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admissible approximate minimizing assume assumptions Banach space bounded set C-problem CG method computationally compute condition conjugate conjugate-gradient method consider constrained problems constraints converges to zero convex functional convex set Corollary criticizing sequence defined Definition derivatives differentiable direction sequence discretization Equation error estimate example exists forcing function given gradient hence Hilbert space hypotheses implies iteration Let x„+1 lim sup limit points linearly independent Lipschitz-continuous local-convergence matrix minimization problems minimizing point minimizing sequence minimum Newton Newton's method nonlinear norm norm-closed numbers positive-definite Proof of Theorem properties Proposition prove Q.E.D. EXERCISE real-valued reflexive space Ritz method satisfying Section self-adjoint sequentially compact sequentially lower semicontinuous solution solve steepest-descent step-size algorithms strongly minimal strongly quasi-convex subset subspace Suppose tends to zero Theorem tion topology unconstrained uniformly continuous valid vector Vf(x weak convergence weak limit points weakly sequentially compact weakly sequentially lower yields