Description:
We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under Lipschitz continuity assumption. The approach is then applied to computing the proximity operator of the sum of weakly convex functions, and particularly to finding the best approximation to the intersection of convex sets.
Description:
This paper aims to maximize the energy efficiency of a downlink cloud radio access network (C-RAN). Here, data is transferred from a baseband unit in the core network to several remote radio heads via a set of edge routers over capacity-limited fronthaul links. The remote radio heads then send the received signals to their users via radio access links. Both data sharing and compression-based strategies are considered for fronthaul data transfer. New mixed-integer nonlinear problems are formulated, in which the ratio of network throughput and total power consumption is maximized. These challenging problem formulations include practical constraints on routing, predefined minimum data rates, fronthaul capacity, and maximum remote radio head transmit power. By employing the successive convex quadratic programming, iterative algorithms are proposed with guaranteed convergence to the Fritz John solutions of the formulated problems. Significantly, each iteration of the proposed algorithms solves only one simple convex program. Numerical examples with practical parameters confirm that the proposed joint optimization designs markedly improve the C-RAN's energy efficiency compared to benchmark schemes. They also show that the fronthaul data-sharing strategy outperforms its compression-based counterpart in terms of energy efficiency, in both single-hop and multi-hop network scenarios.
Description:
This work aims to maximize the energy efficiency of a downlink cloud radio access network (C-RAN), where data is transferred from a baseband unit in the core network to several remote radio heads via a set of edge routers over capacity-limited fronthaul links. The remote radio heads then send the received signals to their users via radio access links. We formulate a new mixed-integer nonlinear problem in which the ratio of network throughput and total power consumption is maximized. This challenging problem formulation includes practical constraints on routing, predefined minimum data rates, fronthaul capacity and maximum RRH transmit power. By employing the successive convex quadratic programming framework, an iterative algorithm is proposed with guaranteed convergence to a Fritz John solution of the formulated problem. Significantly, each iteration of the proposed algorithm solves only one simple convex program. Numerical examples with practical parameters confirm that the proposed joint optimization design markedly improves the C-RAN's energy efficiency compared to benchmark schemes.
Description:
Delamination is a typical failure mode of composite materials caused by weak bonding. It arises when a crack initiates and propagates under a destructive loading. Given the physical law characterizing the properties of the interlayer adhesive between the bonded bodies, we consider the problem of computing the propagation of the crack front and the stress field along the contact boundary. This leads to a hemivariational inequality, which after discretization by finite elements we solve by a nonconvex bundle method, where upper- C 1 criteria have to be minimized. As this is in contrast with other classes of mechanical problems with non-monotone friction laws and in other applied fields, where criteria are typically lower- C 1 , we propose a bundle method suited for both types of nonsmoothness. We prove its global convergence in the sense of subsequences and test it on a typical delamination problem of material sciences.
Description:
The Douglas–Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater’s condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas–Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection–projection.
Description:
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.
Description:
The Douglas–Rachford algorithm is a simple yet effective method for solving convex feasibility problems. However, if the underlying constraints are inconsistent, then the convergence theory is incomplete. We provide convergence results when one constraint is an affine subspace. As a consequence, we extend a result by Spingarn from halfspaces to general closed convex sets admitting least-squares solutions.
Description:
In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive . At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions.