VPTPD: Variable-Preconditioned Primal-Dual Methods

• VPTPD is a family of primal-dual methods that integrate transformed saddle-point systems with variable preconditioners to enhance convergence and conditioning.
• These methods utilize operator-based metrics such as diagonal, Hessian-derived, and Schur-complement preconditioners to keep proximal subproblems tractable.
• Applications span from nonsmooth convex optimization to nonlinear PDEs, demonstrating significant speedups, stability, and mesh-independence in numerical simulations.

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Variable-Preconditioned Transformed Primal-Dual (VPTPD) denotes a family of primal-dual algorithms in which transformed saddle-point dynamics are combined with nonstationary preconditioners or variable metrics. Across the recent literature, the name appears in several closely related settings: variable-wise diagonal preconditioning for primal-dual splitting, transformed primal-dual methods with evolving Schur-complement surrogates, structure-preserving JKO solvers for generalized Wasserstein gradient flows, and duality-based solvers for nonlinear PDEs (Naganuma et al., 2023, Chen et al., 2023, Zeng et al., 18 Sep 2025, Chen et al., 17 Oct 2025). The common design pattern is the replacement of fixed scalar step-sizes by operators such as block-diagonal, diagonal, Hessian-derived, Schur-complement, or antisymmetric preconditioners, with the aim of preserving tractable proximal subproblems while improving conditioning, convergence speed, or mesh-independence. This suggests that “VPTPD” is best understood as a methodological family rather than a single canonical recursion.

1. Scope, nomenclature, and problem classes

The VPTPD label is attached to several algorithmic variants that share a transformed primal-dual structure but target different mathematical models.

Variant	Problem class	Characteristic preconditioner
P-PDS with OVDP (Naganuma et al., 2023)	Nonsmooth convex optimization in split form	Variable-wise diagonal preconditioners from operator-norm bounds
TPDv (Chen et al., 2023)	Affine constrained optimization in Hilbert spaces	Time-evolving SPD metric operators and Schur-complement proxies
VPTPD for generalized Wasserstein gradient flows (Zeng et al., 18 Sep 2025)	Structure-preserving JKO discretizations	Diagonal Hessian-based primal metric and Schur-complement dual metric
DualTPD (Chen et al., 17 Oct 2025)	Nonlinear PDEs via Fenchel–Rockafellar duality	Variable block preconditioners for nonlinear and Schur blocks
Variable-metric primal-dual Douglas–Rachford (Lorenz et al., 2023)	Saddle-point monotone inclusions	Degenerate block-diagonal varying preconditioners
Time-varying preconditioned primal-dual dynamics (Apidopoulos et al., 31 May 2025)	Convex optimization of the form $f(x)+g(Ax)$	Time-dependent symmetric/skew preconditioners, including antisymmetric regimes

In the split convex setting, the central problem class is

$$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$

with associated dual blocks $y_j$ and linear operators $L_{j,i}$ (Naganuma et al., 2023). In affine-constrained optimization, the prototype is

$\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$

with $f$ continuously differentiable, strongly convex, and Lipschitz differentiable in a variable metric (Chen et al., 2023). In generalized Wasserstein gradient flows, the variational structure arises from the JKO minimization

$$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$

which, after Benamou–Brenier reformulation and discretization, becomes an equality-constrained finite-dimensional optimization problem (Zeng et al., 18 Sep 2025). In nonlinear PDE applications, Fenchel–Rockafellar duality is used to rewrite energies of the form $\min_{u\in\mathcal V}F(Du)+G(u)$ as nonlinear saddle systems with separated differential and nonlinear blocks (Chen et al., 17 Oct 2025).

2. Mathematical structure and transformed saddle systems

A recurring feature of VPTPD methods is the conversion of optimization problems into saddle-point systems in which primal and dual variables can be updated under tailored metrics.

For split convex problems, introducing dual variables $y_j$ yields the convex-concave saddle formulation

$$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$

which is then attacked by preconditioned primal-dual splitting (Naganuma et al., 2023). In affine-constrained settings, the Lagrangian

$$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$0

leads to the KKT system

$$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$1

and the transformed primal-dual construction uses an SPD operator $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$2 to rewrite the system so that the dual block acquires a Schur-complement term $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$3 (Chen et al., 2023, Zeng et al., 18 Sep 2025).

