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.
Chen S, Min J, Teng J, et al. Inventory and shelf-space optimization for fresh produce with expiration date under freshness-and-stock-dependent demand rate[J]. Journal of the Operational Research Society, 2016, 67(6): 884-896.
Abdelhak Mezghiche · Mustapha Moulai · Lotfi Tadj. Model Predictive Control of a Forecasting Production System with Deteriorating Items. 2015.
Xiaoyan Xu · Yanan Ji · Yiwen Bian · Yanhong Sun. Service outsourcing under co-opetition and information asymmetry. 2016.
Minghui Xu · Xiaode Zuo. An optimal dynamic advertising model with inverse inventory sensitive demand effect for deteriorating items. 2016.
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.
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 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
verication theorem is provided to justify their optimality. Finally, a numerical
example is provided to illustrate the results.
In this paper, a brief introduction of the nonlinear filtering problems and a review of the quasi-implicit Euler method are presented. The major contribution of this paper is that we propose a nonnegativity-preserving algorithm of Yau-Yau method for solving high-dimensional nonlinear filtering problems by applying quasi-implicit Euler method with discrete sine transform. Furthermore, our algorithms are directly applicable on the compact difference schemes, so that the number of spatial points can be substantially reduced and retain the same accuracy. Numerical results indicate that the proposed algorithm is capable of solving up to six-dimensional nonlinear filtering problems efficiently and accurately.
This paper studies the eect of risk-aversion in the competitive
newsvendor game. Multiple newsvendors with risk-averse preferences face a
random demand and the demand is allocated proportionally to their inventory
levels. Each newsvendor aims to maximize his expected utility instead of his
expected prot. Assuming a general form of risk-averse utility function, we
prove that there exists a pure Nash equilibrium in this game, and it is also
unique under certain conditions. We nd that the order quantity of each
newsvendor is decreasing in the degree of risk-aversion and increasing in the
initial wealth. Newsvendors with moderate preferences of risk-aversion make
more prots compared with the risk-neutral situation. We also discuss the joint
eect of risk-aversion and competition. If the eect of risk-aversion is strong
enough to dominate the eect of competition, the total inventory level under
competition will be lower than that under centralized decision-making.
Nonconvex optimization arises in many areas of computational science and engineering.
However, most nonconvex optimization algorithms are only known to have local
convergence or subsequence convergence properties. In this paper, we propose an algorithm
for nonconvex optimization and establish its global convergence (of the whole sequence) to
a critical point. In addition, we give its asymptotic convergence rate and numerically demonstrate
its efficiency. In our algorithm, the variables of the underlying problem are either treated
as one block or multiple disjoint blocks. It is assumed that each non-differentiable component
of the objective function, or each constraint, applies only to one block of variables. The
differentiable components of the objective function, however, can involve multiple blocks of
variables together. Our algorithm updates one block of variables at a time by minimizing a
certain prox-linear surrogate, along with an extrapolation to accelerate its convergence. The
order of update can be either deterministically cyclic or randomly shuffled for each cycle. In
fact, our convergence analysis only needs that each block be updated at least once in every
fixed number of iterations. We show its global convergence (of the whole sequence) to a
critical point under fairly loose conditions including, in particular, the Kurdyka–Łojasiewicz
condition, which is satisfied by a broad class of nonconvex/nonsmooth applications. These
results, of course, remain valid when the underlying problem is convex.We apply our convergence
results to the coordinate descent iteration for non-convex regularized linear regression,
as well as amodified rank-one residue iteration for nonnegative matrix factorization. We show that both applications have global convergence. Numerically,we tested our algorithm on nonnegative matrix and tensor factorization problems, where random shuffling clearly improves
the chance to avoid low-quality local solutions.
This paper is concerned with an optimal strategy for simultaneously trading
of a pair of stocks. The idea of pairs trading is to monitor their price movements and
compare their relative strength over time. A pairs trade is triggered by their prices
divergence and consists of a pair of positions to short the strong stock and to long
the weak one. Such a strategy bets on the reversal of their price strengths. From
the viewpoint of technical tractability, typical pairs-trading models usually assume a
difference of the stock prices satisfies a mean-reversion equation. In this paper, we
consider the optimal pairs-trading problem by allowing the stock prices to follow
general geometric Brownian motions. The objective is to trade the pairs over time to
maximize an overall return with a fixed commission cost for each transaction. The
optimal policy is characterized by threshold curves obtained by solving the associated
HJB equations. Numerical examples are included to demonstrate the dependence of
our trading rules on various parameters and to illustrate how to implement the results