Line Search Methods for Unconstrained Optimisation

line Search Methods forUnconstrained OptimisationLecture 8, Numerical Linear Algebra and OptimisationOxford University Computing Laboratory, MT 2007Dr Raphael Hauser Generic FrameworkFor the purposes of this lecture we consider the Unconstrained minimisationproblem(UCM) minx Rnf(x),wheref C1(Rn,R) with Lipschitz continous gradientg(x). In practice, these smoothness assumptions are sometimes violated,butthe algorithms we will develop are still observed to work well. The algorithms we will construct have the common feature that, startingfrom an initial educated guessx0 Rnfor a solution of (UCM), a sequenceof solutions (xk)N Rnis produced such thatxk x Rnsuch that the first and second order necessary optimality conditionsg(x ) = 0,H(x ) 0 (positive semidefiniteness)are satisfied.

Theorem 4 (Global Convergence of More General Descent Di-rection Methods). Let the gradient g of f ∈ C1 be uniformly Lipschitz continuous on Rn. Then, for the iterates generated by the Generic LSM with B-A steps and search directions defined by Bkpk = −gk, one of the following situations occurs, i) gk = 0 for some finite k, ii) limk→∞ ...

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  Methods, Line, Search, Convergence, Line search methods

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