Scale Linear Programming Problems Research Articles

Transmission Network Dynamic Planning Based on a Double Deep-Q Network With Deep ResNet

Based on a Double Deep-Q Network with deep ResNet (DDQN-ResNet), this paper proposes a novel method for transmission network expansion planning (TNEP). Since TNEP is a large scale and mixed-integer linear programming (MILP) problem, as the transmission network scale and the optimal constraints increase, the numerical calculation and heuristic learning-based methods suffer from heavy computational complexities in calculation and training. Besides, due to the black box characteristic, the solution processes of the heuristic learning-based methods are inexplicable and usually require repeated training. By using DDQN-ResNet, this paper constructs a high-performance and flexible method to solve large-scale and complex-constrained TNEP problem. Firstly, we form a two-objective TNEP model, in which one objective is to minimize the comprehensive cost, and another objective is to maximize the transmission network reliability. The comprehensive cost takes into account the expansion cost, the network loss cost, and the maintenance cost. The transmission network reliability is evaluated by the expected energy not served (EENS) and the electrical betweenness. Secondly, from the TNEP model, the TNEP task is constructed based on the Markov decision process. By abstracting the task, the TNEP environment is obtained for DDQN-ResNet. In addition, to identify the construction value of lines, an agent is establish based on DDQN-ResNet. Finally, we perform the static planning and visualize the reinforcement learning process. The dynamic planning is realized by reusing the training experience. The validity and flexibility of DDQN-ResNet are verified on RTS 24-bus test system.

An interior point-proximal method of multipliers for convex quadratic programming

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

Using groups in the splitting preconditioner computation for interior point methods

Interior point methods usually rely on iterative methods to solve the linear systems of large scale problems. The paper proposes a hybrid strategy using groups for the preconditioning of these iterative methods. The objective is to solve large scale linear programming problems more efficiently by a faster and robust computation of the preconditioner. In these problems, the coefficient matrix of the linear system becomes ill conditioned during the interior point iterations, causing numerical difficulties to find a solution, mainly with iterative methods. Therefore, the use of preconditioners is a mandatory requirement to achieve successful results. The paper proposes the use of a new columns ordering for the splitting preconditioner computation, exploring the sparsity of the original matrix and the concepts of groups. This new preconditioner is designed specially for the final interior point iterations; a hybrid approach with the controlled Cholesky factorization preconditioner is adopted. Case studies show that the proposed methodology reduces the computational times with the same quality of solutions when compared to previous reference approaches. Furthermore, the benefits are obtained while preserving the sparse structure of the systems. These results highlight the suitability of the proposed approach for large scale problems.

A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra's predictor-corrector method for large scale linear programming problems needs to find a ba...

Efficient GPU-based implementations of simplex type algorithms

Recent hardware advances have made it possible to solve large scale Linear Programming problems in a short amount of time. Graphical Processing Units (GPUs) have gained a lot of popularity and have been applied to linear programming algorithms. In this paper, we propose two efficient GPU-based implementations of the Revised Simplex Algorithm and a Primal–Dual Exterior Point Simplex Algorithm. Both parallel algorithms have been implemented in MATLAB using MATLAB’s Parallel Computing Toolbox. Computational results on randomly generated optimal sparse and dense linear programming problems and on a set of benchmark problems (netlib, kennington, Mészáros) are also presented. The results show that the proposed GPU implementations outperform MATLAB’s interior point method.

A Decomposition Algorithm for Solving a Three Level Large Scale linear Programming Problem

This paper presents a three level large scale linear programming problem in which the objective functions at every level are to be maximized. A three level programming problem can be thought as a static version of the Stackelberg strategy. An algorithm for solving a three planner model and a solution method for treating this problem are suggested. At each level we attempt to optimize its problem separately as a large scale programming problem using Dantzig and Wolfe decomposition method. Therefore, we handle the optimization process through a series of sub problems that can be solved independently. Finally, a numerical example is given to clarify the main results developed in this paper.

