Curved Trajectories towards Local Minimum of a Function Introduction in addition to Notation Typical Iterative Methods Current Methods Infinite Series of Solution
Blackwell, Jennifer, News Director has reference to this Academic Journal, PHwiki organized this Journal Curved Trajectories towards Local Minimum of a Function Al Jimenez Mathematics Department Cali as long as nia Polytechnic State University San Luis Obispo, CA 93407 Taylor Series in addition to Rotations Spring, 2008 Introduction in addition to Notation The Problem Derivatives: A local min x is a critical point: Necessary condition: 0 Typical Iterative Methods Sequence is generated from x0 Such that With vk a vector with property a descent direction And pk > 0 typically approximates solution of called the line search or the scalar search Proven to converge as long as smooth functions
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Current Methods Selecting vk has huge effect on convergence rate: Steepest Descent: 1st order Newtons direction: 2nd order, but may not be a descent direction when far from a min Conjugate Directions uses vk-1, vk-2, Quasi-Newton/Variable metric also uses vk-1, vk-2, High order Tensor models fit prior iteration values Number of derivatives available affects method The scalar search Accuracy of scalar minimization Quadratic models: Trust Region Infinite Series of Solution Matrix vector products, but shown with exponents as long as connections with scalar Taylor series. Infinite Series of Solution Define: Then: For p = 1:
Curved Trajectories Algorithm At kth iteration, estimate , then calculate: Select order, modify di , in addition to select pk 2nd order: 3rd order: 4th order: Challenges High order terms accurately approximated from the Gradient in addition to the Hessian Scalar searches along polynomial curved trajectories Per as long as mance as long as large problems Exploit Sparse Hessian Store nonzeros only, no operations on zeros Far from solution: Hessian not positive definite (solved) Hessian modified in addition to use CG step as last resort
Hessian < 0 Changes Cuter Per as long as mance Profiles Cuter Per as long as mance Profiles Current Research Pursuits H in addition to le multiple functions: Pareto optimal points H in addition to le Constraint Functions Explore the family of infinite series as long as combination of composition functions. Rosenbrock Banana Function Algorithm selects x0 x1 x2 x3 f = 24.2 f = 24.2 f = 4 f = 0.5 3D View Trajectories from starting point Rotations Rotations 3D Rotations At point we have h(p) is trajectory in addition to R() is rotation matrix. h(0) = 0 in addition to R(0) = I, in addition to as long as 2 coordinates, counterclockwise At the kth step far from solution we want: But settle as long as pk, k: Rotations (continued) Gives Trajectory angle with the gradient as long as R(0) = I Observations:
Rotation Challenges/Results Select effective k without too much work Using existing strategy to calculate pk, then calculate a k from in addition to G . Then calculate a new pk again using rotated trajectory. Good results with k > 40º indicates elongated ellipse contours, in addition to rotation seems unproductive in this case. Effective when CTA series is convergent in addition to iteration is not close to the minimum point. Functions of more than 2 variables later f (p, ) f (p, ), = 0, -0.1, -0.2, -0.3 = 0 = -0.1 = -0.2 = -0.3
More than Two Coordinates Ignore coordinates with insignificant Newton correction magnitudes. Success achieved by adding the 3rd coordinate to the first two as follows: Calculate the rotation by paring the 3rd coordinate with each of the top 2 coordinates. This results in a rotation matrix: Where the angles 1 , 2 , 3 are each calculated between two coordinates as explained be as long as e. The 4th coordinate is added by pairing rotations with the first 3 coordinates, in addition to so on.
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