Read e-book online A First Course in Optimization Theory PDF

By Rangarajan K. Sundaram

ISBN-10: 0521497701

ISBN-13: 9780521497701

This e-book introduces scholars to optimization idea and its use in economics and allied disciplines. the 1st of its 3 elements examines the life of suggestions to optimization difficulties in Rn, and the way those options can be pointed out. the second one half explores how options to optimization difficulties swap with adjustments within the underlying parameters, and the final half presents an in depth description of the elemental ideas of finite- and infinite-horizon dynamic programming. A initial bankruptcy and 3 appendices are designed to maintain the publication mathematically self-contained.

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Extra info for A First Course in Optimization Theory

Sample text

The specific trust region methods we will present effect a smooth transition from the steepest descent direction to the Newton direction in a way that gives the global convergence properties of steepest descent and the fast local convergence of Newton’s method. The idea is very simple. We let ∆ be the radius of the ball about xc in which the quadratic model mc (x) = f (xc ) + ∇f (xc )T (x − xc ) + (x − xc )T Hc (x − xc )/2 can be trusted to accurately represent the function. ∆ is called the trust region radius and the ball T (∆) = {x | x − xc ≤ ∆} is called the trust region.

1. Let ∇f be Lipschitz continuous with Lipschitz constant L. 1. Assume that the matrices {Hk } are bounded. 31) lim ∇f (xk ) = 0. k→∞ GLOBAL CONVERGENCE 53 Proof. Assume that ∇f (xk ) = 0 for all k and that f is bounded from below. , the step is accepted and the trust region radius is no longer a candidate for expansion), then sk ≥ MT ∇f (xk ) . 32) for the present. 1 imply that aredk ≥ µ0 predk ≥ µ0 ∇f (xk ) σ min( sk , ∇f (xk ) ). 32) to obtain aredk ≥ µ0 σMT ∇f (xk ) 2 . 33) Now since f (xk ) is a decreasing sequence and f is bounded from below, limk→∞ aredk = 0.

So if ω is very small, the convergence will be extremely slow. Similarly, if ω is large, we see that f (x − λ∇f (x)) − f (x) = only if λ< ω 2 x2 (λω − 2) < −αλω 2 x2 2 2(1 − α) . ω So 2(1 − α) 2(1 − α) < βm = λ < . ω ω If ω is very large, many steplength reductions will be required with each iteration and the line search will be very inefficient. These are examples of poor scaling, where a change in f by a multiplicative factor can dramatically improve the efficiency of the line search or the convergence speed.

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A First Course in Optimization Theory by Rangarajan K. Sundaram


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