Steepest Descent Algorithm for M-convex Function Minimization Using Long Step Length
Taihei Oki, Akiyoshi Shioura
TL;DR
This work tackles the minimization of M-convex functions on $\mathbb{Z}^n$ and proposes long-step lengths within the steepest-descent framework to achieve faster convergence. It introduces M-LSD and its variants (including M-LSD2 and ConstM-LSD families) that move along a steepest direction by multiple units while preserving the slope, yielding improved theoretical bounds such as $O\big(n^2 \log L_{\infty}\, \min\{ |\varphi(x_0)|, \tau(x_0)\}\big)$ iterations in the integer setting. The authors also extend these ideas to constrained problems Min$(f,R,k)$ and to M$^\natural$-convex variants, providing analogous long-step algorithms and time bounds, as well as to polyhedral M-convex functions with PM-LSD, establishing finite termination for the continuous case. Collectively, the results broadens the applicability of long-step steepest-descent techniques to a wide range of discrete and polyhedral optimization problems tied to base polyhedra and polymatroids, with potential practical impact on resource allocation and network-flow related tasks.
Abstract
We consider the minimization of an M-convex function, which is a discrete convexity concept for functions on the integer lattice points. It is known that a minimizer of an Mconvex function can be obtained by the steepest descent algorithm. In this paper, we propose an effective use of long step length in the steepest descent algorithm, aiming at the reduction in the running time. In particular, we obtain an improved time bound by using long step length. We also consider the constrained M-convex function minimization and show that long step length can be applied to a variant of steepest descent algorithm as well.