On the Location of the Minimizer of the Sum of Two Strongly Convex Functions

TL;DR

This paper derives bounds on where the minimizer of the sum of two strongly convex functions can be located, considering different gradient constraints, which aids in understanding distributed optimization problems.

Contribution

It provides the first explicit bounds on the minimizer location for sums of strongly convex functions under various gradient constraints.

Findings

Upper bounds on the minimizer region are established.

Different scenarios with gradient constraints are analyzed.

Characterization of the boundaries of the minimizer region is provided.

Abstract

The problem of finding the minimizer of a sum of convex functions is central to the field of distributed optimization. Thus, it is of interest to understand how that minimizer is related to the properties of the individual functions in the sum. In this paper, we provide an upper bound on the region containing the minimizer of the sum of two strongly convex functions. We consider two scenarios with different constraints on the upper bound of the gradients of the functions. In the first scenario, the gradient constraint is imposed on the location of the potential minimizer, while in the second scenario, the gradient constraint is imposed on a given convex set in which the minimizers of two original functions are embedded. We characterize the boundaries of the regions containing the minimizer in both scenarios.