Adapting the interior point method for the solution of linear programs on high performance computers

In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial

Interior-point methods for P∗(κ)-linear complementarity problem based on generalized trigonometric barrier function

Recently, M.~Bouafoa, et al. investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for $P_{*}(\kappa)$ Linear Complementarity Problems (LCPs). It is shown that the iteration bound for primal-dual large-update and small-update interior-point methods based on this function is as good as the currently best known iteration bounds for these type methods. The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.publishedVersio

Kernel-Based Interior-Point Algorithms for the Linear Complementarity Problem

In this thesis, we consider the Linear Complementarity Problem (LCP), which is a well-known mathematical problem with many practical applications. The objective of the LCP is to find a certain vector that will satisfy a set of linear inequalities and (non-linear) complementary equation. A kernel-based primal-dual Interior-Point Method (IPM) for solving LCP was introduced and analyzed. The class of kernel functions used in this thesis is a class of so-called eligible kernel functions that are fairly general. We have shown for a positive semi-definite matrix M, that the algorithm is globally convergent and has very good convergence properties. For some instances of the eligible kernel functions, the complexity of the algorithm, in terms of the number of iterations, considered in this thesis matches the best complexity results obtained in the literature for these types of methods. This is the main emphasis of the thesis. The theoretical concepts were illustrated by basic implementation in MATLAB for the classical kernel function and for the parametric kernel function (Table 3.3). A series of numerical tests were conducted that shows that even these basic implementations have a potential for good performance. Better implementation and more numerical testing would be necessary to draw more definite conclusions

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page