# Accelerating Min-Max Optimization with Application to Minimal Bounding Sphere

## TL;DR

This paper introduces a smoothing technique for min-max optimization problems with strongly convex functions, significantly reducing the computational complexity to achieve small optimality gaps, and applies it to efficiently approximate minimal bounding spheres.

## Contribution

The paper presents a novel smoothing approach that accelerates min-max optimization and improves the computational efficiency for approximating minimal bounding spheres.

## Key findings

- Achieves an arbitrarily small optimality gap in ilde{O}(1/\sqrt{ ext{gap}}) time.
- Provides a ilde{O}(n d /\sqrt{ ext{epsilon}}) algorithm for ext{(1+ ext{epsilon})}-approximate minimal bounding sphere.
- Outperforms state-of-the-art methods in computational complexity for bounding sphere approximation.

## Abstract

We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an arbitrarily small positive optimality gap of $\delta$ in $\tilde{O}(1/\sqrt{\delta})$ computational complexity (up to logarithmic factors) as opposed to the state-of-the-art strong-convexity computational requirement of $O(1/\delta)$. We apply this important result to the well-known minimal bounding sphere problem and demonstrate that we can achieve a $(1+\varepsilon)$-approximation of the minimal bounding sphere, i.e. identify an hypersphere enclosing a total of $n$ given points in the $d$ dimensional unbounded space $\mathbb{R}^d$ with a radius at most $(1+\varepsilon)$ times the actual minimal bounding sphere radius for an arbitrarily small positive $\varepsilon$, in $\tilde{O}(n d /\sqrt{\varepsilon})$ computational time as opposed to the state-of-the-art approach of core-set methodology, which needs $O(n d /\varepsilon)$ computational time.