A graphical inference mechanism

Abstract: This paper describes an inference system which lends itself to graphical representation. An implementation of the system is described, and its application in a legislation based domain is illustrated. The methodology for knowledge elicitation which the system is intended to support is briefly indicated. The algorithm is described, and semantics for the system are given.     

Optimal almost-periodic control problem: a trigonometric approximation approach

The problem of finding an almost-periodic control that is optimal with respect to a certain time-averaged criterion for the dynamic system operated over a long period of time is considered. The existence of the optimal solution, spectral properties of which satisfy certain regularity conditions, is hypothesized. The problem is approximated by a sequence of finite-dimensional optimization problems containing trigonometric sums for the approximation of the state and control variables, and using a Fejér-Riesz type representation for a positive trigonometric sum to handle the instantaneous constraints for these variables. Sufficient conditions for the sequence of approximate optimal solutions of the discretized problems to be an approximately minimizing sequence for the basic problem are given. The constructive character of the proposed approach and its potential applications are pointed out both for dynamic systems affected by irregularly pulsating disturbances and for stationary systems, the non-linear dynamics of which can be exploited by a non-stationary control to improve the averaged system performance.     

Optimal control problem of singular Boolean control networks

In this paper, the optimal control problem of singular Boolean control networks is considered via semi-tensor product. Using an analogous needle variation, for multi-input case, a necessary condition for the existence of optimal control is provided based on the algebraic form of singular Boolean control networks, and the result is specialized to the single-input case. Then, an algorithm is presented to calculate an optimal control. Illustrative examples, including the single-input case, are given to show the feasibility of the theoretical results.     

Optimal shape design as a material distribution problem

Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.     

Optimal control problem via neural networks

This paper attempts to propose a new method based on capabilities of artificial neural networks, in function approximation, to attain the solution of optimal control problems. To do so, we try to approximate the solution of Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. In the proposed error function, we used trial solutions for the trajectory function, control function and the Lagrange multipliers. These trial solutions are constructed by using neurons. Then, we minimize the error function that contains just the weights of the trial solutions. Substituting the optimal values of the weights in the trial solutions, we obtain the optimal trajectory function, optimal control function and the optimal Lagrange multipliers.     

Optimal control of a two-class national economic model

In this paper we extend the one-sector national economic system modelled on the basis of the cyclical growth theory developed by Bergstrom (1967) into a two-class model. This is done by combining and extending the model developed by Pasinetti (1962), Hu (1073), Ahmed and Yeo (1970) and Yeo and Teo (1976). The following economic factors—(i) profit margin, (ii) interest rates, (iii) monetary policy, and (iv) rate of increase of wages—arc considered as control variables. Based on this two-class model and its control variables, the objective of the economy is to maximize the sum of total consumption over a given planning period. The Pontryagin Maximum Principle is used to obtain the forms of the optimal controls. For illustrative purposes, numerical examples are solved using the computer programme of Moore and Too (private communication).     

Optimal tilt lifting of beams as a min-max problem

This paper considers the problem of finding the optimal pickup points for a beam that is to be tilt lifted by a system of slings attached to a crane. Under the assumptions of a rigid beam and inextensible cables, this task is formulated as a min-max optimization problem. As for its numerical solution, we propose an efficient and robust algorithm involving smoothing and an appropriate discretization of the objective function. A typical numerical example is provided to illustrate the approach.     

Optimal control problem in bond graph formalism

This paper presents a new way to derive an optimal control system for a specific optimisation problem, based on bond graph formalism. The procedure proposed concerns the optimal control of linear time invariant MIMO systems and can deal with both cases of the integral performance index, these correspond to dissipative energy minimization and output error minimization. An augmented bond graph model is obtained starting from the bond graph model of the system associated with the optimal control problem. This augmented bond graph, consisting of the original model representation coupled to an optimizing bond graph, supplies, by its bicausal exploitation, the set of differential-algebraic equations that analytically give the solution to the optimal control problem without the need to develop the analytical steps of Pontryagin’s method. The proof uses the Pontryagin Maximum Principle applied to the port-Hamiltonian formulation of the system.     

