Goran Lesaja

Unified Analysis of Kernel-Based Interior-Point Methods for

We present an interior-point method for the

P_*(\kappa)

P_*(\kappa)

-Linear Complementarity Problems

-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

Full Nesterov–Todd step feasible interior-point method for the Cartesian P*κ-SCLCP

P_*(\kappa)

Introducing Interior-Point Methods for Introductory Operations Research Courses and/or Linear Programming Courses

-LCPs based on the entire class of eligible kernel functions. (The title of this article has been corrected.)

Interior-point methods for Cartesian P*κ-linear complementarity problems over symmetric cones based on the eligible kernel functions

In this paper, we present a feasible interior-point method (IPM) for the Cartesian P *(κ)-linear complementarity problem over symmetric cones (SCLCP) that is based on the classical logarithmic barrier function. The method uses Nesterov–Todd search directions and full step updates of iterates. With the appropriate choice of parameters the algorithm generates a sequence of iterates in the small neighbourhood of the central path which implies global convergence of the method. Moreover, this neighbourhood permits the quadratic convergence of the iterates. The iteration complexity of the method is O((1+4κ)√rlog(r/ϵ)) which matches the currently best known iteration bound for IPMs solving the Cartesian P *(κ)-SCLCP.

INTERIOR-POINT ALGORITHMS FOR A CLASS OF CONVEX OPTIMIZATION PROBLEMS

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.

Generalized diamond-α dynamic opial inequalities

An interior-point method (IPM) for Cartesian P *(κ)- linear complementarity problems over symmetric cones (SCLCP) is analysed and the complexity results are presented. The Cartesian P *(κ)- SCLCPs have been recently introduced as the generalization of the more commonly known and more widely used monotone-SCLCPs. The IPM is based on the barrier functions that are defined by a large class of univariate functions called eligible kernel functions, which have recently been successfully used to design new IPMs for various optimization problems. Eligible barrier (kernel) functions are used in calculating the Nesterov–Todd search directions and the default step-size which lead to very good complexity results for the method. For some specific eligible kernel functions, we match the best-known iteration bound for the long-step methods while for the short-step methods the best iteration bound is matched for all cases.

Grouping of Variables to Facilitate SDL Methods in Multivariate Data Sets

In this paper we consider interior-point methods (IPM) for the nonlinear, convex optimization problem where the objective function is a weighted sum of reciprocals of variables subject to linear constraints (SOR). This problem appears often in various applications such as statistical stratified sampling and entropy problems, to mention just few examples. The SOR is solved using two IPMs. First, a homogeneous IPM is used to solve the Karush-Kuhn-Tucker conditions of the problem which is a standard approach. Second, a homogeneous conic quadratic IPM is used to solve the SOR as a reformulated conic quadratic problem. As far as we are aware of it, this is a novel approach not yet considered in the literature. The two approaches are then numerically tested on a set of randomly generated problems using optimization software MOSEK. They are compared by CPU time and the number of iterations, showing that the second approach works better for problems with higher dimensions. The main reason is that although the first approach increases the number of variables, the IPM exploits the structure of the conic quadratic reformulation much better than the structure of the original problem.

CEJOR special issue of Croatian Operational Research Society

We establish some new dynamic Opial-type diamond alpha inequalities in time scales. Our results in special cases yield some of the recent results on Opials inequality and also provide new estimates on inequalities of this type. Also, we introduce an example to illustrate our result.Mathematics Subject Classification 2000: 39A12; 26D15; 49K05