In the previous post in our series on the Bregman algorithm we discussed how to solve convex optimization problems. In this post we want to give reference to some variations and extensions of the Bregman algorithm.
In a previous post we discussed how to solve constrained optimization problems by using the Bregman algorithm. Here we want to extend the approach unconstrained problems.
Let’s start simple. Assume we want to minimize a convex and smooth function $f\colon\mathbb{R}^{n}\to\mathbb{R}$.
In a previous post we discussed how to find a common point in a family of convex sets by using the Bregman algorithm. Actually the algorithm is capable of more. We can use it to solve constrained optimization problems.
In the 1960s Lev Meerovich Bregman developed an optimization algorithm [1] which became rather popular beginning of 2000s. It’s not my intention to present the proofs for all the algorithmic finesse, but rather the general ideas why it is so appealing.
We developed a QR decomposition algorithm, based on the orthogonalisation process of Gram-Schmidt in a series of posts here, here, here, and here. Let’s have a look how good this algorithm performs against built-in implementations from julia and other programming languages.
Problem Formulation We already discussed QR decompositions and showed that using the modified formulation of Gram Schmidt significantly improves the accuracy of the results. However, there is still an error of about $10^3 M_\varepsilon$ (where $M_\varepsilon$ is the machine epsilon) when using the modified Gram Schmidt as base algorithm for the orthogonalisation.