We propose a generic path-following scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic procedure in use, primal, dual or primal-dual. We show convergence in O( p n) iterations. We verify that the primal, dual and primal-dual algorithms satisfy the local quadratic convergence property. The method can be applied to solve the linear programming problem (with a feasible start) and to compute the analytic center of a bounded polytope. The generic path-following scheme easily extends to the logarithmic penalty barrier approach. Keywords Interior point method, method of centers, path-following, linear programming. This work has been completed with the support from the Swiss National Foundation for Scientific Research, grant 12-34002.92. 1 Introduction Shortly after Karmarkar's s...

In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter. Key Words: linear programming, interior point methods, duality theory, parametric analysis, sensitivity analysis, primal-dual algorithm, implementation. Contents 1 Introduction 3 2 A new approach to the theory of linear programming 4 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Problem definition and assumptions : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 The logarithmic barrier approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.1 The barrier problem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.2 Minimizers of the barrier function : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.3 The central path : : : : : : : : : ...

Recently, we have extended SDP by adding a quadratic term in the objective function and give a potential reduction algorithm using NT directions. This paper presents a predictor-corrector algorithm using both Dikin-type and Newton centering steps and studies properties of Dikin-type step.

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed an efficient primal-dual infeasible interior-point algorithm with full Newton steps for linear optimization problems. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates in at most O ( n log n ε steps either by finding an ε-solution or by determining that the primal-dual problem pair has no optimal solution with vanishing duality gap satisfying a condition in terms of ζ.

Mizuno–Todd–Ye algorithm in a larger neighborhood

A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with O ( √ nL)-iteration complexity ∗