1. A Hybrid Optimization Approach for Automated Parameter Estimation Problems Carlos A. Quintero 1 Miguel Argáez 1, Hector Klie 2, Leticia Velázquez 1 and Mary Wheeler 2 Problem Formulation Contact Information Carlos A. Quintero, Graduate Student The University of Texas at El Paso Department of Mathematical Sciences 500 W. University Avenue El Paso, Texas 79968-0514 Email: caquintero@utep.edu Phone: (915) 747-6858 Fax: (915) 747-6502 Abstract We present a hybrid optimization approach for solving automated parameter estimation problems that is based on the coupling of the Simultaneous Perturbation Stochastic Approximation (SPSA)[Ref.1] and a globalized Newton-Krylov Interior Point algorithm (NKIP) presented by Argáez et al.[Refs.2,3]. The procedure generates a surrogate model that yield to use efficiently first order information and applies NKIP algorithm to find an optimal solution. We implement the hybrid optimization algorithm on a simple test case, and present some preliminary numerical results. where the global solution x* is such that We are interested in problem (1) that have many local minima. We consider the global optimization problem in the form: Local Search: NKIP We test the algorithm on the Rastrigin’s Problems [Ref 6]: 3-D view, n=2 Test Case We find the surrogate model f s (x) using an interpolation method with the data,, provided by SPSA. This can be performed in different ways, e.g., radial basis functions, kriging, regression analysis, or using artificial neural networks. In our test case, we optimize the surrogate function where the multiquadric basis functions are given by The interpolation algorithm [Ref.4] characterizes the uncertainty parameters We plot the original model function and the surrogate function: Surrogate Model Hybrid Optimization Scheme Filtering Data Global Search Via SPSA Sampling Surrogate Model Explore Parameter space Interpolate Response surface No Optimal Solution found for Original model? Optimal Solution found for Original model? Local Search Via NKIP Local Search Via NKIP Stop Yes Improved Solution Sensitivity analysis Front view SPSA is a global derivative free optimization method that uses only objective function measurements. In contrast with most algorithms which requires the gradient of the objective function that is often difficult or impossible to obtain. Further, SPSA is especially efficient in high-dimensional problems in terms of providing a good approximate solution for a relatively small number of measurements of the objective function [Ref. 5]. The parameter estimation is first carried out by means of SPSA algorithm. This process may be performed by starting with different initial guesses (multistart). This increases the chances for finding a global solution, and yields to find a vast sampling of the parameter space. Global Search: SPSA Future Work 1.To add a multi-start on SPSA 2.Filter data from SPSA 3.If x* is such that f(x*) does not satisfy an upper bound given by then we use x* as an initial point for SPSA. References 1 Department of Mathematical Sciences The University of Texas at El Paso 2 ICES-The Center for Subsurface Modeling The University of Texas at Austin Multi-Start [1] J. C. Spall. Introduction to stochastic search and optimization: Estimation, simulation and control. John Wiley & Sons, Inc., New Jersey, 2003. [2] M. Argáez, R. Sáenz, and L. Velázquez. A trust–region interior–point algorithm for nonnegative constrained minimization. Technical report, Department of Mathematics, The University of Texas at El Paso, 2004. [3] M. Argáez and R.A. Tapia. On the global convergence of a modified augmented Lagrangian linesearch interior-point Newton method for nonlinear programming. J. Optim. Theory Appl., 114:1–25, 2002. [4]Mark J. L. Orr. Matlab Functions for Radial Basis Function Networks, 1999. [5] H. Klie and W. Bangerth and M. F. Wheeler and M. Parashar and V. Matossian. Parallel well location optimization using stochastic algorithms on the grid computational framework. 9th European Conf. on the Mathematics of Oil Recovery (ECMOR), August, 2004. [6] A.J. Keane and P.B. Nair. Computational Approaches for Aerospace Design: The Pursuit of Excellence. Wiley, England, 2005. f s (x) x*=(0,0) global solution f(x*)=0 Sampling Data from SPSA f s (x) SPSA data points The scheme is to use SPSA as the sampling device to perform a global search of the parameter space and switch to NKIP to perform the local search via a surrogate model. f(x) Further research and numerical experimentation are needed to demonstrate the effectiveness of the hybrid optimization scheme being proposed, especially for solving large application problems of interest of the Department of Defense. Acknowledgments The authors were partially supported by DOD PET project EQM KY06-03. The authors thank IMA for the travel support to attend the Blackwell-Tapia Conference, November 3-4, 2006. where a and b are determined by the sampled points given by SPSA. NKIP is a globalized and derivative dependent optimization method. This method calculates the directions using the conjugate gradient algorithm, and a linesearch is implemented to guarantee a sufficient decrease of the objective function as described in Argáez and Tapia [Ref.4]. This algorithm has been developed for obtaining an optimal solution for large scale and degenerate problems. NKIP algorithm apply to the surrogate model find an optimal solution at x*=(1.94, 0.01) and f s (x*)= 6.26. We consider the optimization problem in the form: