On the Simplex Method using an Artificial Basis

The use of an artificial basis for the simplex method was suggested in an early paper by Dantzig. The idea is based on an observation that certain bases, which differ only in a relatively few columns from the true basis, may be easily inverted. Such artificial bases can then be exploited when carrying out simplex iterations. This idea was originally suggested for solving structured linear programming problems, and several approaches, such as Beale's method of pseudo-basic variables, have indeed been presented in the literature. In this paper, we shall not consider the structure explicitly; rather its exploitation in our case is expected to result directly from the choice of an artificial basis. We shall consider this basis to remain unchanged over a number of simplex iterations. In particular, this basis may be chosen as the true basis which has been most recently reinverted. In such a case our approach yields an interpretation for a basis representation recently proposed by Bisschop and Meeraus who point out very favorable properties regarding the build-up of nonzero elements in the basis representation. Our approach utilizes an auxiliary basis, which is small relative to the true basis, and whose dimension may change from one iteration to another. We shall finally develop an updating scheme for a product form representation of the inverse of such an auxiliary basis

Large-Scale Convex Optimization via Saddle Point Computation

This article proposes large-scale convex optimization problems to be solved via saddle points of the standard Lagrangian. A recent approach for saddle point computation is specialized, by way of a specific perturbation technique and unique scaling method, to convex optimization problems with differentiable objective and constraint functions. In each iteration the update directions for primal and dual variables are determined by gradients of the Lagrangian. These gradients are evaluated at perturbed points which are generated from current points via auxiliary mappings. The resulting algorithm suits massively parallel computing. Sparsity can be exploited efficiently. Employing simulation of parallel computations, an experimental code embedded into GAMS is tested on two sets of nonlinear problems. The first set arises from multi-stage stochastic optimization of the US energy economy. The second set consists of multi-currency bond portfolio problems. In such stochastic optimization problems the serial time appears approximatively proportional to the number of scenarios, while the parallel time seems independent of the number of scenarios. Thus, we observe that the serial time of our approach in comparison with Minos increases slower with the problem size. Consequently, for large problems with reasonable precision requirements, our method appears faster than Minos even in a serial computer

Perturbation Methods for Saddle Point Computation

A general class of iterative methods for saddle point seeking is developed. The directions used are subgradients evaluated at perturbed points. Convergence of the methods is proved and alternative strategies for implementation are discussed. The procedure suggests scalable algorithms for solving large-scale linear programs via saddle points. For illustration, some encouraging tests with the standard Lagrangian of linear programs from the Netlib library are reported

Parallel Solution of Linear Programs Via Nash Equilibria

The linear programming problem is shown to be equivalent to a game in which primal players minimize the augmented Lagrangian function for the primal problem and dual players maximize the augmented Lagrangian function for the dual problem. Based on that, a parallel solution method is developed in which processors carry out under-relaxed Jacobi steps for the players. Strong convergence of the method is proved and the ratio of linear convergence estimated. Computational results are highly encouraging

Steady State Analysis of the Finnish Forest Sector

During the recent years the total cost of round wood for the Finnish forest industry has been in the order of US$1.5 billion annually. The share of stumpage price represents roughly one half whereas harvesting, transportation etc. account for the rest. The purpose of this study is to investigate long term equilibrium prices for wood (and thereby total round wood costs) under various conditions of world market of wood products. In the first part of this paper a (discrete time) dynamic linear model for the forest sector is discussed. The steady state version of it is analyzed in more detail. An application of the steady state forestry model is carried out for the Finnish forests. As a result, alternative sustained yield solutions for the Finnish forests are obtained. In the analysis of the second part, a steady state sectoral model is adopted to carry out a Stackelberg equilibrium analysis for the round wood market. Further elaboration appeared necessary until the steady state model became suitable for this game theoretic analysis. This elaboration involves definitions of objective functions of the key parties (the forestry and the industry) in the forest sector game. A demand function of constant price elasticity is adopted for wood products

