PDTR: Primal-Dual Twice-Reflected Algorithm

• PDTR is a primal-dual algorithm that uses twice-reflected splitting to handle structured monotone inclusions and three-operator composite optimization problems.
• It employs one proximal step for A and one for C⁻¹ alongside evaluations of B, K, and K*, with parameter b balancing reflection aggressiveness and stability.
• PDTR is positioned alongside methods like Condat–Vũ and PDDY, offering competitive empirical convergence and flexible application in both smooth and nonsmooth regimes.

Searching arXiv for the cited PDTR-related papers to ground the article in the referenced preprints. The Primal-Dual Twice-Reflected Algorithm (PDTR) is a primal-dual forward-reflected-backward algorithm for structured monotone inclusion and three-operator composite optimization problems. In the monotone inclusion setting, it addresses problems of the form

$$\text{Find } x \in \mathbb{H}_1 \text{ such that } 0 \in A x + B x + K^* C K x,$$

where $A$ and $C$ are maximally monotone, $B$ is monotone and $L$-Lipschitz, and $K$ is linear with adjoint $K^*$ (Soe et al., 2 Sep 2025). In the composite optimization setting, PDTR is presented for

$$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$

and is positioned alongside Condat–Vũ (CV) and Primal-Dual Davis-Yin (PDDY) as a primal-dual three-operator splitting method that is highly versatile but inherently non-accelerated in its standard form (Sadiev et al., 26 May 2026).

1. Problem class and operator-theoretic formulation

PDTR is associated with a structured monotone inclusion in real Hilbert spaces: $0 \in A x + B x + K^* C K x,$ with $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$ and $A$0 maximally monotone, $A$1 monotone and $A$2-Lipschitz, and $A$3 linear with adjoint $A$4 (Soe et al., 2 Sep 2025). The corresponding primal-dual coupled inclusions are

$A$5

In block form, with $A$6,

$A$7

A parallel formulation appears in composite convex optimization, where the objective is

$A$8

with $A$9 smooth and possibly strongly convex, $C$0 convex and possibly strongly convex, $C$1 convex, and $C$2 linear (Sadiev et al., 26 May 2026). That problem is equivalently expressed as the saddle-point problem

$C$3

with optimality conditions

$C$4

These formulations locate PDTR within the standard monotone-operator and primal-dual saddle-point paradigms. This suggests that PDTR is best understood not as an isolated scheme, but as a structured splitting method for coupled primal and dual variables in which the term $C$5 is handled through a saddle formulation rather than directly.

2. Canonical PDTR iteration

The standard PDTR is obtained as the specialization $C$6 of the extended primal-dual twice-reflected algorithm (EPDTR) (Soe et al., 2 Sep 2025). With stepsizes $C$7, the PDTR iteration is

$C$8

where $C$9, and similarly for $B$0 (Soe et al., 2 Sep 2025). The admissible stepsizes satisfy

$B$1

The algorithm is “twice-reflected” through the appearance of the doubly reflected $B$2-term in the primal update, namely the combination $B$3. In the same source, PDTR is described as having “Double, on $B$4” reflection terms and per-iteration complexity of “1 prox on $B$5, 1 prox on $B$6, 1 $B$7, 1 $B$8, 1 $B$9” (Soe et al., 2 Sep 2025).

In the three-operator composite optimization framework, two closely related forms are given, denoted PDTR-I and PDTR-II (Sadiev et al., 26 May 2026). Their updates are

$L$0

for PDTR-I, and

$L$1

for PDTR-II (Sadiev et al., 26 May 2026).

A plausible implication is that the notation “PDTR” now covers two closely related usages in the literature: one in the monotone inclusion framework with resolvents $L$2 and $L$3, and one in the composite optimization framework with proximal maps of $L$4 and $L$5 and reflected gradient evaluations of $L$6.

