The Problem of Projecting the Origin of Euclidean Space onto the Convex Polyhedron

Abstract

Abstract. This paper is aimed at presenting a systematic survey of the existing now different formulations for the problem of projection of the origin of the Euclidean space onto the convex polyhedron (PPOCP). In the present paper, there are investigated the reduction of the projection program to the problems of quadratic programming, maximin, linear complementarity, and nonnegative least squares. Such reduction justifies the opportunity of utilizing a much more broad spectrum of powerful tools of mathematical programming for solving PPOCP. The paper's goal is to draw the attention of a wide range of research at the different formulations of the projection problem, which remain largely unknown due to the fact that the papers (addressing the subject of concern) are published even though on the adjacent, but other topics, or only in the conference proceedings.Keywords: projection, convex polyhedron, quadratic programming, maximin problem, complementarity problem, nonnegative least squares problem MSC 2010: 90C30, 90C25, 90C20, 90C33, 65K05

Adaptive Conditional Gradient Method

“…Let the index m be the smallest of them. According to Lemma 2.2, the condition ( 9) or (10) is then fulfilled for i m = î. Due to the condition (e1) of the lemma, ε m+1 = ε m .…”

“…Now, let i * = 1. This case corresponds to the fact that the condition (9) [or (10)] (applied in the rule of calculating the step length) was not fulfilled for η i * −1 . Due to Lemma 2.2, we then obtain 1) .…”

“…Due to Lemma 2.2, if i k = î, then for going to the next-(k + 1)th-iteration it is expedient to leave the unchanged value of the normalization parameter, i.e., to put ε k+1 = ε k . During the process of dropping the step size, let the checked condition (9) [or condition (10)] be fulfilled for i k > î. Then, in accordance with (11), there should be increased current value of the parameter for the ε-normalization of the descent direction, for instance, as follows:…”

“…Besides, it is well known from the optimization literature that Frank-Wolfe-type methods are very useful for solving the problem of projecting the origin of the Euclidean space onto a convex polyhedron (see, for instance, [9]). Therefore, it will be especially interesting to study in the future the application of ACGM to solve the problems of projecting onto the convex polyhedron and computing the distance between the convex polyhedra learned, for example, in [10][11][12].…”

Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets

“…Minimum norm problems for the case of polytopes have been well studied in the literature from both theoretical and numerical point of view; see e.g., [21,30,13] and the references therein. The most suitable algorithm for solving (1.1) is perhaps the one suggested by Gilbert [10].…”