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

Ph.DDOCTOR OF PHILOSOPH

A subgradient method based on gradient sampling for solving convex optimization problems

2015-2016 > Academic research: refereed > Publication in refereed journa

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

Inexact subgradient methods for quasi-convex optimization problems

2014-2015 > Academic research: refereed > Publication in refereed journa

Superlinear convergence of an infeasible interior point algorithm on the homogeneous feasibility model of a semi-definite program

In the literature, superlinear convergence of implementable polynomial-time interior point algorithms to solve semi-definite programs (SDPs) can only be shown by (i) assuming that the given SDP is nondegenerate and modifying these algorithms, or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems, when a suitable initial iterate is required as well. Otherwise, these algorithms are not easy to implement even though they can be shown to have polynomial iteration complexities and superlinear convergence. These are besides the assumption of strict complementarity imposed on the given SDP. In this paper, we show superlinear convergence of an implementable interior point algorithm that has polynomial iteration complexity when it is used to solve the homogeneous feasibility model of a primal-dual SDP pair that has no special structure imposed. This is achieved by only assuming strict complementarity and the availability of an interior feasible point to the primal SDP. Furthermore, we do not need to modify the algorithm to show this.Comment: This preprint contains a major error in that Proposition 3.1 in it is wrong, and this affects the main result in the preprin