New Large Neighborhood Interior Point Methods For Semidefinite Optimization

Abstract

<p>In this thesis, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood N(τ1 ,τ2 ,η) and, as usual, we utilize symmetric directions by scaling the Newton equation with special matrices. After defining the "positive part" and the "negative part" of a symmetric matrix, we solve the Newton equation with its right hand side replaced first by its positive part and then by its negative part, respectively. In this way, we obtain a decomposition of the usual Newton direction and use different step lengths for each of them.</p><p>Starting with a feasible point (X^0 , y^0 , S^0) in N(τ1, τ2 , η ), the algorithm terminates in at most 0(η√( κ∞n)log(1/ε) iterations, where κ∞ is a parameter associated with the scaling matrix and ε is the required precision. To our best knowledge, when the parameter η is a constant, this is the first large neighborhood path-following Interior Point Method (IPM) with the same complexity as small neighborhood path-following IPMs for semidefinite optimization that use the Nesterov-Todd direction. In the case when η is chosen to be in the order of √η, our complexity bound coincides with the known bound for classical large neighborhood IPMs.</p><p>To make this thesis more accessible to readers who are new in this area, we start with a brief introduction to IPMs and SDO. The basic concepts and principles of IPMs and SDO are presented in Chapter 2 and 3.</p>