...then 4mn = 20000, whereas m 3 = 125. There are two commonly used basic approaches: the recursive vector formulation based on the Strang recurrences [8] and the explicit matrix formulation proposed in =-=[3]-=-. To simplify the notation in the subsequent considerations, we will assume without the loss of generality that i ≤ ¯m. Then matrices (4) and (5) do not depend on the upper index, which can be omitted...

...red operations. For example, if n = 1000 and m = 5, then 4mn = 20000, whereas m 3 = 125. There are two commonly used basic approaches: the recursive vector formulation based on the Strang recurrences =-=[8]-=- and the explicit matrix formulation proposed in [3]. To simplify the notation in the subsequent considerations, we will assume without the loss of generality that i ≤ ¯m. Then matrices (4) and (5) do...

...conditions Fi+1 − Fi ≤ ε1tis T i gi, (2) s T i gi+1 ≥ ε2s T i gi, (3) with 0 < ε1 < 1/2 and ε1 < ε2 < 1. We restrict our attention to the limited memory variable metric methods from the Broyden class =-=[7]-=-. Let 0 < ¯m < n, i ∈ N and m = min( ¯m, i). Limited memory variable metric methods from the Broyden class use direction vectors s1 = −g1 and si+1 = −Hi+1gi+1, i ∈ N , where matrix ∆ Hi+1 = Hi i+1 is ...

...hich can be omitted. The first approach is applicable only in case all matrices Hj, 1 ≤ j ≤ i, are obtained by the BFGS update (in fact there exists other possible updates realizable in this way, see =-=[10]-=-, but they do not belong to the Broyden class). The recursive vector formulation of the limited 1memory BFGS method is based on the pseudo-product form: if ηj = 1, formula (5) can be written in the f...

...s and conclusions Limited memory variable metric methods from the Broyden class were tested by using 72 unconstrained minimization problems with 1000 variables from the collection TEST25 described in =-=[6]-=- (ten problems 48, 57–58, 60–61, 67–70, 79, which are unsuitable for testing limited memory variable metric methods, were excluded). This collections can be found on http://www.cs.cas.cz/luksan/test.h...