Showing papers by "Regina S. Burachik published in 2018"
TL;DR: In this article, the authors consider the minimum energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator.
Abstract: We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i) Dykstra’s algorithm, (ii) the Douglas–Rachford (DR) method and (iii) the Aragon Artacho–Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.
10 citations
TL;DR: In this article, the Bregman distance is reduced to the gradient of a convex function, and the authors study the properties of this new distance and establish its continuity properties.
Abstract: Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.
7 citations
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TL;DR: In this article, the Steklov regularization is used to compute a global minimizer of univariate coercive functions, and a trajectory is constructed on the surface generated by the regularization parameter.
Abstract: We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions First, we show that the Steklov regularization convexifies a given univariate coercive function Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function For monic quartic polynomials, we prove that this trajectory does generate a global minimizer In the process, we derive some properties of quartic polynomials Comparisons are made with a previous approach which uses a quadratic regularization function We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function