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Numerical Optimization: Difference between revisions
→Conjugate Gradient Methods
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; Theorem: At step k, the residual <math>r_k</math> is orthogonal to <math>\{p_0,...p_{k-1}\}</math> and the current iteration <math>x_{k}</math> | ; Theorem: At step k, the residual <math>r_k</math> is orthogonal to <math>\{p_0,...p_{k-1}\}</math> and the current iteration <math>x_{k}</math> | ||
: minimizes <math>\phi(x)</math> over the set <math>\{x_0 + span(p_0,...,p_{k-1})\}</math> | : minimizes <math>\phi(x)</math> over the set <math>\{x_0 + span(p_0,...,p_{k-1})\}</math> | ||
===Convergence Rate=== | |||
The convergence rate can be estimated by:<br> | |||
<math>\min_{P_k} \max_{1 \leq i \leq k} [1 + \lambda_i P_k(\lambda_i)]^2</math> | |||
==Resources== | ==Resources== | ||
* [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)] | * [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)] | ||