An efficient search direction for linear programming problems

In this paper, we present an auxiliary algorithm, in terms of the speed of obtaining the optimal solution, that is effective in helping the simplex method for commencing a better initial basic feasible solution. The idea of choosing a direction towards an optimal point presented in this paper is new and easily implemented. From our experiments, the algorithm will release a corner point of the feasible region within few iterative steps, independent of the starting point. The computational results show that after the auxiliary algorithm is adopted as phase I process, the simplex method consistently reduce the number of required iterations by about 40%.Scope and purposeRecent progress in the implementations of simplex and interior point methods as well as advances in computer hardware has extended the capability of linear programming with today's computing technology. It is well known that the solution times for the interior point method improve with problem size. But, experimental evidence suggests that interior point methods dominate simplex-based methods only in the solution of very large scale linear programs. If the problem size is medium, how to combine the best features of these two methods to produce an effective algorithm for solving linear programming problems is still an interesting problem. In this research we present a new effective ε-optimality search direction based on the interior point method to start an initial basic feasible solution near the optimal point for the simplex method.     

The Existence of a Short Sequence of Admissible Pivots to an Optimal Basis in LP and LCP

We say an LP (linear programming) is fully nondegenerate if both the primal and the dual problems are nondegenerate. In this paper, we prove the existence of a sequence of | B ∖ B | admissible pivot from any basis B (not necessarily feasible) to the unique optimal basis B , if the given LP has an optimal solution and is fully nondegenerate. Here admissible pivots are those pivots (satisfying certain sign conditions) that exist if the current LP dictionary is not terminal, i.e., neither optimal, inconsistent nor dual inconsistent. A natural extension of the result to LCPs (linear complementarity problems) with sufficient matrices is given. The existence itself does not yield a strongly polynomial pivot algorithm for LPs but provides us with a good motivation to study the class of admissible pivot methods for LPs, as opposed to the narrower class of simplex methods for which the shortest sequence of pivots is not known to be polynomially bounded.     

Linear programming software tools on UNIX System

Linear programming(LP) is one of the most widely used Operations Research/Management Science/Industrial Engineering techniques. Recently, multiple criteria decision making or multiple objective linear programming has been well established as a practical approach to seeking satisfactory solutions to real-world decision problems.

In this paper we develop software tools for solving various linear programming problems such as a traditional LP problem, bicriteria LP problem, and multi-criteria LP problem on UNIX system. In a phase for reading data of various LP problems, we define a BNF(Backus-Nauel form) of various LP problems and implement BNF rules by using the C programming language.

In a phase for computing various LP problems, we use efficient methods for solving LP problems, develop various software tools on UNIX system, and combine each LP tool corresponding to an user request in which the Shell programming is used.

We also demonstrate some real-world LP problems by using LP software tools developed here on an UNIX System. Sanyo MPS 020.     

退化线性规划的一个新的改进的单纯形方法

本文讨论退化线性规划单纯形方法最优解的判定准则和有限主元规则.首先改进简约价值系数向量,提出线性规划单纯形方法最优解的判定准则.并且利用本文的判定准则给出[3]中定理2.3.5(P.84)的一个新的证明.然后提出一种新的混合有限主元规则,在退化情形下通过对单纯形表使用新的混合有限主元规则进行迭代,可以判断当前退化基本可行解或为最优解或给出下次迭代的主元并且跳出循环.最后给出在一组经典的退化线性规划例子下,改进的单纯形方法好的计算表现.     

Multiloop transportation simplex algorithm

In large-scale transportation problems (TPs), various methods have been developed to obtain an optimal solution. One of the methods is the transportation simplex algorithm (TSA), which obtains an optimal solution for TP. Various heuristic methods have been developed to find an initial basic feasible solution for transportation algorithms. These methods differ in terms of computational cost and finding good initial solution. In TSA, the better the basic feasible solution, the less the number of iterations the algorithm will run. At each step, it uses pivoting operation, where a loop involving the nonbasic variable with the largest cost reduction is determined. Then, it eliminates the entering basic variable. However, for large-scale problems, even the best basic feasible solution may result in high number of iterations. In this paper, we present a novel algorithm called multiloop transportation simplex algorithm which finds multiple independent loops during pivoting operation. This causes larger cost reductions in every iteration. We experimentally show that the average number of iterations and runtime to solve the TP are dramatically reduced.     

