Worst-case iteration bounds for log barrier methods for problems with nonconvex constraints

Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints have Lipschitz first and second derivatives. Unfortunately, previous analyses of log barrier methods in this setting implicitly prove guarantees with exponential dependencies on $1/\mu$, where $\mu$ is the barrier penalty parameter. We provide an IPM that finds a $\mu$-approximate Fritz John point by solving $\mathcal{O}( \mu^{-7/4})$ trust-region subproblems. For this setup, the results represent both the first iteration bound with a polynomial dependence on $1/\mu$ for a log barrier method and the best-known guarantee for finding Fritz John points. We also show that, given convexity and regularity conditions, our algorithm finds an $\epsilon$-optimal solution in at most $\mathcal{O}(\epsilon^{-2/3})$ trust-region steps.Comment: Minor edit

On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems

Convergence analysis of an Inexact Infeasible Interior Point method for Semidefinite Programming

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima,Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is prove

Convergence of infeasible-interior-point methods for self-scaled conic programming

Convergence of infeasible-interior-point methods for self-scaled conic programmin

On the Finite Termination of An Entropy Function Based Smoothing Newton Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach