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On the Set of Possible Minimizers of a Sum of Convex Functions

Moslem Zamani, François Glineur, Julien M. Hendrickx

Abstract

Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results cover several types of assumptions on the summands, such as smoothness or strong convexity. Our main tool is the use of necessary and sufficient conditions for interpolating the considered function classes, which leads to shorter and more direct proofs in comparison with previous work. We also address the setting where each summand minimizer is assumed to lie in a unit ball, and prove a tight bound on the norm of any minimizer of the sum.

On the Set of Possible Minimizers of a Sum of Convex Functions

Abstract

Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results cover several types of assumptions on the summands, such as smoothness or strong convexity. Our main tool is the use of necessary and sufficient conditions for interpolating the considered function classes, which leads to shorter and more direct proofs in comparison with previous work. We also address the setting where each summand minimizer is assumed to lie in a unit ball, and prove a tight bound on the norm of any minimizer of the sum.
Paper Structure (8 sections, 12 theorems, 57 equations, 3 figures)

This paper contains 8 sections, 12 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Interpolation Conditions
  3. Notations and Definitions
  4. Interpolation Results
  5. Minimizers of a sum of two convex functions
  6. Minimizers of a sum of $m$ convex functions
  7. Bounds when summand minimizers lie in a ball
  8. Conclusions

Key Result

Theorem II.1

(see taylor2017smooth) Consider the class of functions $\mathcal{F}_{\mu,L}(\mathbb{R}^n)$ with $0\leq\mu< L\leq \infty$. Given a set of triplets $\{(x_i; g_i; f_i)\}_{i\in [m]}$ with $x_i,g_i\in\mathbb{R}^n$ and $f_i\in \mathbb{R}$, there exists a function $f\in\mathcal{F}_{\mu,L}(\mathbb{R}^n)$ sa i.e. a function that interpolates the function and (sub)gradient values as prescribed in $(x_i; g_i

Figures (3)

  • Figure 1: Sets of potential minimizers $x^\star$ from Proposition \ref{['Prop.1']} with $x_1^\star = (-1, 0)$ and $x_2^\star = (1, 0)$.
  • Figure 2: Sets of potential minimizers $x^\star$ from Proposition \ref{['Prop.2']} with $x_1^\star = (-1, 0)$ and $x_2^\star = (1, 0)$.
  • Figure 3: Sets of potential minimizers $x^\star$ from Proposition \ref{['Prop.3']} with $x_1^\star = (-1, 0)$ and $x_2^\star = (1, 0)$.

Theorems & Definitions (21)

  • Theorem II.1
  • Lemma II.2
  • proof
  • Proposition III.1
  • proof
  • Proposition III.2
  • proof
  • Proposition III.3
  • proof
  • Theorem III.4
  • ...and 11 more