A Mathematical Programming Approach to Land Allocation in Regional Planning

This paper deals with the land allocation problem of finding a good locational pattern over time for various activities (such as different types of industries, agriculture, housing, `and recreation) within a region. A mathematical programming model is formulated to support long-range regional development studies at IIASA concerning the Malmo area (Sweden) and the Silistra region (Bulgaria). Estimates for the total volume of different activities within the region is assumed to be available (e.g., as econometric forecasts or in the framework of central planning). The problem is then to determine subregional development plans in order to meet the estimated volume for the activities, taking into account the initial situation as well as land available in the subregions. As criteria for evaluating alternative development paths we consider investment and operating costs, transportation and other communication costs, as well as some environmental aspects. While determining the investment and operating costs, economies of scale play an important role for certain activities. Formally, our model is a dynamic multicriteria optimization problem with integer variables and quadratic objective functions (which may be neither convex nor concave). A solution technique is proposed for this problem. The method, which relies heavily on the network flow structure of the set of constraints, is illustrated using a numerical example. Finally, the implementation of a plan is briefly discussed

A Preliminary Model of Production, Consumption and International Trade in Forest Products

The preliminary version of IIASA's model of production, consumption, and trade in forest products is discussed. The model involves six regions (Northern Europe, Western Europe, USA, Canada, Japan, and the rest of the World) and nine products (sawlogs, pulpwood, sawnwood, panels, pulp, newsprint, other printing and writing paper, packaging paper and board, and recycled paper). The primary purpose of this version is to analyze sensitivity of the model with respect to various parameters and to learn to understand the model behavior in general. We first review the formulation of the spatial equilibrium model, and then discuss the equilibrium conditions in detail. Subsequent sections describe the preliminary data and two sets of numerical demonstrations. The first set deals with the situation in 1980. A base scenario for model validation is discussed first. Stability of the model with respect to price elasticity parameters for demand is shown. The effect of a change in currency exchange rate is demonstrated. The second set of runs simulates the situation around the year 2000. For the base scenario, we update wood resources, production capacity, and demand for each region. This scenario is compared with high- and zero-growth demand scenarios. The effect of tariffs and changes in timber supply are demonstrated. Finally, we wish to stress that this preliminary model employs very tentative data. Therefore, one should draw conclusions from the demonstration scenarios with special care

Experiments with the Reduced Gradient Method for Linear Programming

This article deals with some methods for linear programming which generate a monotonically improving sequence of feasible solutions. Examples of such methods are the simplex method and the reduced gradient method. A larger class of such methods as well as their convergence has been discussed in a recent article by Kallio and Porteus. We have implemented a version of such methods in the SESAME system developed by Orchard-Hays. This version resembles the reduced gradient method except that only a subset of nonbasic variables to be changed is considered at each iteration. We shall try out several modifications of this basic version. These modifications are concerned with the choice of an initial basis and an initial solution, with strategies for finding a feasible solution, as well as with strategies for determining the direction of change for a feasible solution at each iteration. We have experimented with moderate sized nonstructured as well as dynamic problems. Compared with the simplex method, the overall performance of such methods is about equal in the case of linear programs with no particular structure. For dynamic LP we have obtained some encouraging results. Although we have been able to experiment with only a few problems, so far it seems that using specially defined starting basis and initial nonbasic solution allow a reduction by a factor of eight in the computing time of the reduced gradient method. This starting basis is chosen so that its columns are likely to appear also in an optimal basis. For the initial solution, available information, such as current level of activities in real life, may be employed. Of course, our starting basis for dynamic LP may be used also in the simplex method, and, indeed this results in a considerable improvement of efficiency

Bibliography of Soviet and Western European Publications on Large-Scale Linear Programming

This bibliography originates from the Workshop on Large-Scale Linear Programming held at IIASA over the period June 2-6 1980. The Proceedings of this Workshop (edited by G.B. Dantzig, M.A.H. Dempster and M. Kallio, IIASA CP-81-S1) contains a bibliography covering North American and Western European publications. This bibliography is a supplement which covers Eastern European work on large-scale linear programming. As some important work may still be missing, further contributions (with English translations of titles published in other languages) are most welcome. Such contributions should be sent to H. Gasking at IIASA, where a computerized bibliography is maintained. Revised versions will be issued from time to time