Abstracts for Elijah Polak

The EECS Research Summary for 2003

Adaptive Approximations and Exact Penalization for the Solution of Generalized Semi-Infinite Min-Max Problems

Johannes O. Royset1 and Armen Der Kiureghian2
(Professor Elijah Polak)
Norwegian Research Council, (NSF) ECS-9900985, and Space Sciences Laboratory-Lockheed Martin Mini-grant Program

We develop an implementable algorithm for the solution of a class of generalized semi-infinite min-max problems. To this end, first we use exact penalties to convert a generalized semi-infinite min-max problem into a finite family of semi-infinite min-max-min problems. Second, the inner min-function is smoothed and the semi-infinite max part is approximated, using discretization, to obtain a three-parameter family of finite min-max problems. Under a calmness assumption, we show that when the penalty is sufficiently large the semi-infinite min-max-min problems have the same solutions as the original problem, and that when the smoothing and discretization parameters go to infinity the solutions of the finite min-max problems converge to solutions of the original problem, provided the penalty parameter is sufficiently large.

Our algorithm combines tests for adjusting the penalty, the smoothing, and the discretization parameters and makes use of a min-max algorithm as a subroutine. In effect, the min-max algorithm is applied to a sequence of gradually better-approximating min-max problems, with the penalty parameter eventually ceasing to increase, but the smoothing and discretization parameters driven to infinity. A numerical example demonstrates the viability of the algorithm.

1Graduate Student (non-EECS)
2Outside Adviser (non-EECS)

Send mail to the author : (joroyset@ce.berkeley.edu)

Optimization of Cost Functions that Are Defined on the Solution of Differential Algebraic Equations

Michael Wetter1, Van P. Carey2, and Frederick C. Winkelmann3
(Professor Elijah Polak)
(DOE) DE-AC03-76SF00098 and (NSF) ECS-9900985

We are interested in the optimization of a cost function that is evaluated by a thermal building simulation program. Such programs solve a complex system of differential algebraic equations, and can nowadays only be optimized heuristically.

We developed a generalized pattern search method with adaptive precision function evaluations that uses coarse approximations to the cost function in the early iterations, with the approximation precision controlled by a test. Such an approach leads to substantial time savings, and the precision control guarantees that the optimization converges to a stationary point of the infinite precision cost function.

However, implementing the precision control requires knowledge of an error bound function which is hard to obtain for thermal building simulation programs. Therefore, we currently develop simulation models that allow us to obtain such an error bound function, and that reduce computation time for the optimization significantly.

In parallel to the above optimization method, we develop a smoothing method that will allow using existing simulation programs for optimization with guaranteed convergence properties.

1Graduate Student (non-EECS), LBNL, EETD, Simulation Research Group
2Outside Adviser (non-EECS)
3Outside Adviser (non-EECS), LBNL, EETD, Simulation Research Group

More information (http://www.me.berkeley.edu/~mwetter) or

Send mail to the author : (mwetter@lbl.gov)