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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article we consider min-min type of problems or minimization by twogroups of variables. Min-min problems may occur in case if some groups ofvariables in convex optimization have different dimensions or if these groupshave different domains. Such problem structure gives us an ability to split themain task to subproblems, and allows to tackle it with mixed oracles. Howeverexisting articles on this topic cover only zeroth and first order oracles, inour work we consider high-order tensor methods to solve inner problem and fastgradient method to solve outer problem. We assume, that outer problem isconstrained to some convex compact set, and for the inner problem we considerboth unconstrained case and being constrained to some convex compact set. Bydefinition, tensor methods use high-order derivatives, so the time per singleiteration of the method depends a lot on the dimensionality of the problem itsolves. Therefore, we suggest, that the dimension of the inner problem variableis not greater than 1000. Additionally, we need some specific assumptions to beable to use mixed oracles. Firstly, we assume, that the objective is convex inboth groups of variables and its gradient by both variables is Lipschitzcontinuous. Secondly, we assume the inner problem is strongly convex and itsgradient is Lipschitz continuous. Also, since we are going to use tensormethods for inner problem, we need it to be $p$-th order Lipschitz continuous($p > 1$). Finally, we assume strong convexity of the outer problem to be ableto use fast gradient method for strongly convex functions. We need toemphasize, that we use superfast tensor method to tackle inner subproblem inunconstrained case. And when we solve inner problem on compact set, we useaccelerated high-order composite proximal method.

Tensor methods inside mixed oracle for min-min problems