Convergence Analysis of an Inexact Feasible Interior Point Method for Convex Quadratic Programming

In this paper we will discuss two variants of an inexact feasible interior point algorithm for convex quadratic programming. We will consider two different neighbourhoods: a (small) one induced by the use of the Euclidean norm which yields a short-step algorithm and a symmetric one induced by the use of the infinity norm which yields a (practical) long-step algorithm. Both algorithms allow for the Newton equation system to be solved inexactly. For both algorithms we will provide conditions for the level of error acceptable in the Newton equation and establish the worst-case complexity results

N\mathcal{N}IPM-HLSP: An Efficient Interior-Point Method for Hierarchical Least-Squares Programs

Hierarchical least-squares programs with linear constraints (HLSP) are a type of optimization problem very common in robotics. Each priority level contains an objective in least-squares form which is subject to the linear constraints of the higher priority hierarchy levels. Active-set methods (ASM) are a popular choice for solving them. However, they can perform poorly in terms of computational time if there are large changes of the active set. We therefore propose a computationally efficient primal-dual interior-point method (IPM) for HLSP's which is able to maintain constant numbers of solver iterations in these situations. We base our IPM on the null-space method which requires only a single decomposition per Newton iteration instead of two as it is the case for other IPM solvers. After a priority level has converged we compose a set of active constraints judging upon the dual and project lower priority levels into their null-space. We show that the IPM-HLSP can be expressed in least-squares form which avoids the formation of the quadratic Karush-Kuhn-Tucker (KKT) Hessian. Due to our choice of the null-space basis the IPM-HLSP is as fast as the state-of-the-art ASM-HLSP solver for equality only problems.Comment: 17 pages, 7 figure

An infeasible interior point methods for convex quadratic problems

In this paper, we deal with the study and implementation of an infeasible interior point method for convex quadratic problems (CQP). The algorithm uses a Newton step and suitable proximity measure for approximately tracing the central path and guarantees that after one feasibility step, the new iterate is feasible and suciently close to the central path. For its complexity analysis, we reconsider the analysis used by the authors for linear optimisation (LO) and linear complementarity problems (LCP). We show that the algorithm has the best known iteration bound, namely $n log (n+1)$. Finally, to measure the numerical performance of this algorithm, it was tested on convex quadratic and linear problems

An interior-point and decomposition approach to multiple stage stochastic programming

There is no abstract of this repor

Finding a point in the relative interior of a polyhedron

A new initialization or `Phase I' strategy for feasible interior point methods for linear programming is proposed that computes a point on the primal-dual central path associated with the linear program. Provided there exist primal-dual strictly feasible points - an all-pervasive assumption in interior point method theory that implies the existence of the central path - our initial method (Algorithm 1) is globally Q-linearly and asymptotically Q-quadratically convergent, with a provable worst-case iteration complexity bound. When this assumption is not met, the numerical behaviour of Algorithm 1 is highly disappointing, even when the problem is primal-dual feasible. This is due to the presence of implicit equalities, inequality constraints that hold as equalities at all the feasible points. Controlled perturbations of the inequality constraints of the primal-dual problems are introduced - geometrically equivalent to enlarging the primal-dual feasible region and then systematically contracting it back to its initial shape - in order for the perturbed problems to satisfy the assumption. Thus Algorithm 1 can successfully be employed to solve each of the perturbed problems.\ud We show that, when there exist primal-dual strictly feasible points of the original problems, the resulting method, Algorithm 2, finds such a point in a finite number of changes to the perturbation parameters. When implicit equalities are present, but the original problem and its dual are feasible, Algorithm 2 asymptotically detects all the primal-dual implicit equalities and generates a point in the relative interior of the primal-dual feasible set. Algorithm 2 can also asymptotically detect primal-dual infeasibility. Successful numerical experience with Algorithm 2 on linear programs from NETLIB and CUTEr, both with and without any significant preprocessing of the problems, indicates that Algorithm 2 may be used as an algorithmic preprocessor for removing implicit equalities, with theoretical guarantees of convergence

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research