The Influence of an Adjoint Mismatch on the Primal-Dual Douglas-Rachford Method

TL;DR

This work analyzes how adjoint mismatch affects the primal-dual Douglas-Rachford method for convex-concave saddle-point problems. It establishes fixed-point existence and an explicit primal error bound under strong convexity, and introduces an adapted variant to preserve monotonicity. The authors show that, under suitable parameter choices and strong convexity, the mismatched-adjoint method converges linearly, and they provide a practical parameter-selection strategy. Numerical experiments on convex quadratic problems and computerized tomography demonstrate the approach's viability and competitive performance against mismatched-adjoint alternatives, with clearer benefits when the forward/adjoint discretizations differ substantially.

Abstract

The primal-dual Douglas-Rachford method is a well-known algorithm to solve optimization problems written as convex-concave saddle-point problems. Each iteration involves solving a linear system involving a linear operator and its adjoint. However, in practical applications it is often computationally favorable to replace the adjoint operator by a computationally more efficient approximation. This leads to an adjoint mismatch. In this paper, we analyze the convergence of the primal-dual Douglas-Rachford method under the presence of an adjoint mismatch. We provide mild conditions that guarantee the existence of a fixed point and find an upper bound on the error of the primal solution. Furthermore, we establish step sizes in the strongly convex setting that guarantee linear convergence under mild conditions. Additionally, we provide an alternative method that can also be derived from the Douglas-Rachford method and is also guaranteed to converge in this setting. Moreover, we illustrate our results both for an academic and a real-world inspired example.

Figures (10)

• Figure 1: Left: The result after $10,000$ iterations. Middle: The distance between the iterates and the true solution $x^*$. Right: The objective function value over iterations.
• Figure 2: Left: Value of the objective function over the iterations. Right: Distance between the iterates and the true solution $x^*$.
• Figure 3: Different possible behaviors of the component functions $f_1$ and $f_2$ in the definition of $\eta$.
• Figure 4: Convergence of the iteration \ref{['eq:pddrmm-quadratic']}. Here, $\hat{x}$ is the fixed point \ref{['eq:pdrmmm-quadratic-fp-mm']} of the iteration with the incorrect adjoint, and $x^{*}$ is the original primal solution \ref{['eq:pddrmm-quadratic-fp']}. The solid orange line represents the relative distance of the primal iterates $x^{k}$ from the fixed point of the iteration, while the dashed purple line represents the relative distance of the iterates $x^{k}$ from the original primal solution. As predicted, the latter distance remains below the bound given in Theorem \ref{['thm:error-estimate']}.
• Figure 5: Comparison of the convergence of iterates in the adapted and non-adapted primal-dual Douglas-Rachford method with mismatched adjoint to the fixed point. The green line shows the relative distance of the iterates of the adapted method to the fixed point, while the orange line represents the relative distance of the iterates of the non-adapted primal-dual Douglas-Rachford method to the fixed point. In this example, the convergence speed is nearly identical.

Theorems & Definitions (25)

• lemma 1
• proof
• proposition 1: rockafellar_variationalanalysis
• theorem 1
• proof
• theorem 2
• proof
• lemma 2
• proof
• lemma 3