Optimization and Applications

[no abstract available

Underlying paths and local convergence behaviour of path-following interior point algorithm for SDLCP and SOCP

Ph.DDOCTOR OF PHILOSOPH

Asymptotic Behavior of HKM Paths in Interior Point Method for Monotone Semidefinite Linear Complementarity Problem: General Theory

Abstract An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field which in turn defines a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as the solutions of the systems of ODEs. In Then we show that if the given SDLCP has a unique solution, the first derivative of its off-central path, as a function of √ µ, is bounded. We work under the assumption that the given SDLCP satisfies strict complementarity condition

A study of search directions in primal-dual interior-point methods for semidefinite programming

A study of search directions in primal-dual interior-point methods for semidefinite programmin

On the Nesterov-Todd Direction in Semidefinite Programming

On the Nesterov-Todd Direction in Semidefinite Programmin

On Polynomial-time Path-following Interior-point Methods with Local Superlinear Convergence

Interior-point methods provide one of the most popular ways of solving convex optimization problems. Two advantages of modern interior-point methods over other approaches are: (1) robust global convergence, and (2) the ability to obtain high accuracy solutions in theory (and in practice, if the algorithms are properly implemented, and as long as numerical linear system solvers continue to provide high accuracy solutions) for well-posed problem instances. This second ability is typically demonstrated by asymptotic superlinear convergence properties. In this thesis, we study superlinear convergence properties of interior-point methods with proven polynomial iteration complexity. Our focus is on linear programming and semidefinite programming special cases. We provide a survey on polynomial iteration complexity interior-point methods which also achieve asymptotic superlinear convergence. We analyze the elements of superlinear convergence proofs for a dual interior-point algorithm of Nesterov and Tun\c{c}el and a primal-dual interior-point algorithm of Mizuno, Todd and Ye. We present the results of our computational experiments which observe and track superlinear convergence for a variant of Nesterov and Tun\c{c}el's algorithm