Roger Wets

Professor Wets brought the concept of epi-convergence, now the primary tool, to bear in approximation questions in stochastic optimization and other variational problems. The quantitative aspects of this theory originate with the work of Attouch and Wets, which eventually led to the Attouch-Wets topology for functions spaces and hyperspaces. This allowed them to quantify stability issues for the solution of variational problems. They also introduced the concept of epi/hypo-convergence for saddle functions, with applications to Lagrangians and Hamiltonians, to deal with the convergence of solutions and associated dual variables.

Professor Wets has also examined the statistical properties of stochastic optimization problems. He generalized the law of large numbers that justifies approximation schemes for stochastic optimization problems based on random sampling. As an extension of this work, he considered statistical estimation problems [3], which can be viewed as a particular class of stochastic optimization problems: Find the best estimate for the parameters of a probability distribution. The classical approach is to use only the information coming from the samples. Professor Wets proposed incorporating auxiliary prior knowledge (e.g. smoothness of the density function or unimodality) in the form of constraints. This prior information is particularly important when there are relatively few samples.

Professor Wets also developed two standard algorithms in stochastic programming. First, the L-shaped method [1] arose from the observation that certain problems, including simple recourse problems and certain optimal control problems, have linear constraints which are in an "L" shape when formulated as mathematical programs. He designed a decomposition procedure that took advantage of this special matrix structure. Second, the progressive hedging algorithm [5] has its roots in scenario analysis: When there is uncertainty about the parameters in a problem, one approach is to look at each possible scenario, and solve each corresponding optimization problem. However, the major problem with this approach is that no single solution will take into account all the possible effects that may arise from the uncertainty. Rockafellar and Wets proposed modifying the individual scenarios so that the optimal solutions converge to the solution of the stochastic optimization problem. The algorithm first relaxes and then progressively enforces constraints arising from nonanticipativity. In their earlier work, they had shown that a price system could be attached to the nonanticipativity restrictions, which in turn could be used to `progressively' enforce these constraints.

Professor Wets is completing a book with R.T. Rockafellar, entitled `Variational Analysis', which presents a unified framework for variational problems. He also takes an active role in applications ranging from environmental questions related to lake pollution [2] to problems in finance concerning asset/liability management. Among Professor Wets' successful graduate students are Kerry Back, Armand Makowski, Gabriella Salinetti (Rome), and Jinde Wang (Nanjing).

Selected publications

[2] Stochastic optimization models for lake eutrophication management, Oper. Res. (with L. Somlyody), 36 (1988), 660-681.

[3] Asymptotic behavior of statistical estimators and of optimal solutions for stochastic optimization problems (with J. Dupavcova), Ann. Stat. 16 (1988), 1517-1549.

[4] Quantitative stability of variational systems: I. The epigraphical distance (with H. Attouch), Trans. Amer. Math. Soc. 328 (1991), 695-729.

[5] Scenarios and policy aggregation in optimization under uncertainty (with R. T. Rockafellar), Math. Oper. Res. 16 (1991), 119-147.

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