Stochastic approximation methods

This paper discusses the use of the Robbins Monro algorithm and the Kiefer Wolfowitz algorithm in the multidimensional case.Anglai

An interactive algorithm for linear multiple objective decision making problems in a stochastic environment

The authors consider a linear multiple objective decision making problem in a probabilistic framework. A novel scheme for solving such problems is proposed. This scheme consists of two phases: In phase I, the problem is optimized with the random parameters taking their mean values. With reference to the obtained Pareto solution, the decision makers choose critical target levels for the objects containing random parameters. In phase II, with those levels, an equivalent deterministic problem is formulated which takes into account the dispersion of the objectives and the eventual violation of the constraints in the face of different scenarios. This problem is solved by an interactive algorithm. A simplified regional development problem is solved by the developed scheme.Anglai

Optimal sensors' allocation strategies for a class of stochastic distributed systems

The minimum-variance state-estimator for a class of linear distributed parameter systems with both process and measurement disturbances, is derived. A new algorithm is presented for the optimal simultaneous spatial allocation of sensors. The algorithm minimizes recursively the spatial integral of the covariance matrix of the error in the state estimates. For time-invariant systems the algorithm leads to the minimization of the spatial integral of the steady-state error covariance matrix. The influence of the system disturbances and measurement noise on these locations is discussed. An illustrative example is given to demonstrate the numerical performance of the algorithm.Anglai

Combined identification of the input-output and noise dynamics of a closed-loop controlled linear system

The open-loop input-output dynamics and the noise dynamics of a feedback controlled linear system perturbed by coloured noise admitting a Markov representation are identified in state variable form using a two-stage algorithm. This system is equivalent to an augmented system driven by white noise. First the input-output dynamics are identified through a stochastic approximation algorithm using superimposed white noise. Subtracting the model output from the system output yields correlated residuals which are then used to identify the noise dynamics using stochastic realization theory. An innovations representation is obtained that is equivalent to the above defined augmented system. The two stages are combined by a judicious coordinate transformation. The method can be applied on an operating feedback controlled process, regardless of the structure of the unknown suboptimal regulator.Anglai

A new resolution method for the parametric linear complementarity problem

An original method is presented to solve the parametric linear complementarity problem. To solve such a problem requires to identify the values taken by the solution of a linear complementarity problem where some of its right-hand side terms vary in a prescribed domain. This class of problems allows, among others, to deal with linear market equilibrium problems with varying prices: it allows to identify how does the equilibrium point varies when the prices vary over a set of given values. This method is illustrated through an application related to the institution of ecological taxes on the Western European natural gas market

Piecewise linear modelling of a nonlinear system through an adaptive procedure

The general problem of piecewise linear approximation of an unknown function is formulated as the minimization of a functional over the parameters of the approximating function and of the discriminant function characterizing the partition of the independent variable space. Adaptive recurrence algorithms are proposed and applied to the construction of a rainfall river flow predictor model assuming two functioning modes: one for large rainfall and one for small rainfall.Anglai

Optimal point-wise discrete control and controllers' allocation strategies for stochastic distributed systems

The design of point-wise discrete controllers for a class of stochastic distributed-parameter systems is considered. Assuming a fixed set of controllers' positions, the optimal feedback control is derived using a direct approach in which the infinite dimensional space is approximated using a set of orthonormal functions. The resulting optimal cost is minimized again w.r.t. this set of positions, using gradient techniques, to get the optimal locations for the controller. A one-dimensional diffusion process is used to demonstrate the algorithm.Anglai

Multiobjective fuzzy linear programming problems with fuzzy decision variables

In this paper, a multiobjective decision-making process is modeled by a multiobjective fuzzy linear programming problem with fuzzy coefficients for the objectives and the constraints. Moreover, the decision variables are linked together because they have to sum up to a constant. Most of the time, the solutions of a multiobjective fuzzy linear programming problem are compelled to be crisp values. Thus the fuzzy aspect of the decision is partly lost and the decision-making process is constrained to crisp decisions. We propose a method that uses fuzzy decision variables with a joint membership function instead of crisp decision variables. First, we consider lower-bounded fuzzy decision variables that set up the lower bounds of the decision variables. Then, the method is generalized to lower-upper-bounded fuzzy decision variables that set up also the upper bounds of the decision variables. The results are closely related to the special type of problem we are coping with, since we embed a sum constraint in the joint membership function of the fuzzy decision variables. Numerical examples are presented in order to illustrate our method. (C) 2002 Elsevier Science B.V. All rights reserved