Long vectors for quasi-Newton updates

Abstract

This work concerns the derivation of formulae for updating quasi-Newton matrices used in algorithms for computing approximate minima of smooth unconstrained functions. The paper concentrates strictly on the techniques used to derive update formulae. It demonstrates a technique in which problems of finding matrices in ℝ^{n ×n} of minimum Frobenius norm are converted to equivalent problems, using vector representations in ℝⁿ² and ℝ^n(n+1)/2 of these matrices, and then solvingl ₂-minimization problems. These problems are more directly dealt with, and indeed, the paper demonstrates how this technique may be used to handle weighted sparse updates.