A brief history of linear and mixed-integer programming computation

A common type of mathematical optimization is Linear Programming (LP). An LP solution of aquifer influence functions has recently been reported by Gadjica, etal.1 (1987) and Targac, etal.2 Their LP matrices were large and sparse (only 3% of the elements were non-zero) and were solved on main frame computers. Another recent application of LP is equation-of-state matching of laboratory PVT data3. This problem leads to a smaller, denser LP matrix. Three methods of LP solution were investigated on microcomputers: (1) the simplex method, (2) the revised simplex method, and (3) the symmetric method. These methods were run on several LP problems ranging from a small dense matrix to large sparse matrices. The different methods have different characteristics which affect the speed, storage requirements and simplicity of coding. The simplex method is straightforward, but usually is slower and requires more storage than the other methods. The results of this study are tabulated with running times and storage requirements for the various LP methods and microcomputers. The computers range from the IBM XT to the Compaq 386. This information serves as a documentation of the LP codes and should be useful for an engineer interested in using LP codes on a microcomputer.

This new volume provides the information needed to understand the simplex method, the revised simplex method, dual simplex method, and more for solving linear programming problems.Following a logical order, the book first gives a mathematical model of the linear problem programming and describes the usual assumptions under which the problem is solved. It gives a brief description of classic algorithms for solving linear programming problems as well as some theoretical results. It goes on to explain the definitions and solutions of linear programming problems, outlining the simplest geometric methods and showing how they can be implemented. Practical examples are included along the way. The book concludes with a discussion of multi-criteria decision-making methods.Advances in Optimization and Linear Programming is a highly useful guide to linear programming for professors and students in optimization and linear programming.

Engineering Optimization Theory and Practice

Secure multiparty computation is a basic concept of growing interest in modern cryptography. It allows a set of mutually distrusting parties to perform a computation on their private information in such a way that as little as possible is revealed about each private input. The early results of multiparty computation have only theoretical signi??cance since they are not able to solve computationally complex functions in a reasonable amount of time. Nowadays, e??ciency of secure multiparty computation is an important topic of cryptographic research. As a case study we apply multiparty computation to solve the problem of secure linear programming. The results enable, for example in the context of the EU-FP7 project SecureSCM, collaborative supply chain management. Collaborative supply chain management is about the optimization of the supply and demand con??guration of a supply chain. In order to optimize the total bene??t of the entire chain, parties should collaborate by pooling their sensitive data. With the focus on e??ciency we design protocols that securely solve any linear program using the simplex algorithm. The simplex algorithm is well studied and there are many variants of the simplex algorithm providing a simple and e??cient solution to solving linear programs in practice. However, the cryptographic layer on top of any variant of the simplex algorithm imposes restrictions and new complexity measures. For example, hiding the number of iterations of the simplex algorithm has the consequence that the secure implementations have a worst case number of iterations. Then, since the simplex algorithm has exponentially many iterations in the worst case, the secure implementations have exponentially many iterations in all cases. To give a basis for understanding the restrictions, we review the basic theory behind the simplex algorithm and we provide a set of cryptographic building blocks used to implement secure protocols evaluating basic variants of the simplex algorithm. We show how to balance between privacy and e??ciency; some protocols reveal data about the internal state of the simplex algorithm, such as the number of iterations, in order to improve the expected running times. For the sake of simplicity and e??ciency, the protocols are based on Shamir's secret sharing scheme. We combine and use the results from the literature on secure random number generation, secure circuit evaluation, secure comparison, and secret indexing to construct e??cient building blocks for secure simplex. The solutions for secure linear programming in this thesis can be split into two categories. On the one hand, some protocols evaluate the classical variants of the simplex algorithm in which numbers are truncated, while the other protocols evaluate the variants of the simplex algorithms in which truncation is avoided. On the other hand, the protocols can be separated by the size of the tableaus. Theoretically there is no clear winner that has both the best security properties and the best performance.

Hybrid-LP: Finding advanced starting points for simplex, and pivoting LP methods

Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.

