By introducing redundant Klee-Minty examples, we have previously shown that the central path can be bent along the edges of the Klee-Minty cubes, thus having $2^n-2$ sharp turns in dimension $n$. In those constructions the redundant hyperplanes were placed parallel with the facets active at the optimal solution. In this paper we present a simpler … Read more
In this work we present a family of variable metric interior proximal methods for solving optimization problems under nonnegativity constraints. We define two algorithms, in the inexact and exact forms. The kernels are metrics generated by diagonal matrices in each iteration and the regularization parameters are conveniently chosen to force the iterates to be interior … Read more
We consider the question of global convergence of iterative methods for nonlinear programming problems. Traditionally, penalty functions have been used to enforce global convergence. In this paper we review a recent alternative, so-called filter methods. Instead of combing the objective and constraint violation into a single function, filter methods view nonlinear optimization as a biobjective … Read more
We present a new primal-dual algorithm for convex quadratic multicriteria optimization. The algorithm is able to adaptively refine the approximation to the set of efficient points by way of a warm-start interior-point scalarization approach. Results of this algorithm when applied on a three-criteria real-world power plant optimization problem are reported, thereby illustrating the feasibility of … Read more
A first order affine scaling method and two $m$th order affine scaling methods for solving monotone linear complementarity problems (LCP) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has $O(nL^2(\log nL^2)(\log\log nL^2))$ iteration complexity. If the LCP admits a strict complementary solution then both … Read more
We describe a method of generating a warm-start point for interior point methods in the context of stochastic programming. Our approach exploits the structural information of the stochastic problem so that it can be seen as a structure-exploiting initial point generator. We solve a small-scale version of the problem corresponding to a reduced event tree … Read more
We consider a family of linear optimization problems over the n-dimensional Klee-Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2^n-2 … Read more
We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify … Read more
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 $L^p$. It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier $L^\infty$-setting is analyzed, but also a more involved $L^q$-analysis, $q
In this paper we propose two new nonsymmetric primal-dual potential-reduction methods for conic problems. Both methods are based on {\em primal-dual lifting}. This procedure allows to construct a strictly feasible primal-dual pair linked by an exact {\em scaling} relation even if the cones are not symmetric. It is important that all necessary elements of our … Read more