Introduction: New approaches to linear programming

This issue ofAlgorithmica present papers on various aspects of nonlinear methods for solving linear programming problems, inspired by the work of Karmarkar. This introduction describes some of these aspects and briefly mentions other recent developments in the field. A bibliography of recent articles is included.     

New integer linear programming approaches for course timetabling

The most complete form of academic timetabling problem is the population and course timetabling problem. In this problem, there may be multiple classes of each subject, and the decision on which students are to constitute each class is made in concert with the decision on the timetable for each class. In order to solve this problem, it is normally simplified or decomposed in some fashion. One simplification commonly used in practice is known as blocking: it is assumed that the classes can be partitioned into sets of classes (or blocks) that will be timetabled in parallel. This restricts clashing to occur only between classes in the same block, and essentially removes the timetabling aspect of the problem, which can be carried out once the blocks are constituted and the classes populated. The problem of constituting the blocks and populating the classes, known as the course blocking and population problem, is nevertheless a challenging problem, and provides the focus of this paper. We demonstrate, using data provided by a local high school, that integer linear programming approaches can solve the problem in a matter of seconds. Key features include remodelling to remove symmetry caused by students with identical subject selection, and the observation that in practice, only integrality of the block composition variables needs to be enforced; the class population aspects of the model have strong integrality properties.     

新的线性遗传程序设计方法

受其他多种线性编码的遗传程序设计算法的启发,提出一种新的编码方式的遗传程序设计——符号遗传程序设计。该编码方式具有简单、无语法限制并且能够在不增加计算量的情况下将染色体翻译成多个表达式等特点。分析与实验表明该算法具有较高的效率和较强的稳定性。     

On using exterior penalty approaches for solving linear programming problems

In this paper, we investigate three exterior penalty function approaches for solving linear programming problems. These methods are an active set ℓ₂ penalty approach (ASL2), an inequality–equality-based ℓ₂ penalty approach (IEL2), and an augmented Lagrangian approach (ALAG). Particular effective variants are explored for each method, along with comments and experience on alternative algorithmic strategies that were empirically investigated. Our motivation is that heretofore, there has been no empirical evidence on how variants of these exterior point algorithms that have been suitably tuned might perform with respect to solving moderately large-scale linear programming problems. Our experiments using a set of randomly generated problems of varying sizes and density, as well as a set of NETLIB test problems, provide insights into the relative performance of these different approaches derived from the basic ℓ₂ penalty function, and into their viability for solving linear programs. Average rank tests based on the computational results obtained are performed using two different statistics which assess the speed of convergence and the quality or accuracy of the solution. Overall, a particular variant (ALAG2) of ALAG yielded the best performance with respect to the joint criteria of speed of convergence and the quality of the final solution attained. On the other hand, IEL2 invariably achieved the best quality solutions using comparatively designed termination criteria. However, for highly dense linear programming problems, ASL2 appeared to be the best-alternative among the tested algorithms. Moreover, although our implementation was rudimentary in comparison with CPLEX 6.0, a commercial solver, the tested methods were competitive (sometimes favorable) in attaining a final (though only near optimal) solution for the set of large-scale low-density problems.Scope and purposeAlgorithms based on interior point methods for solving linear programming problems have gained popularity due to some recent techniques which eliminate the ill-conditioning that is inherent within barrier function approaches. Yet, there is comparatively less evidence on the performance of exterior penalty function approaches for solving linear programs. This article investigates three exterior penalty function approaches for solving such problems based on the quadratic penalty function, including the augmented Lagrangian function. Several related algorithmic strategies are explored and tested on a set of randomly generated, degenerate problems, as well as on a standard collection of NETLIB problems. We provide insights into the algorithmic performance and suggest avenues for further research in this relatively less explored domain of approaches for solving linear programming problems.     

An introduction to nonlinear programming—II: The linear programming problem

Linear programming problems represent the most thoroughly analyzed and widely solved class of parameter optimization problems. In Part II, we shall restrict our attention to this general class of problems. The characteristics of the admissible region are investigated and established. The Kuhn-Tucker conditions developed in Part I are applied to establish necessary and sufficient conditions that must be satisfied at a minimum. Included in the discussion is a consideration of dual linear programming problems. Then, we direct our attention to the question of determining the solution of specific problems. A general algorithm known as the Simplex Method is described and applied to several examples.     

SISO approaches for linear programming based methods for tuning decentralized PID controllers

Two tuning techniques are proposed to design decentralized PID controllers for weakly coupled and general MIMO systems, respectively. Each SISO loop is designed separately, and the controller parameters are obtained as a solution of a linear programming optimization problem with constraints on the process stability margins. Despite the SISO approach, loop interactions are accounted for either by Gershgorin bands (non-iterative method) or an equivalent open-loop process (iterative method). The tuning results and performance from both methods are illustrated in four simulations of linear processes, and a laboratory-scale application in a Peltier process. Four applications contemplate closed-loop performance comparisons between the proposed techniques and techniques from the literature. One application illustrates the feasibility of the proposed iterative method, based on EOPs, in tuning decentralized PIDs for a 5 × 5 system. Moreover, an analysis of the effect of model uncertainty in the phase and gain margins of the closed-loop process is performed.     

