Note on Steepest Descent Algorithm for Quasi L$^{\natural}$-convex Function Minimization

TL;DR

This paper extends the steepest descent algorithm to a new class of semi-strictly quasi L$^{ atural}$-convex functions, demonstrating its effectiveness and analyzing its iteration complexity and geodesic properties.

Contribution

It introduces semi-strictly quasi L$^{ atural}$-convex functions and proves the steepest descent algorithm's applicability and properties for this class.

Findings

Steepest descent algorithm works for semi-strictly quasi L$^{ atural}$-convex functions.

The iteration count for the algorithm is precisely characterized.

The algorithm exhibits a geodesic property on this function class.

Abstract

We define a class of discrete quasi convex functions, called semi-strictly quasi L$^{\natural}$-convex functions, and show that the steepest descent algorithm for L$^{\natural}$-convex function minimization also works for this class of quasi convex functions. The analysis of the exact number of iterations is also extended, revealing the so-called geodesic property of the steepest descent algorithm when applied to semi-strictly quasi L$^{\natural}$-convex functions.