English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The limited memory BFGS (L-BFGS) method is widely used for large-scaleunconstrained optimization, but its behavior on nonsmooth problems has receivedlittle attention. L-BFGS can be used with or without "scaling"; the use ofscaling is normally recommended. A simple special case, when just one BFGSupdate is stored and used at every iteration, is sometimes also known asmemoryless BFGS. We analyze memoryless BFGS with scaling, using anyArmijo-Wolfe line search, on the function $f(x) = a|x^{(1)}| + \sum_{i=2}^{n}x^{(i)}$, initiated at any point $x_0$ with $x_0^{(1)}\not = 0$. We show thatif $a\ge 2\sqrt{n-1}$, the absolute value of the normalized search directiongenerated by this method converges to a constant vector, and if, in addition,$a$ is larger than a quantity that depends on the Armijo parameter, then theiterates converge to a non-optimal point $\bar x$ with $\bar x^{(1)}=0$,although $f$ is unbounded below. As we showed in previous work, the gradientmethod with any Armijo-Wolfe line search also fails on the same function if$a\geq \sqrt{n-1}$ and $a$ is larger than another quantity depending on theArmijo parameter, but scaled memoryless BFGS fails under a weaker conditionrelating $a$ to the Armijo parameter than that implying failure of the gradientmethod. Furthermore, in sharp contrast to the gradient method, if a specificstandard Armijo-Wolfe bracketing line search is used, scaled memoryless BFGSfails when $a\ge 2 \sqrt{n-1} $ regardless of the Armijo parameter. Finally,numerical experiments indicate that similar results hold for scaled L-BFGS withany fixed number of updates.

Analysis of Limited-Memory BFGS on a Class of Nonsmooth Convex Functions