Linear response algorithms for approximate inference in graphical models

Belief propagation (BP) on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. However, it does not prescribe a way to compute joint distributions over pairs of distant nodes in the graph. In this article, we propose two new algorithms for approximating these pairwise probabilities, based on the linear response theorem. The first is a propagation algorithm that is shown to converge if BP converges to a stable fixed point. The second algorithm is based on matrix inversion. Applying these ideas to gaussian random fields, we derive a propagation algorithm for computing the inverse of a matrix.     

Optimal inventory control in a multi-period newsvendor problem with non-stationary demand

The optimal control of inventory in supply chains plays a key role in the competiveness of a corporation. The inventory cost can account for half of company’s logistics cost. The classical inventory models, e.g., newsvendor and EOQ models, assume either a single or infinite planning periods. However, these models may not be applied to perishable products which usually have a certain shelf life. To optimize the total logistic cost for perishable products, this paper presents a multi-period newsvendor model, and the problem is formulated as a multi-stage stochastic programming model with integer recourse decisions. We extend the progressive hedging method to solve the model efficiently. A numerical example and its sensitivity analysis are demonstrated.     

A graphical model based solution to the facial feature point tracking problem

In this paper a facial feature point tracker that is motivated by applications such as human-computer interfaces and facial expression analysis systems is proposed. The proposed tracker is based on a graphical model framework. The facial features are tracked through video streams by incorporating statistical relations in time as well as spatial relations between feature points. By exploiting the spatial relationships between feature points, the proposed method provides robustness in real-world conditions such as arbitrary head movements and occlusions. A Gabor feature-based occlusion detector is developed and used to handle occlusions. The performance of the proposed tracker has been evaluated on real video data under various conditions including occluded facial gestures and head movements. It is also compared to two popular methods, one based on Kalman filtering exploiting temporal relations, and the other based on active appearance models (AAM). Improvements provided by the proposed approach are demonstrated through both visual displays and quantitative analysis.     

Congestion control as a stochastic control problem with action delays

We consider the design of explicit rate-based congestion control for high-speed communication networks and show that this can be formulated as a stochastic control problem where the controls of different users enter the system dynamics with different delays. We discuss the existence, derivation and the structure of the optimal controller, as well as of suboptimal controllers of the certainty-equivalent type — a terminology that is precisely defined in the paper for the specific context of the congestion control problem considered. We consider, in particular, two certainty-equivalent controllers which are easy to implement, and show that they are stabilizing, i.e., they lead to bounded infinite-horizon average cost, and stable queue dynamics. Further, these controllers perform well in simulations.     

Optimal periodic control implemented as a generalized sampled-datahold output feedback control

A method for converting analog linear quadratic regulation control to a generalized sampled-data hold output-feedback control for a linear periodic system or a linear time-variant system is presented. It is shown that by using such a conversion, one can implement the optimal periodic control scheme in the presence of incomplete and delayed state measurements     

Optimal boundary control of a tracking problem for a parabolic distributed parameter system

A comparative study is made of five methods for calculating the optimal control function for a linear parabolic tracking problem with boundary control. Questions of computational instability, numerical accuracy and economic computer usage are investigated. Open-loop methods based upon the variational equations are shown to have the advantages of efficiency, accuracy and ease of programming. Methods based on the method of lines and the Riccati equation are shown to be less straightforward in use. The fourth-order explicit Runge-Kutta algorithm has little scope for application because of its restricted stability range. The Kalman-Englar algorithm is much more robust but a Crank-Nicholson algorithm for the associated auxiliary equation is not very satisfactory in some cases. The auxiliary variable method may then confer some benefits.     

