Dual Prices for Frank--Wolfe Algorithms
G\'abor Braun, Sebastian Pokutta

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 over a polytope , dual prices for the linear program obtained from linearization at approximately optimal solutions 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 over , which in most implementations already compute the dual prices as a by-product.
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Taxonomy
TopicsAdvanced Optimization Algorithms Research · Stochastic Gradient Optimization Techniques · Optimization and Search Problems
