University of Montreal

Faculty of Arts and Sciences
Department of Computer Science and Operations Research

Main Research Contribution

Felisa J. Vázquez-Abad, Associate Professor

Stochastic Approximation:
I have actively participated in the development of the weak convergence method for showing convergence of learning algorithms to the optimal operation. My main contribution is in [8], a paper where we generalize the results from my PhD Thesis in 1989. To cite the importance of the results, in [7] we have given the first proof of asymptotic optimality of learning automata under a stochastic environment, which had been conjectured in previous work and acknowledged as a difficult problem. Our method, extended in [5], allows the identification of the appropriate adjustments for adaptive control and was used in [4] and [21] for telecommunication networks. Other potential applications include flexible manufacturing ([3] and [17]), evaluation of exotic financial derivatives ([11]) and intelligent transportation systems.

Sensitivity Analysis:
Estimation of gradients of stationary averages is a difficult and important problem both for on-line as well as off-line optimization. The Phantom RPA method was introduced in [9], as a generalization of the phantom finite difference of my PhD Thesis. This paper has been cited recently by many other researchers in the Discrete Event Systems area, and has had an impact in the development of sensitivity estimators. Further generalizations of this method appear in [3], [12] and [70], and novel applications are in [2] and [67].

Surrogate Estimation:
The short paper [20] summarizes the results in Vázquez-Abad and Kushner, ``A surrogate estimation approach for adaptive routing in communication networks'', Brown University Report LCDS/CCS, No. 1990-2. It proposes a method akin to the indirect estimation of Glynn and Whitt, but here one uses the information on the sensitivities w.r.t. some variables in order to approximate sensitivities w.r.t. the desired controls. Estimating global sensitivities via local ones is crucial in decentralization and distribution of the control for large complex systems, as our examples in [20] show, where in addition to decentralizing the control structure, a dramatic reduction in computational effort is achieved with this method. Generalizations of this approach are in [69].

Functional Estimation:
We have extended the idea behind the phantom method, to create a method for functional estimation in [6], extending the results of [19]. We call this new method the Split-and-Merge Simulation Tree. In  [15] and [66] we present the implementation of the method to an age replacement policy problem, with new results for the problem and a theoretical analysis of the algorithmic efficiency modeling
the simulation tree as a branching process.

Simulation Methods:
Evaluation of certain important quantities in Risk Theory has attracted the attention of mathematicians and practitioners for decades. It is only lately that simulation has been accepted as a numerical method for approximating such quantities. In [2] I have proposed a new method for estimating the probability of ruin of an insurance company using Importance Sampling (IS). In the same work, we develop the sensitivity estimators for three simulation methods: IS and two other previously known methods. Two of them are non-trivial applications of the RPA method. In [11] we propose a change of measure approach for the evaluation of Asian Options. In addition to estimating rare events (when the option is out of the money) through a change of measure, we accelerate convergence using stochastic approximation to seek the best change of measure.