Optimization has been one of the most fundamental and extensive contributions of management science/operations research, with an enormous number of contributions and subfields developed by many researchers and practitioners. When the journal Management Science launched in 1954, little was known about optimization, including some results in nonlinear optimization and the simplex method and duality developed for linear programming. However, linear programming computations were limited to problems with at most 101 linear constraints. Then some early contributions by seminal researchers began to develop foundations for the field. I will review a few of these early contributions, focusing on the traveling sales problem and integer programming, decomposition, and column generation. I will summarize some research and applied contributions since then, including the enormous development of computations. I will focus on linear and integer programs with some material on combinatorial optimization. This paper was accepted by David Simchi-Levi, Special Section of Management Science: 65th Anniversary.

Linear bilevel programming with upper level constraints depending on the lower level solution

In Operations Research, linear-fractional programming is considered as the generalization of linear programming problem. While in a linear programming the objective function is a linear function, and in a linear-fractional programming the objective function is the ratio of two linear or non-linear functions. The majority of the algorithm used for solving the linear fractional programming problem relies upon the classical simplex method. In this paper, we have proposed a new algorithm for solving a linear fractional programming problem in which the objective function is a combination of linear fractional function, while constraint functions are in the form of linear inequalities. Our proposed algorithm is based on the extension of the method, which is used to solve linear programming problems with linear constraints. The primary intent behind developing this method is that we did not need to transform the linear fractional programming problem into linear programming problem, and also it helps in finding out the feasible region via a sequence of points in the direction that improves the feasibility of the fractional objective function. Numerical examples are given to illustrate the use of these proposed methods. Lastly, to demonstrate the efficacy of the proposed algorithm, we have compared the findings obtained with other approaches to display our algorithm's efficacy.

Piecewise linear programming via interior points

Multi‐period planning problems in the oil and refinery industry are typically large, sparse, staircase/band diagonal structured and nonlinear optimization problems. Successive linear programming (SLP) type methods have been widely used for solving these planning problems. But, it has long been recognized that the simplex method used in solving linear programs requires a large number of iterations for staircase/band diagonal structured problems. In this paper, we report results of an application of a recently developed interior point method that promises to be many times faster than the simplex method for multi‐period planning problems. However, to facilitate the use of interior point method in the current SLP algorithms a hybrid method combining the interior point method and the simplex method is developed. Therefore, the results determined with this hybrid method are qualitatively equivalent to that obtained with the simplex method alone. The CPU times corresponding to the hybrid method are compared with the CPU times of simplex and dual affine methods. The new hybrid method generates a basic feasible solution of the linear programming problem and is approximately 7 times faster than the simplex method on the tested planning problems. Moreover, the interior point and hybrid methods become faster as the problem size increases.

Linear Programming (LP) is a significant research area in the field of operations research. The simplex algorithm is the most widely used and well-studied method for solving Linear Programming problems (LPs). Many algorithms have been proposed for the solution of LPs. The vast majority of these algorithms belong to three main categories: (i) Simplex-type or pivoting algorithms, (ii) interior-point methods (IPMs) and (iii) exterior point simplex type algorithms (EPSA). The aim of this paper is to present an implementation of a hybrid simplex algorithm that begins to solve the LP using an IPM and after a number of iterations continues with a primal-dual EPSA algorithm. This hybrid approach aims to take advantage of: (i) IPM strengths, which is the fast convergence in the first iterations, and (ii) EPSA strengths, i.e. the fast convergence when making steps in directions that are linear combinations of attractive directions. The idea of combining different types of linear programming algorithms is not new; to the best of our knowledge, this is the first time that interior point methods and exterior point algorithms are combined. The interior point that is calculated by IPM after a number of iterations can lead to such attractive directions. In order to gain an insight into the practical behavior of the proposed algorithm, we have performed some computational experiments over sparse randomly generated optimal LPs. Finally, in the computational study that we have conducted, we investigate the adequate number of iterations that IPM should run in order to decrease the CPU time and the iterations of the proposed algorithm.

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

A simplex algorithm for piecewise-linear fractional programming problems

: So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)