The stochastic -hub center problem with service-level constraints

Small package delivery companies offer services where packages are guaranteed to be delivered within a given time-frame. With variability in travel time, the configuration on the hub-and-spoke delivery network is vital in ensuring a high probability of meeting the service-level guarantee. We present the stochastic p-hub center problem with chance constraints, which we use to model the service-level guarantees. We discuss analytical results, propose solution heuristics, and present the results from computational experiments.     

Enhancements of two-stage stochastic decomposition

This paper presents some enhancements associated with stochastic decomposition (SD). Specifically, we study two issues: (a) Are there any conditions under which the regularized version of SD generates a unique solution? (b) Is there a way to modify the SD algorithm so that a user can trade-off solution times with solution quality? The second issue addresses the scalability of SD for very large scale problems for which computational resources may be limited and the user may be willing to accept solutions that are “nearly optimal”. We show that by using bootstrapping (re-sampling) the regularized SD algorithm can be accelerated without significant loss of optimality. We report computational results that demonstrate the viability of this approach.     

Model predictive control for systems with stochastic multiplicative uncertainty and probabilistic constraints

Robust predictive control handles constrained systems that are subject to stochastic uncertainty but propagating the effects of uncertainty over a prediction horizon can be computationally expensive and conservative. This paper overcomes these issues through an augmented autonomous prediction formulation, and provides a method of handling probabilistic constraints and ensuring closed loop stability through the use of an extension of the concept of invariance, namely invariance with probability p.     

Detectability and observability of discrete-time stochastic systems and their applications

This paper studies detectability and observability of discrete-time stochastic linear systems. Based on the standard notions of detectability and observability for time-varying linear systems, corresponding definitions for discrete-time stochastic systems are proposed which unify some recently reported detectability and exact observability concepts for stochastic linear systems. The notion of observability leads to the stochastic version of the well-known rank criterion for observability of deterministic linear systems. By using these two concepts, the discrete-time stochastic Lyapunov equation and Riccati equations are studied. The results not only extend some of the existing results on these two types of equation but also indicate that the notions of detectability and observability studied in this paper take analogous functions as the usual concepts of detectability and observability in deterministic linear systems. It is expected that the results presented may play important roles in many design problems in stochastic linear systems.     

Robust delayed-state-feedback stabilization of uncertain stochastic systems

Due to time spent in computation and transfer, control input is usually subject to delays. Problems of deterministic systems with input delay have received considerable attention. However, relatively few works are concerned with problems of stochastic system with input delay. This paper studies delayed-feedback stabilization of uncertain stochastic systems. Based on a new delay-dependent stability criterion established in this paper, a robust delayed-state-feedback controller that exponentially stabilizes the uncertain stochastic systems is proposed. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method.     

Stochastic algorithms for robustness of control performances

In recent years, there has been a growing interest in developing statistical learning methods to provide approximate solutions to “difficult” control problems. In particular, randomized algorithms have become a very popular tool used for stability and performance analysis as well as for design of control systems. However, as randomized algorithms provide an efficient solution procedure to the “intractable” problems, stochastic methods bring closer to understanding the properties of the real systems. The topic of this paper is the use of stochastic methods in order to solve the problem of control robustness: the case of parametric stochastic uncertainty is considered. Necessary concepts regarding stochastic control theory and stochastic differential equations are introduced. Then a convergence analysis is provided by means of the Chernoff bounds, which guarantees robustness in mean and in probability. As an illustration, the robustness of control performances of example control systems is computed.     

Geometrically non-linear behavior of structural systems with random material property: An asymptotic spectral stochastic approach

An asymptotic spectral stochastic approach is presented for computing the statistics of the equilibrium path in the post-bifurcation regime for structural systems with random material properties. The approach combines numerical implementation of Koiter’s asymptotic theory with a stochastic Galerkin scheme and collocation in stochastic space to quantify uncertainties in the parametric representation of the load–displacement relationship, specifically in the form of uncertain post-buckling slope, post-buckling curvature, and a family of stochastic displacement fields. Using the proposed method, post-buckling response statistics for two plane frames are obtained and shown to be in close agreement with those obtained from Monte Carlo simulation, provided a fine enough spectral representation is used to model the variability in the random dimension.     

On the collective sort problem for distributed tuple spaces

In systems coordinated with a distributed set of tuple spaces, it is crucial to assist agents in retrieving the tuples they are interested in. This can be achieved by sorting techniques that group similar tuples together in the same tuple space, so that the position of a tuple can be inferred by similarity. Accordingly, we formulate the collective sort problem for distributed tuple spaces, where a set of agents is in charge of moving tuples up to a complete sort has been reached, namely, each of the N tuple spaces aggregate tuples belonging to one of the N kinds available. After pointing out the requirements for effectively tackling this problem, we propose a self-organizing solution resembling brood sorting performed by ants. This is based on simple agents that perform partial observations and accordingly take decisions on tuple movement. Convergence is addressed by a fully adaptive method for simulated annealing, based on noise tuples inserted and removed by agents on a need basis so as to avoid sub-optimal sorting. Emergence of sorting properties and scalability are evaluated through stochastic simulations.     

Adaptive Stochastic Gradient Descent Optimisation for Image Registration

We present a stochastic gradient descent optimisation method for image registration with adaptive step size prediction. The method is based on the theoretical work by Plakhov and Cruz (J. Math. Sci. 120(1):964–973, 2004). Our main methodological contribution is the derivation of an image-driven mechanism to select proper values for the most important free parameters of the method. The selection mechanism employs general characteristics of the cost functions that commonly occur in intensity-based image registration. Also, the theoretical convergence conditions of the optimisation method are taken into account. The proposed adaptive stochastic gradient descent (ASGD) method is compared to a standard, non-adaptive Robbins-Monro (RM) algorithm. Both ASGD and RM employ a stochastic subsampling technique to accelerate the optimisation process. Registration experiments were performed on 3D CT and MR data of the head, lungs, and prostate, using various similarity measures and transformation models. The results indicate that ASGD is robust to these variations in the registration framework and is less sensitive to the settings of the user-defined parameters than RM. The main disadvantage of RM is the need for a predetermined step size function. The ASGD method provides a solution for that issue.     

Semi-linear Stochastic Difference Equations

We consider in this paper a class of vector valued processes that have the form Y _n + 1 = A _n ( Y _n ) + B _n . B _n is assumed to be stationary ergodic and A _n is assumed to have a divisibility property. This class includes linear stochastic difference equations as well as multi-type branching processes (with a discrete or with a continuous state space). We derive explicit expressions for the probability distribution as well as for the two first moments of state vectors at the stationary regime. We then apply this approach to derive two formalisms to describe the infinite server queue. The first is based on a branching process approach adapted to phase type service time distributions. The second is based on a linear stochastic difference equation and is adapted to independent and generally distributed service times with bounded support. In both cases we allow for generally distributed arrival process (not necessarily i.i.d. nor Markovian).