Accelerating Min-Max Optimization with Application to Minimal Bounding Sphere

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.Comment: 12 pages, 1 figure, preprint, [v0] 201

Reviewed by Aurél Galántai References

Uniform approximation of min/max functions by smooth splines. (English summary) J. Comput. Appl. Math. 236 (2011), no. 5, 699–703. The authors construct smooth splines to approximate min{f1,..., fn} uniformly for use in optimization problems. They first construct a smooth spline s(x1,..., xn) to approximate min{x1,..., xn} uniformly, and then approximate min{f1,..., fn} by the composite function s(f1,..., fn). Unfortunately, no formally stated theorems or examples are given