We discuss the convergence of line search methods for minimization. We explain how Newton’s method and the BFGS method can fail even if the restrictions of the objective function to the search lines are strictly convex functions, the level sets of the objective functions are compact, the line searches are exact and the Wolfe conditions … Read more
We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in … Read more
Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space corresponding … Read more
We propose a novel solution approach for polynomial programming problems with equality constraints. By means of a generic transformation, we show that solution schemes for the (typically simpler) problem without equalities can be used to address the problem with equalities. In particular, we propose new solution schemes for mixed binary programs, pure 0-1 quadratic programs, … Read more
This paper extends the full convergence of the steepest descent algorithm with a generalized Armijo search and a proximal regularization to solve quasiconvex minimization problems defined on complete Riemannian manifolds. Previous convergence results are obtained as particular cases of our approach and some examples in non Euclidian spaces are given. CitationJ. Math. Anal. Appl. 341 … Read more
This work addresses the problem of regularized linear least squares (RLS) with non-quadratic separable regularization. Despite being frequently deployed in many applications, the RLS problem is often hard to solve using standard iterative methods. In a recent work [10], a new iterative method called Parallel Coordinate Descent (PCD) was devised. We provide herein a convergence … Read more
Generating set Search Methods (GSS), a class of derivative-free methods for unconstrained optimisation, are in general robust but converge slowly. It has been shown that the performance of these methods can be enhanced by utilising accumulated information about the objective function as well as a priori knowledge such as partial separability. This paper introduces a … Read more
We consider the problem of minimizing a univariate, real-valued function f on an interval [a,b]. When f is a polynomial, we review how this problem may be reformulated as a semidefinite programming (SDP) problem, and review how to extract all global minimizers from the solution of the SDP problem. For general f, we approximate the … 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
The minimization of an objective function over a constraint set can often be simplified if the “active manifold” of the constraints set can be correctly identified. In this work we present a simple subproblem, which can be used inside of any (convergent) optimization algorithm, that will identify the active manifold of a “prox-regular partly smooth” … Read more