PPT-Conjugate Gradient Optimization

CG Method Nonlinear CG Solving Linear System of Equations Preconditioned CG and Regularization Outline Kiss point x g x dx dx g QuasiNewton Condition g g. Gradient descent is an iterative method that is given an initial point and follows the negative of the gradient in order to move the point toward a critical point which is hopefully the desired local minimum Again we are concerned with only local op The min and max of a function. Michael . Sedivy. Daniel . Eiland. Introduction. Given a function F(x), how do we determine the location of a local extreme (min or max value)?. Two standard methods exist :. to. Numerical Analysis . I. MATH/CMPSC 455. Conjugate Gradient Methods. A-Orthogonal Basis. . . form a basis of , where. is the . i-th. row of the identity matrix. They are orthogonal in the following sense:. Conjugate Gradient . 1) CG is a numerical method to solve a linear system of equations . 2) CG is used when A is Symmetric and Positive definite matrix (SPD). 3) CG of . Hestenes. and . Optimization. is the mathematical discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints.. Protein Folding. Generally . speaking the problem of protein folding . for Geometry Processing. Justin Solomon. Princeton University. David . Bommes. RWTH Aachen University. This Morning’s Focus. Optimization.. Synonym(-. ish. ):. . Variational. methods.. This Morning’s Focus. multilinear. gradient elution in HPLC with Microsoft Excel Macros. Aristotle University of Thessaloniki. A. . Department of Chemistry, Aristotle University of . Thessaloniki. B. Department of Chemical Engineering, Aristotle University of Thessaloniki. G.Anuradha. Review of previous lecture-. Steepest Descent. Choose the next step so that the function decreases:. For small changes in . x. we can approximate . F. (. x. ):. where. If we want the function to decrease:. Unconstrained minimization. Steepest descent vs. conjugate gradients. Newton and quasi-Newton methods. Matlab. . fminunc. Unconstrained local minimization. The necessity for one dimensional searches. Applications. Lectures 12-13: . Regularization and Optimization. Zhu Han. University of Houston. Thanks . Xusheng. Du and Kevin Tsai For Slide Preparation. 1. Part 1 Regularization Outline. Parameter Norm Penalties. Michael . Sedivy. Daniel . Eiland. Introduction. Given a function F(x), how do we determine the location of a local extreme (min or max value)?. Two standard methods exist :. F(x) with global minimum D and local minima B and F. Unconstrained minimization. Steepest descent vs. conjugate gradients. Newton and quasi-Newton methods. Matlab. . fminunc. Unconstrained local minimization. The necessity for one dimensional searches. Non-convex optimization. All loss-functions that are not convex: not very informative.. Global optimality: too strong. Weaker notions of optimality?. What is a saddle point?. Different kinds of critical/stationary points. Shi & Bo. What is sparse system. A system of linear equations is called sparse if . only a relatively small . number of . its matrix . elements . . are nonzero. It is wasteful to use general methods .