading and discharging ports. According to this way, the cargo is rejected if the ship cannot meet these time windows. Another way develops soft time windows, where the ship for an additional cost or a penalty may violate the cargo laycan. The context in the previous ways includes a tramp deterministic model. In this paper, the soft time windows are considered within a context of a tramp stochastic model, where both the cargo transport demand and the laycan dates are random variables. The objective is a gross-profit-per-day accompanied with realistic shipping elements embedded into decision support systems known as shipping optimization systems (SOS). The chance-constraint programming and the modified version of the Dantzig–Wolfe decomposition principle known as block-angular linear ratio programming solve the problem. A case study demonstrates that using stochastic soft time windows with stochastic cargo transport demand has considerably improved the gross profit-per-day better than trying to arrive on time by varying ship speed or being within a non-stochastic shipping context.

Decentralized decomposition algorithms for peer-to-peer linear optimization

We propose Decentralized Benders Decomposition and Decentralized Dantzig–Wolfe Decomposition algorithms for large-scale block angular linear programming problems. Our methods allow multiple peer decision makers to cooperate with the aim of solving the problem without the need of a central coordination mechanism. Instead we achieve cooperation by partial information sharing across a strongly connected communication network. Our main goal is to design decentralized solution approaches for decision makers who are unwilling to disclose their local data, but want to solve the global problem collaboratively for mutual benefit. We prove that our proposed methods reach global optimality in a finite number of iterations. We confirm our theoretical results with computational experiments.

Parallel Implementation of a Central Decomposition Method for Solving Large-Scale Planning Problems

We use a decomposition approach to solve three types of realistic problems: block-angular linear programs arising in energy planning, Markov decision problems arising in production planning and multicommodity network problems arising in capacity planning for survivable telecommunication networks. Decomposition is an algorithmic device that breaks down computations into several independent subproblems. It is thus ideally suited to parallel implementation. To achieve robustness and greater reliability in the performance of the decomposition algorithm, we use the Analytic Center Cutting Plane Method (ACCPM) to handle the master program. We run the algorithm on two different parallel computing platforms: a network of PC's running under Linux and a genuine parallel machine, the IBM SP2. The approach is well adapted for this coarse grain parallelism and the results display good speed-up's for the classes of problems we have treated.

An interactive fuzzy satisficing method for multiobjective block angular linear programming problems with fuzzy parameters

In this paper, by considering the experts’ imprecise or fuzzy understanding of the nature of the parameters in the problem-formulation process, large-scale multiobjective block-angular linear programming problems involving fuzzy parameters characterized by fuzzy numbers are formulated. Using the α-level sets of fuzzy numbers, the corresponding nonfuzzy α-programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of an extended Pareto optimality concept, if the decision maker specifies the degree α and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig–Wolfe decomposition method is applicable. Then a linear programming-based interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented along with an illustrative numerical example.

Interactive decision making for large-scale multiobjective linear programs with fuzzy numbers

In this paper, by considering the experts' imprecise or fuzzy understanding of the nature of the parameters in the problem-formulation process, large-scale multiobjective block-angular linear programming problems involving fuzzy numbers are formulated. Through the use of the α-level sets of fuzzy numbers, an extended Pareto optimality concept, called the α-Pareto optimality is introduced. To generate a candidate for the satisficing solution which is also a-Pareto optimal, decision maker is asked to specify the degree α and the reference objective values. It is shown that the corresponding α-Pareto optimal solution can be easily obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method is applicable. Then a linear programming-based interactive decision-making method for deriving a satisficing solution for the decision maker efficiently from an α-Pareto optimal solution set is presented.

Generalized proximal point algorithm for convex optimization

Ha (Ref. 1) recently introduced a generalized proximal point algorithm for solving a generalized equation. In this note, we present a generalized proximal point algorithm for convex optimization problems based on Ha's work. The idea behind this algorithm is that, instead of adding a quadratic term to all the variables, we add a quadratic term to a subset of the variables. We extend the criteria for approximate solutions given by Rockafellar (Ref. 2) and Auslender (Ref. 3) and present convergence results. Finally, we show how this algorithm can be applied to solve block-angular linear and quadratic programming problems.

Fuzzy Programming for Large-Scale Multiobjective Linear Programming Problems with Fuzzy Parameters

In this paper, a large-scale multiobjective block-angular linear programming problem involving fuzzy parameters is formulated by considering the experts' vague understanding of the nature of the parameters in the problem-formulation process. Using the α-level sets of fuzzy numbers, the corresponding nonfuzzy α-programming problem is introduced and the fuzzy goals of the decision maker are quantified by eliciting the membership functions. Through the introduction of an extended Pareto optimality concept, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method is applicable. Then a linear programming-based interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker from an extended Pareto optimal solution set is presented.

Bundle-based decomposition for large-scale convex optimization: Error estimate and application to block-angular linear programs

Robinson has proposed the bundle-based decomposition algorithm to solve a class of structured large-scale convex optimization problems. In this method, the original problem is transformed (by dualization) to an unconstrained nonsmooth concave optimization problem which is in turn solved by using a modified bundle method. In this paper, we give a posteriori error estimates on the approximate primal optimal solution and on the duality gap. We describe implementation and present computational experience with a special case of this class of problems, namely, block-angular linear programming problems. We observe that the method is efficient in obtaining the approximate optimal solution and compares favorably with MINOS and an advanced implementation of the Dantzig—Wolfe decomposition method.

