Masayuki Shida

Local convergence of predictor--corrector infeasible-interior-point algorithms for SDPs and SDLCPs

An example of an SDP (semidefinite program) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno—Todd—Ye type predictor—corrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno—Todd—Ye type predictor—corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions.

A Predictor-Corrector Interior-Point Algorithm for the Semidefinite Linear Complementarity Problem Using the Alizadeh--Haeberly--Overton Search Direction

This paper proposes a globally convergent predictor-corrector infeasible-interior-point algorithm for the monotone semidefinite linear complementarity problem using the Alizadeh--Haeberly--Overton search direction, and shows its quadratic local convergence under the strict complementarity condition.

Existence and Uniqueness of Search Directions in Interior-Point Algorithms for the SDP and the Monotone SDLCP

Various search directions used in interior-point algorithms for the semidefinite program (SDP) and the monotone semidefinite linear complementarity problem (SDLCP) are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace. This observation provides a unified geometric view over the existence of those search directions.

Lagrangian Dual Interior-Point Methods for Semidefinite Programs

This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguishing features of the method are full use of the BFGS quasi-Newton method in the corrector procedure and an application of the conjugate gradient method with an effective preconditioning matrix induced from the BFGS quasi-Newton method in the predictor procedure. Some preliminary numerical results are reported.

A note on the Nesterov-Todd and the Kojima-Shindoh-hara search directions in semidefinite programming

This short note shows that the Nesterov-Todd search direction used in primal-dual interior-point methods for semidefinite programs belongs to the family of search directions proposed by Kojima, Shindoh and Hara.

Centers of monotone generalized complementarity problems

Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product of x, y ∈ E, and M a maximal monotone subset of E × E. This paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with C and M: Find x, y ∈ M ∩ C × C* such that = 0. Here C* = {y ∈ E : ≥ 0 for all x ∈ C} denotes the dual cone of C. The main result of the paper unifies and extends some results established for monotone complementarity problems in Euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices.

Manifold Structure of the Karush–Kuhn–Tucker Stationary Solution Set with Two Parameters

This paper deals with a 2-dimensional parameter family of nonlinear programs: minimize

Change of generalized indices of stationary solutions. to multiparametric optimization

h_0 ( x,t )

Search directions in the SDP and the monotone SDLCP : generalization and inexact computation

subject to the equality constraints

Monotone Semidefinite Complementarity Problems

h_i (x,t) = 0\, (i = 1,\ldots ,l)