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Heinz H. Bauschke

Publications and source records attributed to Heinz H. Bauschke.

At least 73 records · Page 4Linked to original sources
arXiv2019

Numerical explorations of feasibility algorithms for finding points in the intersection of finite sets

Projection methods are popular algorithms for iteratively solving feasibility problems in Euclidean or even Hilbert spaces. They employ (selections of) nearest point mappings to generate sequences that are designed to approximate a point in the intersection of a collection of constraint sets. Theoretical properties of projection methods are fairly well understood when the underlying constraint sets are convex; however, convergence results for the nonconvex case are more complicated and typically only local. In this paper, we explore the perhaps simplest instance of a feasibility algorithm, namely when each constraint set consists of only finitely many points. We numerically investigate four constellations: either few or many constraint sets, with either few or many points. Each constellation is tackled by four popular projection methods each of which features a tuning parameter. We examine the behaviour for a single and for a multitude of orbits, and we also consider local and global behaviour. Our findings demonstrate the importance of the choice of the algorithm and that of the tuning parameter.

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arXiv2018

On circumcenter mappings induced by nonexpansive operators

We introduce the circumcenter mapping induced by a set of (usually nonexpansive) operators. One prominent example of a circumcenter mapping is the celebrated Douglas--Rachford splitting operator. Our study is motivated by the Circumcentered--Douglas--Rachford method recently introduced by Behling, Bello Cruz, and Santos in order to accelerate the Douglas--Rachford method for solving certain classes of feasibility problems. We systematically explore the properness of the circumcenter mapping induced by reflectors or projectors. Numerous examples are presented. We also present a version of Browder's demiclosedness principle for circumcenter mappings.

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arXiv2018

Regularizing with Bregman-Moreau envelopes

Moreau's seminal paper, introducing what is now called the Moreau envelope and the proximity operator (also known as the proximal mapping), appeared in 1965. The Moreau envelope of a given convex function provides a regularized version which has additional desirable properties such as differentiability and full domain. Fifty years ago, Attouch proposed using the Moreau envelope for regularization. Since then, this branch of convex analysis has developed in many fruitful directions. In 1967, Bregman introduced what is nowadays known as the Bregman distance as a measure of discrepancy between two points generalizing the square of the Euclidean distance. Proximity operators based on the Bregman distance have become a topic of significant research as they are useful in the algorithmic solution of optimization problems. More recently, in 2012, Kan and Song studied regularization aspects of the left Bregman-Moreau envelope even for nonconvex functions. In this paper, we complement previous works by analyzing the left and right Bregman-Moreau envelopes and by providing additional asymptotic results. Several examples are provided.

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arXiv2018

On sums and convex combinations of projectors onto convex sets

The projector onto the Minkowski sum of closed convex sets is generally not equal to the sum of individual projectors. In this work, we provide a complete answer to the question of characterizing the instances where such an equality holds. Our results unify and extend the case of linear subspaces and Zarantonello's results for projectors onto cones. A detailed analysis in the case of convex combinations is also carried out. We establish the partial sum property for projectors onto convex cones, and we also present various examples as well as a detailed analysis in the univariate case.

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arXiv2018

On the minimal displacement vector of compositions and convex combinations of nonexpansive mappings

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. Five years ago, it was shown that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis-Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results.

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arXiv2018

Proximal Averages for Minimization of Entropy Functionals

In their 2016 article `Meetings with Lambert $\mathcal{W}$ and Other Special Functions in Optimization and Analysis', Borwein and Lindstrom considered the minimization of an entropy functional utilizing the weighted average of the negative Boltzmann--Shannon entropy $x\mapsto x\log x -x$ and the energy $x \mapsto x^2/2$; the solution employed the Fenchel-Moreau conjugate of the sum. However, in place of a traditional arithmetic average of two convex functions, it is also, perhaps even more, natural to use their proximal average. We explain the advantages and illustrate them by computing the analogous proximal averages for the negative entropy and energy. We use the proximal averages to solve entropy functional minimization problems similar to those considered by Borwein and Lindstrom, illustrating the benefits of using a true homotopy. Through experimentation, we discover computational barriers to obtaining solutions when computing with proximal averages, and we demonstrate a method which appears to remedy them.

