We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

This paper is a follow-up of the work [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)] where an NCP-function and a descent method were proposed for the nonlinear complementarity problem. An unconstrained reformulation was formulated due to a merit function based on the proposed NCP-function. We continue to explore properties of the merit function in this paper. In particular, we show that the gradient of the merit function is globally Lipschitz continuous which is important from computational aspect. Moreover, we show that the merit function is <i>SC</i> ¹ function which means it is continuously differentiable and its gradient is semismooth. On the other hand, we provide an alternative proof, which uses the new properties of the merit function, for the convergence result of the descent method considered in [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)].

The surveys in the field of nonlinear filtering (NLF) are enumerous. Most of them are application-oriented and served as the tutorials for the practioners. The local approaches, including Kalman filter and its invariants, have already been studied from various point of views, due to its off-the-shelf implementation and wide applications. However, it cannot give good estimation of the states in highly nonlinear system or with non-Gaussian initial conditional density functions. Moreover, while the local methods only approximate the mean and variance, the global ones seek the way to directly obtain the conditional density function of the states. Consequently, all the statistical information is acquired. In this survey, we shall briefly go through the local approaches and put emphases on the existing three major global approaches: finite-dimensional NLF, sequential Monte Carlo methods (particle filter) and the {Yau-Yau's} on- and off-line solver of Duncan-Mortensen-Zakai's equation {\cite{YY_08}}. The discussions are mainly from the mathematical point of view.

This paper focuses on pricing and vertical cooperative advertising decisions in a two-tier supply chain. Using a Stackelberg game model where the manufacturer acts as the game leader and the retailer acts as the game follower, we obtain closed-form equilibrium solution and explicitly show how pricing and advertising decisions are made. When market demand decreases exponentially with respect to the retail price and increases with respect to national and local advertising expenditures in an additive way, the manufacturer benefits from providing percentage reimbursement for the retailer’s local advertising expenditure when demand price elasticity is large enough. Whether the manufacturer benefits from cooperative advertising is also closely related to supply chain member’s relative advertising efficiency. In the decision for adopting coop advertising strategy, it is critical for the manufacturer to identify how market demand depends on national and local advertisements. The findings from this research can enhance our understanding of cooperative advertising decisions in a two-tier supply chain with price-dependent demand.

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.