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We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward–backward algorithms with semi-algebraic problem’s data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.

Wind energy is important for achieving net-zero greenhouse gas emissions but also contributes to global bat mortality. Current strategies to minimize bat mortality due to collision with wind-turbine blades fall broadly into two categories: curtailment (limiting turbine operation during high-risk periods) and deterrence (discouraging bat activity near turbines). Recently, there has been interest in combining these strategies to achieve greater reductions in bat fatalities than either strategy might achieve in isolation. To investigate the effectiveness of combining curtailment with ultrasonic deterrent minimization strategies, we deployed six ultrasonic deterrents at nacelle height on 16 experimental turbines at Avangrid Renewables’ Blue Creek Wind Energy Facility. We rotated between four conditions (normal operations, curtailment only, deterrent only, curtailment and deterrent) randomly assigned to four wind turbines each night between 15 June and 3 October 2017. We found that bat mortality at wind turbines was independent of wind speed. The effectiveness of ultrasonic acoustic deterrents varied between high-frequency-calling species (eastern red bats) and low-frequency-calling species (hoary bats, silver-haired bats, and big brown bats). When deterrents were active, mortality was twice as high for eastern red bats compared to the control. Conversely, deterrents had a weak dampening effect on bat mortality for low-frequency species. We found no additive effects on mortality reduction for turbines operating both curtailment and deterrents compared to either approach in isolation. Our findings suggest that ultrasonic acoustic deterrents may not be effective for both high and low frequency echolocating bats. The increase in fatalities of eastern red bats is alarming and underscores the importance of considering site- and species-specific effects of minimization solutions.

Solvents are important in most industrial and domestic applications. The impact of solvent losses and emissions drives efforts to minimise them or to avoid them completely. Since the 1990s, this has become a major focus of green chemistry, giving rise to the idea of the ‘green’ solvent. This concept has generated a substantial chemical literature and has led to the development of so-called neoteric solvents. A critical overview of published material establishes that few new materials have yet found widespread use as solvents. The search for less-impacting solvents is inefficient if carried out without due regard, even at the research stage, to the particular circumstances under which solvents are to be used on the industrial scale. Wider sustainability questions, particularly the use of non-fossil sources of organic carbon in solvent manufacture, are more important than intrinsic ‘greenness’. While solvency is universal, a universal solvent, an alkahest, is an unattainable ideal.

As a convex relaxation of the rank minimization model, the nuclear norm minimization (NNM) problem has been attracting significant research interest in recent years. The standard NNM regularizes each singular value equally, composing an easily calculated convex norm. However, this restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, which adaptively assigns weights on different singular values. As the key step of solving general WNNM models, the theoretical properties of the weighted nuclear norm proximal (WNNP) operator are investigated. Albeit nonconvex, we prove that WNNP is equivalent to a standard quadratic programming problem with linear constrains, which facilitates solving the original problem with off-the-shelf convex optimization solvers. In particular, when the weights are sorted in a non-descending order, its optimal solution can be easily obtained in closed-form. With WNNP, the solving strategies for multiple extensions of WNNM, including robust PCA and matrix completion, can be readily constructed under the alternating direction method of multipliers paradigm. Furthermore, inspired by the reweighted sparse coding scheme, we present an automatic weight setting method, which greatly facilitates the practical implementation of WNNM. The proposed WNNM methods achieve state-of-the-art performance in typical low level vision tasks, including image denoising, background subtraction and image inpainting.

This paper presents a review of the most common power converters and torque ripple minimisation approaches for switched reluctance motors (SRMs). Unlike conventional three-phase AC motors, namely squirrel cage induction motors and permanent magnet synchronous motors, which require a typical three-phase inverter for operation, the switched reluctance motor requires a different topology power converter for reliable and efficient operation. In addition, due to the non-linear, discrete nature of SRM torque production, torque ripple is severely pronounced, which is undesirable in servo applications like electric vehicles. Hence, deploying a proper torque control function for smooth and quiet motor operation is crucial. This paper sheds light over the most popular SRM power converters as well as torque ripple minimisation methods, and it suggests an optimal SRM drive topology for EV applications.

Electrification of transport and the deployment of plug-in electric vehicles (PEVs) shift emissions from tail-pipes to bulk power systems (BPSs). Coordinated distributed energy resources and PEV charging can mitigate the impact of this shift. This study presents an analysis of photovoltaic (PV) solar parking lots that address this benefit. Real-world charging data, solar data, and electricity tariffs are used to determine the microgrid system that minimises the cost of retrofitting an existing parking lot with PV and PEV infrastructure coupled. The result is a load scheduling algorithm that takes into account tariffs and insolation to reduce costs while ensuring customer satisfaction. The techno-economic feasibility of PV infrastructure in the microgrid is determined by minimising the net present cost (NPC) in two case studies: Victoria, BC, and Los Angeles, CA. Relatively low solar irradiation and electricity prices make it economically infeasible to install solar panels in Victoria even though the operational costs are reduced by 11%. In Los Angeles, high time-of-use prices, together with abundant solar radiation, make PV retrofitting economically feasible with any array capacity. At the current solar infrastructure price, coordinated charging in this region yields 8–16% savings on NPC and smaller feeder size requirements with greater load growth opportunities.

The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems (the code can be downloaded from http://www.columbia.edu/~sm2756/FPCA.htm for non-commercial use). Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10 −5 in about 3 min by sampling only 20% of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms.

In this paper we study general l p regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the l p minimization problems. We extend some existing iterative reweighted l 1 ( IRL 1 ) and l 2 ( IRL 2 ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous ϵ -approximation to ‖ x ‖ p p . Using this result, we develop new IRL 1 methods for the l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter ϵ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that ϵ be dynamically updated and approach zero. Our computational results demonstrate that the new IRL 1 method and the new variants generally outperform the existing IRL 1 methods (Chen and Zhou in 2012 ; Foucart and Lai in Appl Comput Harmon Anal 26:395–407, 2009 ).

In the period 1991-2015, algorithmic advances in Mixed Integer Optimization (MIO) coupled with hardware improvements have resulted in an astonishing 450 billion factor speedup in solving MIO problems. We present a MIO approach for solving the classical best subset selection problem of choosing k out of p features in linear regression given n observations. We develop a discrete extension of modern first-order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with n in the 1000s and p in the 100s in minutes to provable optimality, and finds near optimal solutions for n in the 100s and p in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than Lasso and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.

Financial Management for Health-System Pharmacists, 2nd edition, serves as a guidebook to support the management of enterprise pharmacy finance across business and care continuums. The 2nd edition engages the reader with a mix of chapters, some new to this edition, along with a trove of new health-system pharmacy financial business cases. As leaders look to transform their organizations, the principles and practices provided give the reader the knowledge and guidance to craft a new path forward as they look to improve the provision of pharmacy and patient-care services.