This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1 Yf/n); each iteration reduces the duality gap by at least Yf/n. Here n denotes the size of the probl...

The interior trust region algorithm for convex quadratic programming is further developed. This development is motivated by the barrier function and the \center" path-following methods, which create a sequence of primal and dual interior feasible points converging to the optimal solution. At each iteration, the gap between the primal and dual objective values (or the complementary slackness val...

In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in [2]. We show that the corresponding large-update algorithm improves the ite...

We consider a homogeneous self-dual interior point algorithm for solving multistage stochastic linear programs. The algorithm is particularly suitable for the so-called “scenario formulation” of the problem, whose constraint system consists of a large block-diagonal matrix together with a set of sparse nonanticipativity constraints. Due to this structure, the major computational work required b...

In this paper, we study polynomial-time interior-point algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for conic linear programs introduced by self-concordant barrier functions, and develop a precise iterati...

We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semideenite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the rst computational results of interior-point algorithms for approximating the maximum cut semideenite programs with dimen...

Abstract. This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L. It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L-setting is analyzed, but also a more involved L-analysis, q < ∞, is pr...

Symmetricity of an optimal solution of Semi-Definite Program (SDP) with certain symmetricity is discussed based on symmetry property of the central path that is traced by a primal-dual interior-point method. A symmetric SDP is defined by operators for rearranging elements of matrices and vectors, and the solution on the central path is proved to be symmetric. Therefore, it is theoretically guar...

Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J.Peng et al. [14, 15] introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximity for linear optimization (LO) problems. They have also exte...