Improving probabilistic inference in graphical models with determinism and cycles

Many important real-world applications of machine learning, statistical physics, constraint programming and information theory can be formulated using graphical models that involve determinism and cycles. Accurate and efficient inference and training of such graphical models remains a key challenge. Markov logic networks (MLNs) have recently emerged as a popular framework for expressing a number of problems which exhibit these properties. While loopy belief propagation (LBP) can be an effective solution in some cases; unfortunately, when both determinism and cycles are present, LBP frequently fails to converge or converges to inaccurate results. As such, sampling based algorithms have been found to be more effective and are more popular for general inference tasks in MLNs. In this paper, we introduce Generalized arc-consistency Expectation Maximization Message-Passing (GEM-MP), a novel message-passing approach to inference in an extended factor graph that combines constraint programming techniques with variational methods. We focus our experiments on Markov logic and Ising models but the method is applicable to graphical models in general. In contrast to LBP, GEM-MP formulates the message-passing structure as steps of variational expectation maximization. Moreover, in the algorithm we leverage the local structures in the factor graph by using generalized arc consistency when performing a variational mean-field approximation. Thus each such update increases a lower bound on the model evidence. Our experiments on Ising grids, entity resolution and link prediction problems demonstrate the accuracy and convergence of GEM-MP over existing state-of-the-art inference algorithms such as MC-SAT, LBP, and Gibbs sampling, as well as convergent message passing algorithms such as the concave–convex procedure, residual BP, and the L2-convex method.     

Optimal control of a solar collector loop using a distributed-lumped model

This paper considers the optimal control of a solar collector loop described by a bilinear distributed parameter model for the collector fluid temperature and a bilinear lumped parameter model for the storage fluid temperature. The objective is to control the collector fluid velocity so as to maximize the net energy collected over a fixed time period. Necessary conditions for optimality, given by a set of equations whose solution yields the optimal control, are derived. It is shown that the optimal control is an open-loop, bang-bang control which depends on two terms: a measurable quantity which depends on the state of the collector fluid, and a quantity which depends on a future knowledge of the weather data. It is also shown that for the case in which only two switches occur during the period of operation, the optimal control depends only on the temperature difference across the collector. Thus, one can construct a feedback on/off controller for the system provided that it is known a priori that only two switches will occur during the time interval under consideration.     

Optimal periodic control problem: a modified trigonometric approximation and quasi-stationarity conditions

The constrained optimal periodic control problem for a system described by differential equations and endowed with inertial controllers is considered, A sequence of discretized problems using trigonometric polynomials is proposed to approximate the problem. Instantaneous constraints for the state and control are handled by a new and more precise approach that imposes only a small number of non-linear but easily computable constraints. The convergence conditions for a sequence of optimal solutions of discretized problems are derived. The inclusion in the approximating scheme of various quasi-stationarity conditions for the control and state variables is analysed. Extension of a new approximating approach for inertialess and smooth problems is also discussed.     

Using model uncertainty for robust optimization in approximate inference control

Recently, the optimization-by-inference approach has been proposed as a new means for solving high-dimensional optimization problems quickly. Approximate Inference COntrol (AICO) is one of the most successful and promising methods that implement the optimization-by-inference approach. AICO is able to solve stochastic optimal control problems and has already been successfully used in many applications. However, it is known that the iterative inference of AICO sometimes fails to converge to the optimal solution. To make the optimization more robust, in this paper, we propose to take model uncertainty into account. In AICO, the cost function to be minimized is accurate around a particular state of a given stochastic system, but the accuracy is uncertain in regions far from that state. Because using such an uncertain function is harmful to the convergence, we modify AICO, so that it does not use the function in uncertain regions. Our method is easy to implement and does not add much computational time to the original AICO. Experiments using two different scenarios show that our method substantially improves AICO in terms of the rate at which the algorithm produces convergent results.     

Optimal control problem for stochastic evolution equations in Hilbert spaces

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results.