An experimental comparison of the new goal programming and the linear programming approaches in the two-group discriminant problems

The aim of this article is to consider a new linear programming and two goal programming models for two-group classification problems. When these approaches are applied to the data of real life or of simulation, our proposed new models perform well both in separating the groups and the group–membership predictions of new objects. In discriminant analysis some linear programming models determine the attribute weights and the cut-off value in two steps, but our models determine simultaneously all of these values in one step. Moreover, the results of simulation experiments show that our proposed models outperform significantly than existing linear programming and statistical approaches in attaining higher average hit-ratios.     

Bicriterion linear programming

Basically leaning on the concept of “best” compromise, the technique seeks the optimal solution by fair relaxations of the objectives commensurate with their degrees of importance until the optimum feasible compromise is reached. The concept is made operational by deriving a linear constraint (referred to as the compromise constraint) to be added into the original set. An offspring of both objective functions, the compromise constraint cuts the original feasible region and forces both objectives to settle on a common point along this added restriction. The resulting singular equivalent of the bicriterion problem optimizes any one of the two objective functions and their equivalent sum (referred to as the supergoal) subject to the new set of constraints. Post optimality analysis is employed to minimize the computational effort usually done by a computer. A very attractive feature of this new practical technique is its ability to search for the optimum in any point in the feasible region, even other than the vertices of the convex set.     

Rule-based intelligence to support linear programming analysis

Increased size and complexity of linear programs make it difficult to understand results and manage the model. This difficulty is overcome, at least partially, with new advances in computing environments and techniques from artificial intelligence and operations research. A software system, called ANALYZE, is presented within the larger context of developing an artificially intelligent environment for mathematical programming modeling and analysis. With heuristics governed by rule-based reasoning and with syntax-driven translations into English, the ANALYZE system enables a form of intelligence to support analysis. Particular problems that illustrate this capability are explanations of dual prices, diagnoses of infeasibilities, and reasoning about redundancy.     

A sequential linear programming approach to solve mixed integer programming problems

The optimal operation of pumps in a large water supply system under time-of-use electricity rates is formulated as a mixed integer programming (MIP) problem. The problem is solved using an iterative linear programming (LP) scheme. The scheme is applied to an actual world problem, the City of Inglewood Water Supply System. Computational results are presented and termination criteria for the solution scheme are discussed.     

A linear programming approach to max-sum problem: a review

The max-sum labeling problem, defined as maximizing a sum of binary (i.e., pairwise) functions of discrete variables, is a general NP-hard optimization problem with many applications, such as computing the MAP configuration of a Markov random field. We review a not widely known approach to the problem, developed by Ukrainian researchers Schlesinger et al. in 1976, and show how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product. In particular, we review Schlesinger et al.'s upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to the constraint satisfaction problem, the fact that this minimization is dual to a linear programming relaxation of the original problem, and the three kinds of consistency necessary for optimality of the upper bound. We revisit problems with Boolean variables and supermodular problems. We describe two algorithms for decreasing the upper bound. We present an example application for structural image analysis.     

Multi-media instructional approach to teaching linear programming

This paper describes different strategies employed in converting a lecture-oriented mathematical programming course to a Personalized Self-Paced Instructional (PSI) format. This is an elective course for students in science, engineering and management. A multi media instructional approach is used in the PSI system which combines traditional lectures, self-paced and individualized learning assisted by interactive computer programs and video taped instructional materials. This unique PSI system for mathematical programming provides maximum learning opportunity and flexibility to students. The author's experiences with the PSI system and the students' evaluation of the self-paced system are also discussed.     

Investment planning for the development of a national resource —linear programming based approaches

This paper will review and generalize the work of Gearing, Swart and Var dealing with the development of a mathematical model to aid the government of Turkey in determining the “best” allocation of the capital budget for tourism among a large group of competing proposals. One of the more unique aspects of this work was the development of a measure of benefit for particular allocation plans which allowed for subjective information to be an integral part of the investment planning model.The computational procedures developed heretofore to derive investment strategies from the model have been based on dynamic programming, integer programming or combined dynamic programming-integer programming approaches. This paper will show how various methods based on linear programming yield satisfactory answers to many questions regarding development policies. The primary advantage of these linear programming methods is that they can be implemented with standard linear programming computer packages which are generally available, and hence eliminate the high cost of developing special purpose computer codes.     

Spectral approach to linear programming bounds on codes

We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte’s linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. Examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in projective spaces.     

Application of linear programming to structural system reliability

Reliability of framed structures comprised of ductile components is studied assuming that loads and component strengths are random variables. A bounded variable simplex algorithm and duality in linear programming are used to determine the coefficients of strength terms in the failure mode expression (FME). The method is extended to identify an FME for a structure under simultaneous multiple load variables. It is shown that in the case of single load, Phase 1 of the simplex algorithm can be skipped since a trivial basic feasible solution is always available. Enumeration of FMEs is carried out using sensitivity and simulation approaches in conjunction with LP methods. The second moment format is used to find a measure of reliability for a structure. The significant modes are used in a Monte Carlo simulation to accurately obtain mean and variance of the safety margin of the system and consequently a safety index for the system is computed. Numerical examples are given to illustrate the methods presented in the paper.