This paper considers regularized block multiconvex optimization, where the feasible set and objective function are generally nonconvex but convex in each block of variables. It also accepts nonconvex blocks and requires these blocks to be updated by proximal minimization. We review some interesting applications and propose a generalized block coordinate descent method. Under certain conditions, we show that any limit point satisfies the Nash equilibrium conditions. Furthermore, we establish global convergence and estimate the asymptotic convergence rate of the method by assuming a property based on the Kurdyka-Lojasiewicz inequality. The proposed algorithms are tested on nonnegative matrix and tensor factorization, as well as matrix and tensor recovery from incomplete observations. The tests include synthetic data and hyperspectral data, as well as image sets from the CBCL and ORL databases. Compared to the existing state-of-the-art algorithms, the proposed algorithms demonstrate superior performance in both speed and solution quality. The MATLAB code of nonnegative matrix/tensor decomposition and completion, along with a few demos, are accessible from the authors' homepages.

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This paper is concerned with a mean-reversion trading rule. In contrast to most market models treated in the literature, the underlying market is solely determined by a two-state Markov chain. The major advantage of such Markov chain model is its striking simplicity and yet its capability of capturing various market movements. The purpose of this paper is to study an optimal trading rule under such a model. The objective of the problem under consideration is to nd a sequence stopping (buying and selling) times so as to maximize an expected return. Under some suitable conditions, explicit solutions to the associated HJ equations (variational inequalities) are obtained. The optimal stopping times are given in terms of a set of threshold levels. A verication theorem is provided to justify their optimality. Finally, a numerical example is provided to illustrate the results.

It is well known that the nonlinear filter has important applications in military, engineering and commercial industries. In this paper, we propose efficient and accurate numerical algorithms for the realization of the Yau-Yau method for solving nonlinear filtering problems by using finite difference schemes. The Yau-Yau method reduces the nonlinear filtering problem to the initial-value problem of Kolmogorov equations. We first solve this problem by the implicit Euler method, which is stable in most cases, but costly. Then, we propose a quasi-implicit Euler method which is feasible for acceleration by fast Fourier transformations. Furthermore, we propose a superposition technique which enables us to deal with the nonlinear filtering problem in an off-time process and thus, save a large amount of computational cost. Next, we prove that the numerical solutions of Kolmogorov equations by our schemes are always nonnegative in each iteration. Consequently, our iterative process preserves the probability density functions. In addition, we prove convergence of our schemes under some mild conditions. Numerical results show that the proposed algorithms are efficient and promising.

This paper studies the inventory management problem of dual channels operated by one vendor. Demands of dual channels are inventory-level-dependent.We propose a multi-period stochastic dynamic programming model which shows that under mild conditions, the myopic inventory policy is optimal for the infinite horizon problem. To investigate the importance of capturing demand dependency on inventory levels, we consider a heuristic where the vendor ignores demand dependency on inventory levels, and compare the optimal inventory levels with those recommended by the heuristic. Through numerical examples, we show that the vendor may order less for dual channels than those recommended by the heuristic, and the difference between the inventory levels in the two cases can be so large that the demand dependency on inventory levels cannot be neglected. In the end, we numerically examine the impact of different ways to treat unmet demand and obtain some managerial insights.

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In this paper, we study the Kurdyka–Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is 1/2 . The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is 1/2 . This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with l_1 regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent,our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.

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In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Holder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for many important iterative methods in solving broad mathematical problems such as convex feasibility problems and variational inequality problems. These include Krasnoselskii–Mann iterations, the cyclic projection algorithm, forward-backward splitting and the Douglas–Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the Holder regularity properties are automatically satisfied, from which sublinear convergence follows.