# Roger Wets

** Distinguished Professor****Optimization and probability**

Ph.D., 1965, University of California, Berkeley**Refereed publications:** Via Math Reviews

**Web Page: ** http://www.math.ucdavis.edu/~rjbw/

Email: rjbw@math.ucdavis.edu

Office: MSB 2238

Phone: 601-4444 x4013

Professor Roger J-B Wets' research is in stochastic optimization and related fields. He is best known for developing procedures to solve stochastic programming problems. Because stochastic optimization problems include parameters defined only through a probability distribution, they are inherently infinite-dimensional optimization problems. Thus, algorithms to solve such problems must rely on approximation schemes. Professor Wets has worked both on the approximation theory for variational problems and on its computational implementation for stochastic optimization problems.

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

**[1]**Stochastic programming: solution techniques and approximation schemes, in Mathematical Programming: the state of the art (Bonn, 1982), 566-603, Springer, Berlin-New York, 1983.

**[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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