3. Derivation from generalized forward-reflected-backward splitting

A central structural result is that PDTR is a specialization of the generalized forward-reflected-backward (GFRB) method when the problem is lifted to a product Hilbert space and equipped with a particular metric (Soe et al., 2 Sep 2025). For inclusions of the form $L$7, with $L$8 maximally monotone and $L$9 monotone and $K$0-Lipschitz, the GFRB step in a general metric $K$1 is

$K$2

To recover PDTR or EPDTR, one sets

$K$3

and chooses the metric

$K$4

which is positive definite if $K$5 (Soe et al., 2 Sep 2025). Under this construction, “the PDTR is a specialization of GFRB applied in a particular metric and product Hilbert space,” and when $K$6, it precisely matches the PDTR scheme (Soe et al., 2 Sep 2025).

The same source states that the reflection terms in the $K$7 block arise naturally from the higher-order reflections in GFRB, while the use of the resolvent $K$8 produces the primal and dual updates with correct coupling. This places PDTR within a broader reflected-splitting genealogy: FRB is a specific case of GFRB with $K$9, or $K^*$0 in the PDTR context, whereas PDTR imports this reflected mechanism into a primal-dual block structure.

4. Extended PDTR and parameterization by $K^*$1

The extended primal-dual twice-reflected algorithm (EPDTR) is defined by the iteration

$K^*$2

for $K^*$3, with initialization $K^*$4, $K^*$5, scalar $K^*$6, and stepsizes satisfying

$K^*$7

(Soe et al., 2 Sep 2025).

The standard PDTR is recovered by setting $K^*$8. For general $K^*$9, the step-size domain shrinks according to the factor $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$0. The source explicitly states that “the step-size admissible domain shrinks as the reflection parameter $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$1 increases, reflecting a tradeoff between aggressiveness of reflection and stability” (Soe et al., 2 Sep 2025). It also records practical guidance: setting $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$2 yields the “standard” PDTR; nonzero $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$3 gives the extended GFRB/EPDTR variant; larger $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$4 can accelerate convergence but shrinks the step-size region; and one should ensure that $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$5 remains positive definite.

The same parameterization has an analogue in the broader GFRB setting, where the iteration can be written with a parameter $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$6, and FRB appears as the case $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$7 (Soe et al., 2 Sep 2025). This suggests that EPDTR should be interpreted as a metric-induced reflected family rather than merely an ad hoc modification of PDTR.

5. Convergence properties, rates, and complexity

For the monotone inclusion formulation, convergence is established under the condition

$$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$8

in which case the sequence $$\min_{x \in \mathbb{R}^{d_x}} f(x) + g(x) + h(\mathsf{K}x),$$9 converges to a solution of the coupled primal-dual inclusion; for $0 \in A x + B x + K^* C K x,$0, the region is maximized as

$0 \in A x + B x + K^* C K x,$1

(Soe et al., 2 Sep 2025). The same source states that both the primal and dual iterates converge simultaneously.

Regarding rates, the paper on GFRB and EPDTR does not give “a precise rate bound for PDTR in the general setting,” but it states that, in analogous settings, “the best possible linear rate is often determined by the spectral radius arising from the iteration matrix, with sharpness proofs in low-dimensional examples” (Soe et al., 2 Sep 2025). It further states that, for a particular example, the rate of convergence of GFRB is tight to FRB, and in another example, the rate of convergence of GFRB is improved compared to existing algorithms by choosing proper initial parameters. It also states that the step-size region of PDTR is “optimal up to a factor of 2” compared to standard FBF and that the algorithm inherits the improved empirical convergence of reflected methods (Soe et al., 2 Sep 2025).

In the accelerated three-operator framework, standard PDTR is described as non-accelerated, while accelerated variants APDTR-I and APDTR-II are derived through a unified Bregman primal-dual framework (Sadiev et al., 26 May 2026). For all APDTR and accelerated CV methods, convergence is proven via a Lyapunov function

$0 \in A x + B x + K^* C K x,$2

with the signs depending on the algorithmic form (Sadiev et al., 26 May 2026).

