By L. R. A. Casse, W. D. Wallis

ISBN-10: 3540080538

ISBN-13: 9783540080534

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**Example text**

13) implies that (∇fk+1 − ∇fk )T pk ≤ αk L pk 2 . By combining these two relations, we obtain αk ≥ c2 − 1 ∇fkT pk . 6a), we obtain fk+1 ≤ fk − c1 1 − c2 (∇fkT pk )2 . 12), we can write this relation as fk+1 ≤ fk − c cos2 θk ∇fk 2 , where c c1 (1 − c2 )/L. By summing this expression over all indices less than or equal to k, we obtain k fk+1 ≤ f0 − c cos2 θj ∇fj 2 . 15) j 0 Since f is bounded below, we have that f0 − fk+1 is less than some positive constant, for all k. 15), we obtain ∞ cos2 θk ∇fk 2 < ∞, k 0 which concludes the proof.

In general, nonlinear conjugate gradient directions are much more effective than the steepest descent direction and are almost as simple to compute. These methods do not attain the fast convergence rates of Newton or quasiNewton methods, but they have the advantage of not requiring storage of matrices. An extensive discussion of nonlinear conjugate gradient methods is given in Chapter 5. All of the search directions discussed so far can be used directly in a line search framework. They give rise to the steepest descent, Newton, quasi-Newton, and conjugate gradient line search methods.

We now deﬁne the terminology associated with different types of convergence, for reference in later chapters. Let {xk } be a sequence in IR n that converges to x ∗ . We say that the convergence is Q-linear if there is a constant r ∈ (0, 1) such that xk+1 − x ∗ ≤ r, xk − x ∗ for all k sufﬁciently large. 2. Overview of Algorithms This means that the distance to the solution x ∗ decreases at each iteration by at least a constant factor. 5)k converges Q-linearly to 1. The preﬁx “Q” stands for “quotient,” because this type of convergence is deﬁned in terms of the quotient of successive errors.

### Combinatorial Mathematics IV by L. R. A. Casse, W. D. Wallis

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