In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the classical logarithmic function, the prototype selfregular function, and non-self-regular kernel functions as special cases. The analysis of the algorithms in the paper follows the same line of arguments as in Bai et al. (SIAM J. Optim. 15:101-128, 2004), where a variety of non-self-regular kernel functions were considered including the ones with linear and quadratic growth terms. However, the important case when the growth term is between linear and quadratic was not considered. The goal of this paper is to introduce such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. They match the currently best known iteration bounds for the prototype self-regular function with quadratic growth term, the simple nonself-regular function with linear growth term, and the classical logarithmic kernel function. In order to achieve these complexity results, several new arguments had to be used.
Abstract. We present an interior-point method for the P * (κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several specific eligible kernel functions. For some of them we match the best known iteration bounds for the long-step method, while for the short-step method the iteration bounds are of the same order of magnitude. As far as we know, this is the first paper that provides a unified approach and comprehensive treatment of interior-point methods for P * (κ)-LCPs based on the entire class of eligible kernel functions.
In recent years the introduction and development of Interior-Point Methods has had a profound impact on optimization theory as well as practice, influencing the field of Operations Research and related areas. Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming. Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory Operations Research and/or Linear Programming courses, Industrial Engineering courses and Math Modeling courses. The objective of this paper is to discuss the ways of simplifying the introduction of Interior-Point Methods for students who have various backgrounds or who are not necessarily mathematics majors.
Finding spanning trees under various constraints is a classic problem with applications in many fields. Recently, a novel notion of dense ( sparse ) tree, and in particular spanning tree (DST and SST respectively), is introduced as the structure that have a large (small) number of subtrees, or small (large) sum of distances between vertices. We show that finding DST and SST reduces to solving the discrete optimization problems. New and efficient approaches to find such spanning trees is achieved by imposing certain conditions on the vertex degrees which are then used to define an objective function that is minimized over all spanning trees of the graph under consideration. Solving this minimization problem exactly may be prohibitively time consuming for large graphs. Hence, we propose to use genetic algorithm (GA) which is one of well known metaheuristics methods to solve DST and SST approximately. As far as we are aware this is the first time GA has been used in this context.We also demonstrate on a number of applications that GA approach is well suited for these types of problems both in computational efficiency and accuracy of the approximate solution. Furthermore, we improve the efficiency of the proposed method by using Kruskal s algorithm in combination with GA. The application of our methods to several practical large graphs and networks is presented. Computational results show that they perform faster than previously proposed heuristic methods and produce more accurate solutions. Furthermore, the new feature of the proposed approach is that it can be applied recursively to sub-trees or spanning trees with additional constraints in order to further investigate the graphical properties of the graph and/or network. The application of this methodology on the gene network of a cancer cell led to isolating key genes in a network that were not obvious from previous studies.
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