In the smooth regime, if $0 \in A x + B x + K^* C K x,$3 is $0 \in A x + B x + K^* C K x,$4-strongly convex, $0 \in A x + B x + K^* C K x,$5 is $0 \in A x + B x + K^* C K x,$6-strongly convex, and $0 \in A x + B x + K^* C K x,$7 is $0 \in A x + B x + K^* C K x,$8-smooth, the Lyapunov function satisfies

$0 \in A x + B x + K^* C K x,$9

and the best parameter choices produce the accelerated rate

$A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$0

(Sadiev et al., 26 May 2026). In the nonsmooth regime, if $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$1 is strongly convex and smooth and $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$2 is injective on the relevant subspace, the paper gives the rate

$A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$3

(Sadiev et al., 26 May 2026).

6. Relation to FRB, CV, PDDY, Chambolle-Pock, and accelerated variants

PDTR is explicitly compared with GFRB and FRB in the monotone inclusion literature. The comparison table in the GFRB paper gives the following summary (Soe et al., 2 Sep 2025):

Method	Per iteration complexity	Step-size range
PDTR	1 prox on $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$4, 1 prox on $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$5, 1 $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$6, 1 $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$7, 1 $A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$8	$A: \mathbb{H}_1 \rightrightarrows \mathbb{H}_1$9
GFRB	1 prox, 1 evaluation of $A$00	$A$01
FRB	1 prox, 1 $A$02	$A$03

The accompanying interpretation is that PDTR “encompasses the structure” through splitting of $A$04 via a dual variable and $A$05, while GFRB is more general and FRB is a specific case (Soe et al., 2 Sep 2025).

Within composite optimization, PDTR is placed next to Condat–Vũ (CV) and PDDY as a three-operator primal-dual method (Sadiev et al., 26 May 2026). The same source states that all of these methods share the goal of computing proximal steps for $A$06 and $A$07 via $A$08 and Fenchel duality together with the gradient of a smooth function $A$09, but PDTR differs in that it “reflects” the forward steps in both the primal and dual directions in a specific way. It also states that the APDTR methods can be viewed through the Chambolle-Pock algorithm applied to primal-dual reformulations with Bregman divergences replacing Euclidean distance terms, thereby yielding accelerated PDTR and accelerated CV families (Sadiev et al., 26 May 2026).

The same paper further states that APDTR-I and APDTR-II achieve the same theoretical iteration complexity as the accelerated Condat–Vũ algorithms, and that they are “optimally accelerated for the problem class” (Sadiev et al., 26 May 2026). It also relates APDTR to CV, PDDY, and Chambolle-Pock more directly: APDTR can be viewed as a “reflection” of CV; PDDY uses Douglas–Rachford splitting applied to a monotone inclusion; and the Bregman acceleration recipe generalizes Chambolle-Pock ideas from two operators to three (Sadiev et al., 26 May 2026).

7. Implementation guidance, empirical behavior, and limitations

The practical implementation profile stated for PDTR in the monotone inclusion setting is concise: each iteration requires one evaluation of $A$10, one evaluation of $A$11, and one evaluation each of $A$12, $A$13, and $A$14 (Soe et al., 2 Sep 2025). The metric

$A$15

is critical, and positive definiteness is required. The same source recommends starting with $A$16 or small, and maximizing $A$17 within the admissible region for best progress per iteration.

The empirical claims reported are that the proposed GFRB-based approach is validated on several state-of-the-art problems and demonstrates better numerical performance compared to existing methods (Soe et al., 2 Sep 2025). More specifically, PDTR variants with increased flexibility via extended GFRB often outperform classical FRB and FBF-type methods both in iteration count and computation time, especially when $A$18 is expensive (Soe et al., 2 Sep 2025). In the accelerated composite-optimization framework, APDTR is described as particularly suited when neither $A$19 nor $A$20 are zero and both are nonsmooth and/or strongly convex, settings in which standard accelerated PGD methods are inapplicable and classical methods such as CV and PDDY lack acceleration (Sadiev et al., 26 May 2026).

A stated limitation of the monotone inclusion analysis is that current analysis and algorithms require knowledge of $A$21 and $A$22 for admissible stepsizes (Soe et al., 2 Sep 2025). The same source notes that parameter-free PDTR, based on linesearch or adaptive estimation, is an interesting direction for future research. This suggests that one unresolved issue is not the basic splitting architecture of PDTR, but the extent to which it can be made adaptive without sacrificing the established convergence guarantees.