Student Research Project

Title: Solving nonlinear optimization problems with nondifferentiable objective function by pseudo gradient search.

Adviser: Zhenyuan Wang

Description: In an optimization problem, when the objective function is not differentiable, such as the least square error involving nonlinear integrals, the gradient search fails. In this case, we may replace gradient with a pseudo gradient to determine the optimal search direction. The pseudo gradient can be obtained via a statistical learning based given data for the objective attribute and relevant arguments of the objective function. Once the optimal search direction is determined, the optimal step length can be determined by statistical learning as well. Similar to the gradient search, its advantage is fast convergence, while easily falling in some local extremum is its disadvantage. Hence, choosing a suitable initial point is rather important. With some valid initialization method, such as a genetic algorithm, the pseudo gradient search can be applied in nonlinear multiregression, classification, and decision making widely.

References:

[1] F. S. Hillier and G. J. Lieberman, Introduction to Operations Research (Eighth edition), McGraw Hill, 2004.

[2] M. Smith, Neural Networks for Statistical modeling, Van Nostrand, 1993.

[3] J. Wang and Z. Wang, Using neural networks to determine Sugeno measures by statistics, Neural Networks 10, No. 1 (1997), 183-195.

[4] Z. Wang, A new genetic algorithm for nonlinear multiregressions based on generalized Choquet integrals, Proc. FUZZ-IEEE2003, 819-821.

[5] Z. Wang and G. J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.

Prerequisites: MATH 4300/8306, Math 4310/8316, MATH 8520/9110, programming language (i.e., C++)

Requirements: Constructing a mathematical model. Develop the relevant algorithm and programming it. Running the program(s) for testing data. Completing a research paper on this topic that can be submitted to some international conference or international journal before April 2005. Presenting the paper at the MAM in the spring of 2005.