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In this paper, we consider a class of structured fractional minimization problems where the numerator part of the objective is the sum of a convex function and a Lipschitz differentiable (possibly) nonconvex function, while the denominator part is a convex function. By exploiting the structure of the problem, we propose a first-order algorithm, namely, a proximal-gradient-subgradient algorithm with backtracked extrapolation (PGSA_BE) for solving this type of optimization problem. It is worth pointing out that there are a few differences between our backtracked extrapolation and other popular extrapolations used in convex and nonconvex optimization. One of such differences is as follows: if the new iterate obtained from the extrapolated iteration satisfies a backtracking condition, then this new iterate will be replaced by the one generated from the non-extrapolated iteration. We show that any accumulation point of the sequence generated by PGSA_BE is a critical point of the problem regarded. In addition, by assuming that some auxiliary functions satisfy the Kurdyka-Łojasiewicz property, we are able to establish global convergence of the entire sequence, in the case where the denominator is locally Lipschitz differentiable, or its conjugate satisfies the calmness condition. Finally, we present some preliminary numerical results to illustrate the efficiency of PGSA_BE. [ABSTRACT FROM AUTHOR]

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm. [ABSTRACT FROM AUTHOR]

CG_DESCENT is a state-of-the-art algorithm to solve large-scale unconstrained minimization problems. However, research activities on CG_DESCENT in some other scenarios are relatively fewer. In this paper, by combining with the projection method of Solodov and Svaiter, we extend CG_DESCENT to solve large-scale nonlinear convex constrained monotone equations. The proposed method does not require the Jacobian information, even though it does not store any matrix at each iteration. It thus has the potential to solve large-scale non-smooth problems. Under some mild conditions, we show that the proposed method converges globally. Primary numerical results illustrate that the proposed method works quite well. Moreover, we also extend this method to solve the -norm regularized problems to decode a sparse signal in compressive sensing. Performance comparisons show that the proposed method is practical, efficient and competitive with the compared ones. [ABSTRACT FROM AUTHOR]

Abstract: Previous works describing the generalization performance of bipartite ranking algorithms are usually based on the assumption of (0–1) loss or the area under the receiver operating characteristic (ROC) curve. In this paper we go far beyond this classical framework by investigating the generalization performance of bipartite ranking algorithms with convex losses over reproducing kernel Hilbert spaces. Based on the McDiarmid inequality and Rademacher complexity, we establish the upper bound on the generalization error for a bipartite ranking algorithm. The theoretical analysis is different from the previous results on error analysis and shows the attractive uniform convergence property of regularized bipartite ranking algorithms. [Copyright &y& Elsevier]

Abstract: We consider the minimization of a smooth convex function regularized by the composite prior models. This problem is generally difficult to solve even if each subproblem regularized by one prior model is convex and simple. In this paper, we present two algorithms to effectively solve it. First, the original problem is decomposed into multiple simpler subproblems. Then, these subproblems are efficiently solved by existing techniques in parallel. Finally, the result of the original problem is obtained by averaging solutions of subproblems in an iterative framework. The proposed composite splitting algorithms are applied to the compressed MR image reconstruction and low-rank tensor completion. Numerous experiments demonstrate the superior performance of the proposed algorithms in terms of both accuracy and computation complexity. [Copyright &y& Elsevier]

Abstract: In recent years, the issue of energy consumption in parallel and distributed computing systems has attracted a great deal of attention. In response to this, many energy-aware scheduling algorithms have been developed primarily using the dynamic voltage–frequency scaling (DVFS) capability which has been incorporated into recent commodity processors. Majority of these algorithms involve two passes: schedule generation and slack reclamation. The former pass involves the redistribution of tasks among DVFS-enabled processors based on a given cost function that includes makespan and energy consumption, while the latter pass is typically achieved by executing individual tasks with slacks at a lower processor frequency. In this paper, a new slack reclamation algorithm is proposed by approaching the energy reduction problem from a different angle. Firstly, the problem of task slack reclamation by using combinations of processors’ frequencies is formulated. Secondly, several proofs are provided to show that (1) if the working frequency set of processor is assumed to be continues, the optimal energy will be always achieved by using only one frequency, (2) for real processors with a discrete set of working frequencies, the optimal energy is always achieved by using at most two frequencies, and (3) these two frequencies are adjacent/neighbouring when processor energy consumption is a convex function of frequency. Thirdly, a novel algorithm to find the best combination of frequencies to result the optimal energy is presented. The presented algorithm has been evaluated based on results obtained from experiments with three different sets of task graphs: 3000 randomly generated task graphs, and 600 task graphs for two popular applications (Gauss–Jordan and LU decomposition). The results show the superiority of the proposed algorithm in comparison with other techniques. [Copyright &y& Elsevier]

Abstract: This paper presents variable neighborhood search (VNS) for the problem of finding the global minimum of a nonconvex function. The variable neighborhood search, which changes systematically neighborhood structures in the search for finding a better solution, is used to guide a set of standard improvement heuristics. This algorithm was tested on some standard test functions, and successful results were obtained. Its performance was compared with the other algorithms, and observed to be better. [Copyright &y& Elsevier]