Soondal Park

Shortest paths in a network with time-dependent flow speeds

The Shortest Path Problem in Time-dependent Networks, where the travel time of each link depends on the time interval, is not realistic since the model and its solution violate the Non-passing Property (NPP:often referred to as FIFO) of real phenomena. Furthermore, solving the problem needs much more computational and memory complexity than the general shortest path problem. A new model for Time-dependent Networks where the flow speeds of each link depend on time interval, is suggested. The model is more realistic since its solution maintains the NPP. Solving the problem needs just a little more computational complexity, and the same memory complexity, as the general shortest path problem. A solution algorithm modified from Dijkstras label setting algorithm is presented. We extend this model to the problem of Minimum Expected Time Path in Time-dependent Stochastic Networks where flow speeds of each link change statistically on each time interval. A solution method using the Kth-shortest Path algorithm is presented.

A label-setting algorithm for finding a quickest path

The quickest path problem is to find a path to send a given amount of data from the source to the destination with minimum transmission time. To find the quickest path, existing algorithms enumerate nondominated paths with distinct capacity, and then determine a quickest path by comparing their transmission time. In this paper, we propose a label-setting algorithm for finding a quickest path by transforming a network to another network where an important property holds that each subpath of a quickest path is also a quickest path. The proposed algorithm avoids enumerating non-dominated paths whose transmission time is greater than the minimum transmission time. Although the computational complexity of the proposed algorithm is the same as that of existing algorithms, experimental results show that our algorithm is efficient when a network has two or more non-dominated paths.

An ε-sensitivity analysis in the primal–dual interior point method

Abstract This paper presents a method of sensitivity analysis on the cost coefficients and the right-hand sides for most variants of the primal–dual interior point method. We first define an e -optimal solution to describe the characteristics of the final solution obtained by the primal–dual interior point method. Then an e -sensitivity analysis is defined to determine the characteristic region where the final solution remains the e -optimal solution as a cost coefficient or a right-hand side changes. To develop the method of e -sensitivity analysis, we first derive the expressions for the final solution from data which are commonly maintained in most variants of the primal–dual interior point method. Then we extract the characteristic regions on the cost coefficients and the right-hand sides by manipulating the mathematical expressions for the final solution. Finally, we show that in the nondegenerate case, the characteristic regions obtained by e -sensitivity analysis are convergent to those obtained by sensitivity analysis in the simplex algorithm.

POSITIVE SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING

Positive sensitivity analysis (PSA) is a sensitivity analysis method for linear programming that finds the range of perturbations within which positive value components of a given optimal solution remain positive. Its main advantage is that it is applicable to both an optimal basic and nonbasic optimal solution. The first purpose of this paper is to present some properties of PSA that are useful for establishing the relationship between PSA and sensitivity analysis using optimal bases, and between PSA and sensitivity analysis using the optimal partition. We examine how the range of PSA varies according to the optimal solution used for PSA, and discuss the relationship between the ranges of PSA using different optimal solutions. The second purpose is to clarify the relationship between PSA and sensitivity analysis using an optimal basis, and the relationship between PSA and sensitivity analysis using the optimal partition. We show that sensitivity analysis using the optimal partition is a special case of PSA, and its properties can be derived from the properties of PSA. The comparison among the three sensitivity analysis methods will lead to a better understanding of the difference among sensitivity analysis methods.

Finding a path in the hierarchical road networks

In a hierarchical road network, all roads can be classified into two groups according to their attributes, such as their speed limit and the number of lanes. By splitting the whole network into high-level and low-level subnetworks, the size of the road network to be searched can be reduced, and more human-oriented paths can be obtained. In this paper, we define a convenient path in the hierarchical road network and propose an algorithm for finding that path. The proposed algorithm produces a convenient and approximately shortest path with little computational effort by reducing the number of nodes to be considered. The difference between the length of the paths produced by the proposed algorithm and by existing algorithms and that of the shortest path of the whole network is discussed. Some experimental results are given to show the efficiency of the proposed algorithm.

Solving linear systems in interior-point methods

Abstract There are two approaches to solve the linear systems in interior-point methods: the normal equation approach and the augmented system approach. We integrated the two methods by applying matrix partitioning to the augmented system approach. Specifically, we show the Schur complement method which is applied to problems with dense columns is a special case of the augmented system approach. We will use this property for the integrated approach. If we use the integrated approach, we can solve linear systems maintaining sparsity of matrices without respect of the existence of dense columns. Scope and purpose Interior-point methods require a step to solve the linear systems for computing a new direction at every iteration. Generally, we solve the linear systems by applying Cholesky factorization. When there is a dense column, we can not exploit the sparsity of matrices. The most popular way of treating such a dense column employs the Schur complement method or the augmented system approach. The Schur complement method is faster than the augmented system approach, but suffers from numerical unstability. We present a fast and numerically stable approach by integrating former approaches.

An [epsilon]-sensitivity analysis for semidefinite programming

Abstract We extend the concept of ϵ -sensitivity analysis developed for linear programming to that for semidefinite programming. First, the notion of ϵ -optimality for a given semidefinite programming problem is defined, and then a generic ϵ -sensitivity analysis for semidefinite programming is introduced. Based on the definitions, we develop an implementation of the generic ϵ -sensitivity analysis under perturbations of either the cost parameters or the right-hand side.

LPAKO: A Simplex-based Linear Programming Program

LPAKO is a public domain simplex-based linear programming program which can solve large-scale, sparse linear programming problems. It has been widely used in many applications and shows better performance than other public domain simplex-based programs. Several aspects considered in the development of LPAKO are described in this article such as the construction of initial basis, LU factorization of basis matrix, pricing rule, presolving/postsolving and other miscellaneous issues. At the end of the article, we introduce H. Mittelmans benchmark result which compares the performance of LPAKO with those of several simplex-based programs. We also compare LPAKO with CPLEX on the NETLIB test set.

Numerical aspects in developing LP softwares, LPAKO and LPABO

We have developed two public domain linear programming programs for several years, LPAKO and LPABO, which can solve large-scale sparse LP problems stably and fast. In this paper, several important numerical aspects which were considered in developing LPAKO and LPABO are presented. Common issues are scaling, tolerances and presolving. For the LPAKO, LU factorization and pivoting rule are important aspects. In case of LPABO, Cholesky factorization, ordering and dense column handling are important. In the end of this paper, several issues to be considered in the future development are proposed.

ON THE PROPERTIES OF ∊-SENSITIVITY ANALYSIS FOR LINEAR PROGRAMMING

∊-Sensitivity analysis (∊-SA) is a kind of method to perform sensitivity analysis for linear programming. Its main advantage is that it can be directly applied for interior-point methods with a little computation. In this paper, we discuss the property of ∊-SA analysis and its relationship with other sensitivity analysis methods. First, we present a new property of ∊-SA, from which we derive a simplified formula for finding the characteristic region of ∊-SA. Next, based on the simplified formula, we show that the characteristic region of ∊-SA includes the characteristic region of Yildirim and Todds method. Finally, we show that the characteristic region of ∊-SA asymptotically becomes a subset of the characteristic region of sensitivity analysis using optimal partition. Our results imply that ∊-SA can be used as a practical heuristic method for approximating the characteristic region of sensitivity analysis using optimal partition.