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arXiv2018

On circumcenters of finite sets in Hilbert spaces

A well-known object in classical Euclidean geometry is the circumcenter of a triangle, i.e., the point that is equidistant from all vertices. The purpose of this paper is to provide a systematic study of the circumcenter of sets containing finitely many points in Hilbert space. This is motivated by recent works of Behling, Bello Cruz, and Santos on accelerated versions of the Douglas--Rachford method. We present basic results and properties of the circumcenter. Several examples are provided to illustrate the tightness of various assumptions.

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arXiv2018

On the asymptotic behaviour of the Aragon Artacho-Campoy algorithm

Aragón Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone operators. In this paper, we complete this analysis by demonstrating that the underlying curve converges to the nearest zero of the sum of the two operators. We also provide a new interpretation of the underlying operators in terms of the resolvent and the proximal average.

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arXiv2018

Intriguing maximally monotone operators derived from nonsunny nonexpansive retractions

Monotone operator theory and fixed point theory for nonexpansive mappings are central areas in modern nonlinear analysis and optimization. Although these areas are fairly well developed, almost all examples published are based on subdifferential operators, linear relations, or combinations thereof. In this paper, we construct an intriguing maximally monotone operator induced by a certain nonexpansive retraction. We analyze this operator, which does not appear to be assembled from subdifferential operators or linear relations, in some detail. Particular emphasis is placed on duality and strong monotonicity.

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arXiv2018

The Douglas--Rachford algorithm for a hyperplane and a doubleton

The Douglas--Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which --- somewhat surprisingly --- depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.

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arXiv2018

Projecting onto the intersection of a cone and a sphere

The projection onto the intersection of sets generally does not allow for a closed form even when the individual projection operators have explicit descriptions. In this work, we systematically analyze the projection onto the intersection of a cone with either a ball or a sphere. Several cases are provided where the projector is available in closed form. Various examples based on finitely generated cones, the Lorentz cone, and the the cone of positive semidefinite matrices are presented. The usefulness of our formulae is illustrated by numerical experiments for determining copositivity of real symmetric matrices.

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arXiv2018

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

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 Aragón 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.

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arXiv2017

The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings

Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., the have or "almost have" fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover, we present various examples illustrating that our bounds are sharp.

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arXiv2017

A class of multi-marginal c-cyclically monotone sets with explicit c-splitting potentials

Multi-marginal optimal transport plans are concentrated on c-splitting sets. It is known that, similar to the two-marginal case, c-splitting sets are c-cyclically monotone. Within a suitable framework, the converse implication was very recently established by Griessler. However, for an arbitrary cost c, given a multi-marginal c-cyclically monotone set, the question whether there exists an analogous explicit construction to the one from the two-marginal case of c-splitting potentials is still open. When the margins are one-dimensional and the cost belongs to a certain class, Carlier proved that the two-marginal projections of a c-splitting set are monotone. For arbitrary products of sets equipped with cost functions which are sums of two-marginal costs, we show that the two-marginal monotonicity condition is a sufficient condition which does give rise to an explicit construction of c-splitting potentials. Our condition is, in principle, easier to verify than the one of multi-marginal c-cyclic monotonicity. Various examples illustrate our results. We show that, in general, our condition is sufficient; however, it is not necessary. On the other hand, we conclude that when the margins are one-dimensional equipped with classical cost functions, our condition is a characterization of c-splitting sets and extends classical convex analysis.

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arXiv2016

The resolvent order: a unification of the orders by Zarantonello, by Loewner, and by Moreau

We introduce and investigate the resolvent order, which is a binary relation on the set of firmly nonexpansive mappings. It unifies well-known orders introduced by Loewner (for positive semidefinite matrices) and by Zarantonello (for projectors onto convex cones). A connection with Moreau's order of convex functions is also presented. We also construct partial orders on (quotient sets of) proximal mappings and convex functions. Various examples illustrate our results.

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arXiv2016

On the Douglas-Rachford algorithm

The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. More than a decade ago, however, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and it is weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Several examples illustrate the results.

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arXiv2016

Affine nonexpansive operators, Attouch-Théra duality and the Douglas-Rachford algorithm

The Douglas-Rachford splitting algorithm was originally proposed in 1956 to solve a system of linear equations arising from the discretization of a partial differential equation. In 1979, Lions and Mercier brought forward a very powerful extension of this method suitable to solve optimization problems. In this paper, we revisit the original affine setting. We provide a powerful convergence result for finding a zero of the sum of two maximally monotone affine relations. As a by product of our analysis, we obtain results concerning the convergence of iterates of affine nonexpansive mappings as well as Attouch-Théra duality. Numerous examples are presented.

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