cgtaya.blogg.se - Mathematica gradient

We’ve introduced a line search algorithm in the previous article: Armijo Line Search.

The residual rₖ now should be the gradient of the objective function f.

Instead, we would like to have αₖ as the approximation of the minimizer using a line search algorithm.

The steplength αₖ is currently the exact minimizer of f along pₖ.We can do this by changing two things in the previous algorithm: Can we adapt the approach to minimize general convex functions or even general nonlinear functions f? In the previous section, we’ve implemented Conjugate Gradient as a minimization algorithm for the convex quadratic function f. Conjugate Gradient for Nonlinear Optimization Problem The Conjugate Gradient method is recommended only for large problems otherwise, Gaussian elimination or other factorization algorithms such as the singular value decomposition are to be preferred, since they are less sensitive to rounding errors. In fact, if A has only r distinct eigenvalues, then the Conjugate Gradient iteration will terminate at the solution in at most r iterations. That’s why we don’t need to safeguard our algorithm from infinite loop (using max iteration for instance) in LinearCG function. What can we learn from these examples? The most obvious one is that the iteration needed for the conjugate gradient algorithm to find the solution is the same as the dimension of matrix A.