Quadratic conjugate gradient method

The conjugate gradients method deals with our simple quadratic function/(x,y) = + 2i/... 

Other solution procedures for quadratic programming problems include conjugate gradient methods and the Dantzig-Wolfe method (see Dantzig 1963), which uses a modification of the simplex algorithm for linear programming. 

Methods that use the gradient to evaluate certain specific directions to find the minimum of a quadratic function efficiently. These methods are called methods of conjugate directions or even conjugate gradient methods. 

The conjugate gradient method has been adapted for iterative minimization of the quadratic function [28-30]. It is one of the most efficient iterative methods for identifying the extremes of a function of n variables, assuming that the gradient of the function can be calculated. This non-stationary method... 

To overcome these difficulties, new methods have been developed which combine the advantages of the steepest descent and the Newton-Raphson methods but avoid their pitfalls. Such methods are both stable and fast, namely quadratically, converging. Among these the most suitable for energy calculations in molecular biology is the so-called "conjugate gradient" method (Fletcher and Reeves, 1964). 

Figure 5.6 Gradient minimizers applied to a quadratic cost fimction. Results at left from steepest descent method and those at right from conjugate gradient method.

Conjugate gradient method applied to quadratic cost functions... 

We now consider the origin of the excellent performance of the conjugate gradient method for a quadratic cost function, defined in terms of a symmetric, positive-definite matrix A and a vector b,... 

In Chapter 5, we considered file conjugate gradient method which solves Ax = bhy minimizing the quadratic cost function... 

When one refers to the CG method, one often means the linear conjugate gradient, that is, the implementation for the convex quadratic form. In this case, minimizing IxTAx + bTx is equivalent to solving the linear system Ax = -b. Consequently, the conjugate directions pfe, as well as the lengths kh, can be computed in closed form. 

The best known Krylov subspace method is the method of Conjugate Gradients (CG) by Hestenes and Stiefel [70]. If A is S5mimetric positive definite, the solution of the problem Ax = b corresponds to determining a local minimum of the quadratic function ... 

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