Linear Programming Boosting via Column Generation

We examine linear program (LP) approaches to boosting and demonstrate their efficient solution using LPBoost, a column generation based simplex method. We formulate the problem as if all possible weak hypotheses had already been generated. The labels produced by the weak hypotheses become the new feature space of the problem. The boosting task becomes to construct a learning function in the label space that minimizes misclassification error and maximizes the soft margin. We prove that for classification, minimizing the 1-norm soft margin error function directly optimizes a generalization error bound. The equivalent linear program can be efficiently solved using column generation techniques developed for large-scale optimization problems. The resulting LPBoost algorithm can be used to solve any LP boosting formulation by iteratively optimizing the dual misclassification costs in a restricted LP and dynamically generating weak hypotheses to make new LP columns. We provide algorithms for soft margin classification, confidence-rated, and regression boosting problems. Unlike gradient boosting algorithms, which may converge in the limit only, LPBoost converges in a finite number of iterations to a global solution satisfying mathematically well-defined optimality conditions. The optimal solutions of LPBoost are very sparse in contrast with gradient based methods. Computationally, LPBoost is competitive in quality and computational cost to AdaBoost.     

An improved initial basis for the Simplex algorithm

A lot of research has been done to find a faster (polynomial) algorithm that can solve linear programming (LP) problems. The main branch of this research has been devoted to interior point methods (IPM). The IPM outperforms the Simplex method in large LPs. However, there is still much research being done in order to improve pivoting algorithms. In this paper, we present a new approach to the problem of improving the pivoting algorithms: instead of starting the Simplex with the canonical basis, we suggest as initial basis a vertex of the feasible region that is much closer to the optimal vertex than the initial solution adopted by the Simplex. By supplying the Simplex with a better initial basis, we were able to improve the iteration number efficiency of the Simplex algorithm in about 33%.     

The ‘Idiot’ crash quadratic penalty algorithm for linear programming and its application to linearizations of quadratic assignment problems

ABSTRACT

We provide the first meaningful documentation and analysis of the ‘Idiot’ crash implemented by Forrest in Clp that aims to obtain an approximate solution to linear programming (LP) problems for warm-starting the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is used. Later the technique corresponds closely to a classical quadratic penalty method. We discuss the extent to which it can be used to obtain fast approximate solutions of LP problems, in particular when applied to linearizations of quadratic assignment problems.     

Fuzzy synthetic rating and a satisfying solution for Lee–Tanaka’s LP problem

Fuzzy synthetic rating is a mapping to capture the relationship between the characteristics of an object in a group and its overall fuzzy rating. This paper presents a general treatment on the problems of fuzzy synthetic rating based on factor space, fuzzy clustering and Lee–Tanaka’s idea of transferring the learning task of fuzzy regression NN by means of solving a kind of linear programming, called Lee–Tanaka’s LP problem by the author here. The author presents a satisfying solution for the LP problem. The satisfying solution is not necessarily an optimal solution in the traditional sense. That is, the fuzzy optimal methodology could be applied even when the feasible region is empty.     

Symmetric approximate linear programming for factored MDPs with application to constrained problems

A weakness of classical Markov decision processes (MDPs) is that they scale very poorly due to the flat state-space representation. Factored MDPs address this representational problem by exploiting problem structure to specify the transition and reward functions of an MDP in a compact manner. However, in general, solutions to factored MDPs do not retain the structure and compactness of the problem representation, forcing approximate solutions, with approximate linear programming (ALP) emerging as a promising MDP-approximation technique. To date, most ALP work has focused on the primal-LP formulation, while the dual LP, which forms the basis for solving constrained Markov problems, has received much less attention. We show that a straightforward linear approximation of the dual optimization variables is problematic, because some of the required computations cannot be carried out efficiently. Nonetheless, we develop a composite approach that symmetrically approximates the primal and dual optimization variables (effectively approximating both the objective function and the feasible region of the LP), leading to a formulation that is computationally feasible and suitable for solving constrained MDPs. We empirically show that this new ALP formulation also performs well on unconstrained problems.