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