Monte Carlo application and gradient appliance for solving large scale linear programming problems: Essence and laboriousness

We propose two recently developed iterative algorithms for solving large scale linear programming problems: the first one is based on the gravitation centers method, which in its turn widely uses Monte Carlo statistical test method, and the second one is realized on the basis of the constant step gradient method. Though the developed iterative algorithms are approximate, it is rather worthwhile in most cases to apply them when solving large scale linear problems, as they are characterized by positive properties, such as simplicity of the algoritm and the program, high-speed performance, and besides the appropriate software is free of such an unpleasant, specific for the simplex method event as a loop. We present some computational results. We assess the both developed algorithms with respect to a high-speed performance criterion and compare their high-speed performance with the one of the simplex method.

USING COLUMN GENERATION TECHNIQUE TO ESTIMATE PROBABILITY STATISTICS IN TRANSITION MATRIX OF LARGE SCALE MARKOV CHAIN WITH LEAST ABSOLUTE DEVIATION CRITERIA

The Least Absolute Deviation (LAD) method is the one of many methods used to estimate transition probability matrix of Markov Chain. It can be formu lated as a Linear Programming Problem (LP) and solved using its regular state of the art software. However, when the Markov Chain has a large number of states and historical state probabilities data, the corresponding LP size can be very large reaching c omputer hardware/software limitation. The aim of this study is to apply the Column Generation (CG) technique t o solve this large scale LP and to evaluate its extension b eyond direct hardware/software capabilities. In thi s study, the sample state probabilities data were simulated stat istically and two methods were used to solve the pr oblem. The first method was using ‘linprog’ function in MATLAB to solve the related LP that all decision vari ables were considered simultaneously while the others was the CG method expected to require a much less percentage of all variables. As result effectivenes s, both methods solved all test problems resulting equal LADs each. The CG method required more average time. Nevertheless, less than 30% of decision variables were considered in the CG method. The lesser percentages were found as the problem size grew. Moreover, larger size problems beyond direct use of software were solved using the proposed approach.

Computing a hybrid preconditioner approach to solve the linear systems arising from interior point methods for linear programming using the conjugate gradient method

(ProQuest: ... denotes formulae omitted.)1. IntroductionThe development of sophisticated software to solve linear optimization problems by interior point methods has started since the early works on this subject. There are three main research lines aimed at improving the efficiency of such methods for solving large-scale problems: reduction of the total number of iterations, techniques to obtain a fast iteration and specific methods for particular classes of problems.This work addresses the second one. Iterative methods are used to solve the linear systems of equations which are the most expensive step at each iteration of interior point methods. Since such systems are very ill-conditioned near a solution, the design of specially tailored preconditioners is an important implementation issue. On the other hand, since the early linear systems do not present the same features, it is advisable to adopt hybrid preconditioners that begin as a generic preconditioner and adapt during the course of the iteration, becoming ever more specialized as convergence takes place (Bocanegra et al., 2007).During the initial iterations a controlled Cholesky factorization is adopted (Campos & Birkett, 1998). Its major advantage is the control parameter that allows the preconditioner to vary all way from a diagonal preconditioner to the full Cholesky factorization, if desired. At the onset of convergence, a splitting preconditioner is used (Oliveira & Sorensen, 2005). Its major advantage is its excellent behavior near a solution of the linear program. However, this desirable feature has a price: the preconditioner could be very expensive to compute. A careful implementation must be performed in order to achieve competitive numerical results regarding both: speed and robustness. An effective implementation of the splitting preconditioner depends crucially upon finding a suitable set of linearly independent columns to form a nonsingular matrix, to be factored, from the constraint matrix.There are several techniques for finding such a set of columns such as the delayed update form for the LU factorization, the symbolic dependent columns, the symbolic independent columns, the combination of symbolic dependent and independent columns and strongly connected components. Some are well known and already applied in other contexts (Coleman & Pothen, 1987; Duff & Reid, 1986; El-Bakry et al., 1994). Others were developed to compute the splitting preconditioner (Oliveira, 1997; Oliveira & Sorensen, 2005). Among the techniques used is the study of the nonzero structure of the constraint matrix to speed up the numerical factorization, such as using key columns, symbolically dependent and independent columns, finding strongly connected components (Oliveira & Sorensen, 2005). Other implementation issues, include ways for changing preconditioners, are also discussed in (Ghidini et al., 2012; Velazco et al., 2010).The choice of the controlled factorization is justified due to the possibility of computing an inexpensive preconditioner in the initial interior point iterations and, as the linear systems become more ill conditioned, the controlled preconditioner can be improved with just the change of a parameter value. Numerical experiments illustrating the effectiveness of such strategies in order to solve large scale linear programming problems are presented in Bocanegra et al. (2007) and Velazco et al. (2010).This work is organized as follows: Section 2 presents the predictor-corrector interior point method, defines its search directions and explains how to solve the resulting linear systems of equations. The controlled Cholesky factorization and splitting preconditioners are discussed in this section. Sections 3 and 4 study several techniques in order to achieve an efficient implementation of the splitting preconditioner. In Section 5 the numerical experiments are shown and discussed. Conclusions follow in Section 6. …

