Dual Prices for Frank--Wolfe Algorithms

TL;DR

This paper shows that dual prices in Frank--Wolfe algorithms can be interpreted as a convex sensitivity analysis tool, linking linearization at approximate solutions to the rate of change in optimal value.

Contribution

It introduces a convex form of sensitivity analysis for constrained convex minimization problems using dual prices in Frank--Wolfe algorithms.

Findings

Dual prices provide a rate of change in optimal value similar to linear programming.

Dual prices can be obtained as a by-product of linear minimization in Frank--Wolfe.

This interpretation enhances understanding of Frank--Wolfe algorithm behavior.

Abstract

In this note we observe that for constrained convex minimization problems $\min_{x \in P}f(x)$ over a polytope $P$, dual prices for the linear program $\min_{z \in P} \nabla f(x) z$ obtained from linearization at approximately optimal solutions $x$ have a similar interpretation of rate of change in optimal value as for linear programming, providing a convex form of sensitivity analysis. This is of particular interest for Frank--Wolfe algorithms (also called conditional gradients), forming an important class of first-order methods, where a basic building block is linear minimization of gradients of $f$ over $P$, which in most implementations already compute the dual prices as a by-product.