Load balancing in the parallel optimization of block-angular linear programs

Decomposition algorithms for block-angular linear programs give rise to a natural, coarse-grained parallelism that can be exploited by processing the subproblems concurrently within a distributed-memory environment. The parallel efficiency of the distributed approach, however, is critically dependent on the duration of the inherently serial master phase relative to that of the bottleneck subproblem. This paper investigates strategies for improving efficiency in distributed Dantzig—Wolfe decomposition by better balancing the load between the master and subproblem processors. We report computational experience on an Intel iPSC/2 hypercube multiprocessor with test problems having dimensions up to about 30 000 rows, 87 000 columns, and 200 coupling constraints.

Parallel solution of large-scale, block-angular linear programs

Many important large-scale optimization problems can be formulated as linear programs with a block-angular structure. This structure lends itself naturally to parallel solutions and is used to great advantage in the solution method described. To demonstrate the efficiency of the method, it has been implemented and computationally tested on both a shared-memory vector multiprocessor (CRAY-2) and a local-memory hypercube (NCUBE/seven) with 64 processors. Computational results for problems with as many as 24,000 rows and 74,000 columns (1,024 blocks and 1.4 million nonzero elements) are presented. A problem of this size was solved on the NCUBE in less than four minutes and the CRAY-2 in 37 seconds.

Parallel bundle-based decomposition for large-scale structured mathematical programming problems

In this paper, we present parallel bundle-based decomposition algorithms to solve a class of structured large-scale convex optimization problems. An example in this class of problems is the block-angular linear programming problem. By dualizing, we transform the original problem to an unconstrained nonsmooth concave optimization problem which is in turn solved by using a modified bundle method. Further, this dual problem consists of a collection of smaller independent subproblems which give rise to the parallel algorithms. We discuss the implementation on the CRYSTAL multi-computer. Finally, we present computational experience with block-angular linear programming problems and observe that more than 70% efficiency can be obtained using up to eleven processors for one group of test problems, and more than 60% efficiency can be obtained for relatively smaller problems using up to five processors for another group of problems.

On the resemblance between the Kornai-Liptak and cross decomposition techniques for block-angular linear programs

In this paper the resemblance is demonstrated between the master- and subproblems generated by the Kornai-Liptak algorithm and the subproblems obtained by using the cross decomposition method on linear optimization problems with block-angular structure. The significance of the similarity between these two algorithms becomes apparent considering the main disadvantage attributed to cross decomposition. In cross decomposition a master problem has to be solved from time to time since the subproblems alone do not always give a converging sequence of primal and dual solutions. But if the cross decomposition algorithm is modified in such a way that the successive primal and dual subproblem solutions are taken into consideration with equal weights, this results in the Kornai-Liptak algorithm for which convergence is guaranteed.

Decomposition of linear programs using parallel computation

This paper describes DECOMPAR: an implementation of the Dantzig-Wolfe decomposition algorithm for block-angular linear programs using parallel processing of the subproblems. The software is based on a robust experimental code for LP decomposition and runs on the CRYSTAL multicomputer at the University of Wisconsin-Madison. Initial computational experience is reported. Promising directions in future development of this approach are discussed.

Horizontal mixed decomposition

In this contribution a new decomposition approach for block-angular linear programming problems is presented. Historically, decomposition methods are either price or resource directive. The present paper integrates the two in the sense that part of the common constraints are coordinated by prices, while, at the same time, the other part of the common constraints are coordinated by direct allocations. We find increasing lower and decreasing upper bounds for the optimal value and globally feasible solutions with improving value can easily be obtained. The resulting algorithm has an appealing economic interpretation in terms of mixed price-budget oriented planning in a two-level organization.

Trade and development in the world economy: Methodological features of Project DYNAMICO

The paper reports on the methodological features and early application of the model underlying the DYNAMICO Projects, which has been developed at the United Nations Secretariat to study the interaction between trade and development. In order to investigate alternative development strategies, scenarios are calculated year by year for the global economy and the ten regions covered in the model. Each year is represented by a block-angular linear programming problem, where the systemwide (or coupling) constraints represent the world market clearance conditions for nine tradable commodities. Each subblock of relations represents the economy of a given region. The Dantzig-Wolfe decomposition method is used to solve the system. Section 2 reviews the specifications of a typical regional subproblem. Each regional subproblem contains the following material balances; labor, land, and capital requirements; investment functions; the most important macroeconomic definitions; and several other restrictions to simulate policy constraints concerning both domestic and external economic activity. Decomposition methods may, in general, be viewed as resource-allocation procedures valuable to capture some of the essential features of decentralized decision-making. In the literature, two classes of such resource-allocation procedures have been proposed: primal (relying on quantity signals), and dual (relying on prices). Section 3 discusses some of the theory underlying the Dantzig-Wolfe method, which is a price-directive one. Section 4 describes in detail the particular solution method implemented in the model. At first, in a prelink phase, each regional subproblem is solved a number of times on the basis of

A dual decomposition method for block-diagonal linear programs

In this paper, a decomposition algorithm for solving block-angular linear programs which is based on the dual simplex method will be presented. It can be considered to be a dualisation ofMuller-Merbach's direct decomposition method orKaul's extension of the generalized upper bounding technique for linear programs combined with the relaxation principle formulated byGeoffrion and applied to block-angular programs byRosen.