Large Scale Linear Programming in the Windows and Linux computer operating systems

In this study, calculations necessary to solve the large scale linear progra mming problems in two operating systems, Linux and Windows 7 (Win), are compared using two different methods. Relying on the interior-point methods, linear-programming interior point solvers (LIPSOL) software was used for the first method and re lying on an augmented Lagrangian method-based algorithm, the second method used the generalized derivative. The performed calculations for various problems show the produced random in the Linux operating system (OS) and Win OS indicate the efficiency of the perfo rmed calculations in the Linux OS in terms of the accuracy and using of the optimum memory.

Identification of redundant constraints in large scale linear programming problems with minimal computational effort

Identification of redundant constraints in large scale linear programming problems with minimal computational effort

Converging linear particle swarm optimization and intelligent genetic algorithm for a simple multi-echelon distribution and multi-product inventory control in a supply chain model

Inventory management in supply chain networks involves keeping track of hundreds of items spread across multiple locations with complex interrelationships between them. However, it is not computationally reasonable and feasible to consider each item individually during the decision making process. This research optimizes the various decision variables of the supply chain by considering the cost components such as ordering material cost and processing costs for various products at all plants, distribution costs from each plant to all distributors and from each distributor to all retailers inventory carrying costs at all plants, distributors and retailers for the simple multi-echelon supply chain model (MESCM) using two evolutionary computations (ECs), intelligent genetic algorithm (IGA) embedded to Genetic Algorithm for Numerical Optimization with Linear Constraints II (GENOCOP II) and converging linear particles swarm optimization (CLPSO). This simple MESCM is constructed as a large scale linear programming problem. Finally, a simplified rear-word industry case of multi-echelon distribution and multi-product inventory control problem in this simple MESCM are solved using the existing software package, LINGO, and the two proposed ECs in order to validate their effectiveness and determine the optimal inventory solutions for and comparison. The performance of these two ECs which one can adequately obtain the optimal solutions and their computing accuracy is the major works of this research.

Limit Analysis of Masonry Structures Accounting for Uncertainties in Constituent Materials Mechanical Properties~!2008-03-30~!2008-04-22~!2008-06-12~!

The uncertainty often observed in experimental strengths of masonry constituent materials makes critical the selection of the appropriate inputs in the finite elements limit analysis of complex masonry buildings, as well as requires modeling the building ultimate load as a random variable. The most direct approach to solve limit analysis problems in presence of random input parameters is the use of extensive Monte Carlo (MC) simulations. Nevertheless, when MC methods are used to estimate the collapse load cumulative distribution of a masonry structure, large scale linear programming problems must be numerically tackled several times, so precluding the practical utilization of large scale MC simulations. To reduce the computational cost of a traditional MC approach, in the present paper direct computer calculations are replaced with inexpensive Response Surface (RS) models. In particular, RS models are utilized for the limit analysis of a masonry structure in- and out-of-plane loaded, assuming input mechanical properties as random parameters. Two different RS models are analyzed, derived respectively from small scale (20 replicates) MC and Latin Hypercube (LH) simulations. The accuracy of the estimated RS models, as well as the good estimations of the collapse load cumulative distributions obtained via polynomial RS models in comparison with large scale MC simulations, show how the proposed approach could be a useful tool in problems of technical interest.

