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Minimization of Nonconvex Locally Lipschitz Functions Using Mollifier Subdifferentials and Uniform Approximations of Generalized Second Order Derivative

Yousefpour Sadat Mahalleh, Rohollah | 2009

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 39506 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezameddin
  7. Abstract:
  8. Here, we first ivestigate some available nonsmooth algorithms indentilying their drawbacks, and then provide some guidelines to avert these drawbacks. We divide the algorithms to two main classes, first order and second order. The main drawback of the second order class of algorithm is that there does not exist any suitable method to approximate the generalized second order derivative. To solve this problem, we construct a uniform approximation for the generalized Hessian matrix of an SC1 function. Using the discrete gradient and the extended second order derivative, we define the discrete Hessian matrix. We construct a sequence of sets, where each set is composed of discrete Hessian matrices. We first show some new properties of SC1 functions. Then, we prove that for SC1 functions the sequence of the set of discrete Hessian matrices is uniformly convergent to the generalized Hessian matrix. There are several algorithms established based on Clarke subdifferential in the first order. The common drawback in these class of algorithms is in thier need to find the accurate search direction by computing many subdifferentials to obtian a good approximation for the Clarke subdifferential. We present a new algorithm to avert this drawback. First, we approximate the Clarke subdifferential using the mollifier subdifferential and using this approximation, we give a dynamic algorithm to find a search direction. This algorithm finds the search direction without computing too many subdifferentials. Then, we implement our algorithm in MATLAB 2007 sofware environment. The numerical results show that our algorithm is more robust and more efficient than the GS algorithm, currently pbeing an effective algorithm for minimization of nonconvex Lipschitz functions
  9. Keywords:
  10. Clarke Subdifferential ; SC1 Function ; Nonsmooth Algorithm ; Search Direction ; Dynamic Algorithm ; Generalized Second Order Differential

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