Iteration complexity of the Difference-of-Convex Algorithm for unconstrained optimization: a simple proof

TL;DR

This paper provides a straightforward proof of the worst-case iteration complexity for the Difference of Convex functions Algorithm (DCA) in unconstrained optimization, demonstrating a convergence rate of o(1/k) for the gradient norms.

Contribution

It introduces a simple proof for the iteration complexity of DCA, establishing the convergence rate and providing an example showing the rate's optimality.

Findings

Convergence rate of o(1/k) for DCA in unconstrained optimization

Proof simplicity improves understanding of DCA's complexity

Example indicates the rate cannot be improved

Abstract

We propose a simple proof of the worst-case iteration complexity for the Difference of Convex functions Algorithm (DCA) for unconstrained minimization, showing that the global rate of convergence of the norm of the objective function's gradients at the iterates converge to zero like o(1/k). A small example is also provided indicating that the rate cannot be improved.