Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity

TL;DR

The paper addresses the convergence of the Chambolle–Pock algorithm for composite inclusions 0 ∈ Ax + L^⊤ BLx with possibly nonmonotone A and B. It casts CPA as a relaxed PPPA and uses a V-oblique weak Minty framework to derive step-size and relaxation bounds that depend on the singular values of L, not just its operator norm, and to accommodate semimonotone operators. A generalized (M,R)-semimonotone operator class is developed to certify the oblique Minty condition, unifying and extending monotone results and enabling convergence proofs for nonmonotone and strongly monotone settings; the approach yields tight bounds demonstrated by linear examples and recovers known results as special cases. The resulting CPA convergence theory accommodates rank-deficient L without switching schemes and provides practical look-up tables for step-sizes, broadening applicability to large-scale structured problems in convex/variational analysis. Overall, the work advances CPA’s theoretical scope and provides tools for verifying convergence in broader nonmonotone regimes with concrete parameter guidance.

Abstract

The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method, has gained popularity over the last decade due to its success in solving large-scale convex structured problems. This work extends its convergence analysis for problems with varying degrees of (non)monotonicity, quantified through a so-called oblique weak Minty condition on the associated primal-dual operator. Our results reveal novel stepsize and relaxation parameter ranges which do not only depend on the norm of the linear mapping, but also on its other singular values. In particular, in nonmonotone settings, in addition to the classical stepsize conditions, extra bounds on the stepsizes and relaxation parameters are required. On the other hand, in the strongly monotone setting, the relaxation parameter is allowed to exceed the classical upper bound of two. Moreover, we build upon the recently introduced class of semimonotone operators, providing sufficient convergence conditions for CPA when the individual operators are semimonotone. Since this class of operators encompasses traditional operator classes including (hypo)- and co(hypo)-monotone operators, this analysis recovers and extends existing results for CPA. Tightness of the proposed stepsize ranges is demonstrated through several examples.

Figures (5)

• Figure 1: Range of the stepsizes $\gamma$ and $\tau$ for \ref{['eq:CP']}.
• Figure 2: Definition of $\eta^\prime$ in \ref{['ass:CP:relaxation:rule']}.
• Figure 3: Convergence of the sequence $\seq{s^k} = \bigl( X_1^\top x^k - Y_1^\top y^k , \; X_{2:}^\top x^k , \; Y_{2:}^\top y^k \bigr)_{k \in \N}$ from \ref{['ex:CP:second']} for $n = 3$, $\ell_2 = 1/2$, $\ell_3 = 1/5$ and $\lambda = 2.1$. (a) Norm of the sequence $\seq{z^k} = \seq{x^k, y^k}$. This sequence does not converge, since $\lambda$ has been selected larger than two (see \ref{['it:CP:full']}). (b) Norm of the sequence $\seq{X_1^\top x^k - Y_1^\top y^k}$, which converges to zero. (c) Visualization of the primal sequences $\seq{x^k}$ and $\seq{X_{2:}^\top x^k}$. It can be seen that although $\seq{x^k}$ does not converge (its first coordinate diverges), its projection onto the 2-dimensional space spanned by the columns of $X_{2:}$ does converge to zero (marked by a green dot). (d) Visualization of the dual sequences $\seq{y^k}$ and $\seq{Y_{2:}^\top y^k}$. Analogous to the primal setting, $\seq{y^k}$ diverges while $\seq{Y_{2:}^\top y^k}$ converges to zero.
• Figure 5: Range of the stepsizes $\gamma$ and $\tau$ for \ref{['eq:CP']} involving semimonotone operators.
• Figure 6: Definition of $\eta^\prime$ in \ref{['ass:CP:relaxation:rule:semi']}.

Theorems & Definitions (9)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof