Stat Optim Inf Comput

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to |-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.CC999999/ImCDC/Intramural CDC HHSUnited States/2022-01-01T00:00:00Z34141814PMC820532010747vault:3716

Unified Analysis of Kernel-Based Interior-Point Methods for \u3cem\u3eP\u3c/em\u3e *(κ)-LCP

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

Interior-Point Algorithms Based on Primal-Dual Entropy

We propose a family of search directions based on primal-dual entropy in the context of interior point methods for linear programming. This new family contains previously proposed search directions in the context of primal-dual entropy. We analyze the new family of search directions by studying their primal-dual affine-scaling and constant-gap centering components. We then design primal-dual interior-point algorithms by utilizing our search directions in a homogeneous and self-dual framework. We present iteration complexity analysis of our algorithms and provide the results of computational experiments on NETLIB problems

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Complexity analysis of primal-dual algorithms for the semidefinite linear complementarity problem

In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented. The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path. We provide an unified analysis for both long and small-update primal-dual algorithms. Finally, the iteration bounds for these algorithms are obtained

Interior Point Methods 25 Years Later

Interior point methods for optimization have been around for more than 25 years now. Their presence has shaken up the field of optimization. Interior point methods for linear and (convex) quadratic programming display several features which make them particularly attractive for very large scale optimization. Among the most impressive of them are their low-degree polynomial worst-case complexity and an unrivalled ability to deliver optimal solutions in an almost constant number of iterations which depends very little, if at all, on the problem dimension. Interior point methods are competitive when dealing with small problems of dimensions below one million constraints and variables and are beyond competition when applied to large problems of dimensions going into millions of constraints and variables. In this survey we will discuss several issues related to interior point methods including the proof of the worst-case complexity result, the reasons for their amazingly fast practi-cal convergence and the features responsible for their ability to solve very large problems. The ever-growing sizes of optimization problems impose new requirements on optimizatio