Nonlinear Optimization by Sven O. Krumke

By Sven O. Krumke

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13) F (x∗ + t(x(k+1) − x∗ ))dt. 13) and the continuity of F that limk→ ∞ Gk = F (x∗ ). In particular, lub2 (Gk ) ≤ c for some constant c independent from k. 14) F(x (k+1) ) ≤ lub2 (Gk ) x(k+1) − x∗ ≤ c x(k+1) − x∗ . 15) =: (1 − ck ) x (k) ∗ −x . By (a) we have limk→ ∞ ck = 0. 15)) cck x(k) − x∗ (1 − ck ) x(k) − x∗ cck k→ ∞ = → 0. 2 Quasi Newton Methods I: Systems of Nonlinear Equations 53 Assume now conversely that (c) is satisfied. 12) = yk − Bk sk , sk so by (c) we have limk→ ∞ dk = 0. 13) which satisfy F(x(k+1) ) = Gk (x(k+1) − x∗ ).

The matrix ∇2 f(x∗ ) is positive definite. 4. 22a) or x(k+1) := x(k) − λk Hk gk . 22b) Here, Bk ≈ H(x(k) )−1 is an approximation to the Hessian of f at x(k) and λk > 0 is chosen by line-search: f(x(k+1) ) ≈ min f(x(k) + λk dk ), dk := −B−1 k gk ≈ −Hk gk . λ≥0 We use again the abbreviations B := Bk B+ := Bk+1 x := x(k) s := sk = x+ − x x+ := x(k+1) y := yk = g(x+ ) − g(x). In designing update formulae we need to keep the following goals in mind: 1. We want to satisfy the Quasi-Newton condition B+ y = s.

The problem of minimizing a strictly convex quadratic function f(x) = 12 xT Ax + bT x + c with positive definite A is equivalent to solving the linear system Ax = −b. g. [SB91a, SB91b]. However, even for sparse A the lower triangular matrix L is usually dense. The cg-algorithm is able to exploint sparseness since we only need matrix-vector multiplications. Thus, the running time and even more the storage requirements are much less for the cg-algorithm. 2. In practice the cg-algorithm does usually not terminate after n iterations due to roundoff errors.

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