In this paper we consider a simple estimation problem on the special orthogonal group $SO(n)$ and indicate a possible way to construct approximate filters which is much in the same spirit of the “wide sense” approach to linear filtering theory. Our interest is mainly motivated by applications to computer vision.

The goal of this paper is to study the entire class of linear second order multi-point methods. We characterize, as a three parameter family, those methods with good numerical properties. We will examine the error analysis of the class of second order methods and will study in some detail the statistics of switching between two methods. We characterize the average value obtained by switching and construct the covariance matrix. Two examples are done in some detail.

We propose a randomized search method called Stochastic Model Reference Adaptive Search (SMRAS) for solving stochastic optimization problems in situations where the objective functions cannot be evaluated exactly, but can be estimated with some noise (or uncertainty), e.g., via simulation. The method generalizes the recently proposed Model Reference Adaptive Search (MRAS) for deterministic optimization, which is motivated by the well-known Cross-Entropy (CE) method. We prove global convergence of SMRAS in a general stochastic setting, and carry out numerical studies to illustrate its performance. An emphasis of this paper is on the application of SMRAS for solving static stochastic optimization problems; its various applications for solving dynamic decision making problems can be found in H. S. Chang, M. C. Fu, J. Hu, and S. I. Marcus, Simulation-based Algorithms for Markov Decision Processes, Springer-Verlag, London, 2007.

We consider the routing problem in wireline, packet-switched communication networks. We cast our optimal routing problem in a multicommodity network flow optimization framework. Our cost function is related to the congestion in the network, and is a function of the flows on the links of the network. The optimization is over the set of flows in the links corresponding to the various destinations of the incoming traffic. We separately address the single commodity and the multicommodity versions of the routing problem. We consider the dual problems, and using dual decomposition techniques, we provide primal-dual algorithms that converge to the optimal solutions of the problems. Our algorithms, which are subgradient algorithms to solve the corresponding dual problems, can be implemented in a distributed manner by the nodes of the network. For online, adaptive implementations of our algorithms, the nodes in the network need to utilize ‘locally available information’ like estimates of queue lengths on outgoing links. We show convergence to the optimal routing solutions for synchronous versions of the algorithms, with perfect (noiseless) estimates of the queueing delays. Every node of the network controls the flows on the outgoing links using the distributed algorithms. For both the single commodity and multicommodity cases, we show that our algorithm converges to a loop-free optimal solution. Our optimal solutions also have the attractive property of being multipath routing solutions.

It is well known that the filtering theory has important applications in both military and commercial industries. The Kalman–Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman–Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation algebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner in theory.