Optimality Conditions for Nonsmooth Optimization and Mordukhovich Subdifferentials
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Date
2012-01
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Addis Ababa University
Abstract
The differentiability assumptions plays a vital role in nonlinear programming, because most
of methods of finding the optimum point in non linear programming starts by finding the
gradient of the function and then the stationary points. For unconstrained optimization
problems, checking the Positive definiteness of the Hessian matrix at stationary points, one can
conclude whether those stationary points are optimum points or not if the objective function is
differentiable. Similarly, if the objective function and functions in the constraint set are
differentiable, the well known optimality condition called Karush Kuhn Tucker (KKT)
condition leads to find the optimum point(s) of the given optimization problem. But, since
finding the gradient of the function for non-differentiable functions is not possible, we treat the
problem by finding the subgradient, the directional derivative, finding the Mordukhovich
normal cone depending on the convexity of the function. Consequently, the optimization
procedures for the optimization problems on which functions in the problem are not
differentiable is different from the optimization procedures for the optimization problems in
which the objective function as well as functions in constraints are differentiable. This project
focuses on finding the optimality conditions for optimizations problems without any
differentiability assumptions. The subgradient and directional derivative approach are used to
solve nonsmooth optimization problem of convex type; and the Mordukhovich exremal
principle is applied to solve nonsmooth optimization problems of non convex type
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Optimality Conditions, for Nonsmooth Optimization