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DEPARTMENT OF MATHEMATICS (UW)

UNIVERSITY OF WASHINGTON

Abstracts of Invited Talks

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• John Dennis (Rice University and University of Washington)

  Optimization using Surrogates for Engineering Design

  This talk will outline the surrogate management framework for general nonlinear programming without derivatives. The point of this talk is to show young optimizers how useful their work can be.

  This line of research was motivated by industrial applications, indeed, by a question I was asked by Paul Frank of Boeing Phantom Works. His group was often asked for help in dealing with very expensive low dimensional design problems from all around the company. Everyone there was dissatisfied with the common practice of substituting inexpensive surrogates for the expensive ``true'' objective and constraint functions in the optimal design formulation. We had been asked this question before, but this time the ideas behind the surrogate management framework occurred to us, and we hope to demonstrate in this talk just how simple that answer is.

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• Michael P. Friedlander (University of British Columbia)

  MINOS and Knossos with Second Derivatives

  For problems with nonlinear constraints, Knossos currently uses MINOS or SNOPT to solve a sequence of linearly constrained subproblems. The convergence theory is largely independent of how the subproblems are solved. We note that MINOS can be developed to use second derivatives of the objective function (using a straightforward modified-Newton method on each reduced Hessian). By implementing such a version of MINOS, we obtain a second-derivative version of Knossos for general NLPs. (Joint work with Michael Saunders.)

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• Terry Rockafellar (University of Washington)

  Convex Analysis in Finance

  Although convex analysis has come to be widely appreciated for its importance in understanding and solving problems of optimization, its applications in finance are relatively new. In truth, many of the researchers who work on problems about "portfolios", "insurance", and other such matters have little background even in optimization, not to speak of convexity. Nonetheless, significant inroads are being made in dealing with notions or risk and other aspects of uncertainty in the outcome of investments. This talk will explain the basic framework and the reasons why convex analysis is needed for good progress in this field.

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• Yanfang Shen (University of Washington)

  Annealing Adaptive Search with Hit-and-Run Sampling Methods for Global Optimization

  Stochastic algorithm, such as simulated annealing and genetic algorithms, have been widely applied to solve global optimization problems. To understand the behavior of simulated annealing, the theoretical performance of annealing adaptive search (AAS) is analyzed. We show that for a large class of continuous/discrete global optimization problems, the expected number of improving points generated by AAS grows linearly in dimension, and the expected number of function evaluations can also be linear when our adaptive cooling schedule is employed. This eliminates the need to heuristically choose a cooling schedule for simulated annealing. AAS assumes points can be exactly sampled according to a sequence of Boltzmann distributions. A Markov chain Monte Carlo (MCMC) sampler is used to implement AAS and performance bounds are derived in terms of the choice of cooling schedule and the rate of convergence of the MCMC sampler to a Boltzmann distribution. We develop and analyze the performance of several MCMC samplers, based on Hit-and-Run, that can be applied to discrete and mixed continuous/discrete domains. We conclude by embedding the family of Hit-and-Run samplers into the AAS framework to provide robust global optimization algorithms. Numerical results are presented. (Joint work with Zelda Zabinsky.)

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• Jane Ye (University of Victoria)

  Quasiconvex Programming with Locally Starshaped Constraint Region and Applications to Quasiconvex MPEC

  A quasiconvex programming problem is a mathematical programming problem where the objective function is quasiconvex. We derive some necessary and sufficient conditions for quasiconvex programming problem with a locally starshaped constraint region. Our optimality conditions are different from the usual optimality conditions in that the limiting subdifferential of the objective function is replaced by a normal cone operator. Such an optimality condition has advantage over the usual one in that it becomes sufficient even when the objective function is only quasiconvex but not pseduconvex. As a special case we derive the corresponding results for the class of Quasiconvex-Quasiaffine MPEC which is a class of mathematical program with complementarity constraints where the objective function is quasiconvex, the inequality constraint is quasiconvex and the rest of constraints are quasiaffine.

For a printable version of the abstracts click HERE.