Resolution of some optimization problems via kernel functions

Abstract

This thesis topic falls within the realm of linear optimization and semidefinite optimization. The objective is to study primal-dual interior-point methods for solving linear optimization problems. These methods are based on introducing new hyperbolic kernel functions to determine new class of search directions. The analysis of the complexity will be established, and an extension to the semidefinite optimization case will be addressed. In particular, we investigate the concept of feasible and infeasible interior-point methods that rely on kernel functions to define the search directions. We first deal with feasible primal-dual interior-point methods for solving linear optimization problems. These methods are based on new hyperbolic kernel functions. Then, we extend primal-dual feasible interior-point methods to solve semidefinite optimization problems. After that, we present a full-Newton step infeasible interior-point method for solving linear optimization problems based on a hyperbolic kernel function.