Detecting “dense” columns in interior point methods for linear programs

During the iterations of interior point methods symmetric indefinite systems are decomposed by LD?L T factorization. This step can be performed in a special way where the symmetric indefinite system is transformed to a positive definite one, called the normal equations system. This approach proved to be efficient in most of the cases and numerically reliable, due to the positive definite property. It has been recognized, however, that in case the linear program contains "dense" columns, this approach results in an undesirable fill---in during the computations and the direct factorization of the symmetric indefinite system is more advantageous. The paper describes a new approach to detect cases where the system of normal equations is not preferable for interior point methods and presents a new algorithm for detecting the set of columns which is responsible for the excessive fill---in in the matrix AA T . By solving large---scale linear programming problems we demonstrate that our heuristic is reliable in practice.

Techniques that strive to combat the influence of degeneracy in linear programming problems

Degeneracy can cause enormous problems when solving large scale linear programming problems. This is not only because there is a possibility that the problem can cycle, but also because a large number of iterations can be executed that do not improve the objective. In this article a procedure which utilises derived reduced costs is discussed. The derived reduced cost of a non- basic variable is defined in such a way that it makes the introduction to the non-basic variable into the basis unattractive if such a decision fails to improve the objective. The procedure deliberately strives to combat degeneracy using derived reduced costs, but it also utilises the advantageous properties of the classical gradient methods.

Analysis of Hybrid Heuristic Model Reduction Algorithms for Solving Linear Programming Problems

Structural redundancies in large scale mathematical programming models are nothing uncommon and large scale Linear Programming Problems (LPP) are no exception. The presence of such embedded redundancies is hardly disputed and can play havoc with LP solution procedures and greatly hamper the solution effort. It is necessary to understand the modelling implications of these embedded redundancies as well as to exploit them during actual computation. The latter goal places heavy emphasis on efficient as well as effective identification techniques for economic application to large scale models. These embedded redundancies may be due to poor modelling and occasionally by bad source data.

Linear time algorithms for linear programming

A linear time algorithm for the solution of asymmetric large scale linear programming problems is presented.

Strategies for Creating Advanced Bases for Large-Scale Linear Programming Problems

The performance of the revised sparse simplex method, or its variants, for the solution of large scale linear programming problems can be considerably enhanced if an advanced starting basis is introduced instead of the traditional all-logical initial basis. We consider three such procedures known as crash procedures to obtain advanced starting points for the revised sparse simplex: 1) We extend traditional crash procedures whereby triangularity of the basis is always maintained and take advantage of the problem characteristics; we apply additional criteria for determining the entering structural variables and their position in the basis. 2) We allow the triangularity to be slightly “violated,” and make some cheap pivot steps to find a better basis. 3) We use the iterative successive-over-relaxation technique to compute a non-extreme point solution and make a crossover to a basic solution. In this article, we first review the work done by other researchers. Then we present an analysis of issues, followed by a description of the techniques and our computational experience.

HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method

HOPDM is an implementation of the primal-dual interior point method for solving large scale linear programming (LP) problems. HOPDM stands for Higher Order Primal Dual Method. Its newest version 2.12 (of May 1995) is a robust and efficient code that compares favourably with today's commercial LP solvers.

SOLVING LARGE SCALE LINEAR PROGRAMMING PROBLEMS USING AN INTERIOR POINT METHOD ON A MASSIVELY PARALLEL SIMD COMPUTER

Abstract The interior point method (IPM) is now well established as a competitive technique for solving very large scale linear programming problems. The leading variant of the interior point method is the primal dual—predictor corrector algorithm due to Mehrotra. The main computational steps of this algorithm are the repeated calculation and solution of a large sparse positive definite system of equations We describe an implementation of the predictor corrector IPM algorithm on MasPar, a massively parallel SIMD computer. At the heart of the implementation is a parallel Cholesky factorization algorithm for sparse matrices. Our implementation uses a new scheme of mapping the matrix onto the processor grid of the MasPar, that results in a more efficient Cholesky factorization than previously suggested schemes The IPM implementation uses the parallel unit of MasPar to speed up the factorization and other computationally intensive parts of the IPM. An important part of this implementation is the judicious div...