The same transformation principle appears in the nonlinear PDE literature. In the TPDv framework, the continuous transformed flow is written with time-dependent SPD operators

$$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$4

and an approximation $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$5 where $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$6 (Chen et al., 2023). In DualTPD, after applying the triangular transform

$$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$7

the nonlinear saddle system becomes an upper-block form with $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$8 and elliptic Schur complement $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$9 (Chen et al., 17 Oct 2025).

A related but distinct continuous-time formulation considers the monotone inclusion associated with

$y_j$0

and replaces the un-preconditioned Arrow–Hurwicz flow by

$y_j$1

where $y_j$2 is a time-varying $y_j$3 block preconditioner with symmetric and skew parts (Apidopoulos et al., 31 May 2025). This places VPTPD in direct contact with continuous-time primal-dual dynamics and with the preconditioning logic behind Chambolle–Pock-type schemes.

3. Variable preconditioners and metric design

The defining feature of VPTPD is the explicit design of nonconstant preconditioners from operator norms, local Hessians, Schur complements, or problem-specific block structure.

In the OVDP construction for preconditioned primal-dual splitting, one defines

$y_j$4

for $y_j$5 (Naganuma et al., 2023). The three stated special cases are $y_j$6 (OVDP2), $y_j$7 (OVDP1), and $y_j$8 (OVDP3). The construction uses only upper bounds on operator norms, not explicit matrix entries, and because it is variable-wise rather than element-wise, it “keeps any proximity operator analytically computable” (Naganuma et al., 2023).

In TPDv for affine-constrained optimization, the preconditioners are evolving metric operators $y_j$9 and $L_{j,i}$0, together with a Schur-complement proxy $L_{j,i}$1. The discrete update

$L_{j,i}$2

shows that the dual metric is an interpolation of the previous preconditioner and a new Schur-complement approximation (Chen et al., 2023).

In the generalized Wasserstein setting, the paper defines a regularized smooth objective

$L_{j,i}$3

and sets

$L_{j,i}$4

(Zeng et al., 18 Sep 2025). The stated rationale is that $L_{j,i}$5 “approximates the inverse Hessian so as to reduce condition-number, yet remains diagonal for cheap inversion and separability,” while $L_{j,i}$6 is “its associated Schur-complement to precondition the dual” (Zeng et al., 18 Sep 2025).

In DualTPD, the nonlinear and elliptic blocks are both preconditioned:

$L_{j,i}$7

The nonlinear block may use a mass matrix inverse or Jacobian inverse, and the Schur block may be approximated by geometric multigrid or PCG with AMG/Hiptmair–Xu auxiliary-space (Chen et al., 17 Oct 2025).

The continuous-time preconditioned dynamics literature adds a further design axis: asymptotically antisymmetric preconditioners. There, convergence analysis requires uniform positive definiteness of the symmetric part $L_{j,i}$8 and, for the general problem, the asymptotic condition

$L_{j,i}$9

as stated in the source (Apidopoulos et al., 31 May 2025). In the exact-antisymmetric case, one takes $\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$0.

4. Iterative schemes, proximal steps, and transformed updates

The update rules vary with the application, but they retain a recognizable primal-dual structure with variable metrics.

For block-split nonsmooth optimization, the preconditioned primal-dual splitting updates are

$\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$1

or, blockwise,

$\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$2

(Naganuma et al., 2023). The implementation computes $\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$3 and $\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$4 before the two prox evaluations.

For affine constraints with a transformed primal-dual step, the TPDv discretization performs four operations per iteration: a slave-step for $\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$5, an update of the dual-metric preconditioner $\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$6, a dual update, and an over-relaxation for $\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$7 (Chen et al., 2023). The generalized Wasserstein VPTPD method modifies this further into a semi-implicit-explicit proximal iteration:

$\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$8

$\min_{x\in V}\; f(x)\qquad\text{s.t.}\quad Bx=b,$9

with optional over-relaxations

$f$0

(Zeng et al., 18 Sep 2025).

A central subroutine in that setting is the generalized proximal operator

$f$1

which becomes cellwise separable for $f$2 (Zeng et al., 18 Sep 2025). The resulting scalar equation $f$3 is solved by Newton’s method,

$f$4

with an interval bracketing the unique root and a tailored initial guess; the paper states “global Newton convergence and bound-preservation” (Zeng et al., 18 Sep 2025).

The Douglas–Rachford variant proceeds from the non-stationary proximal-point iteration with block-diagonal preconditioner

$f$5

and factorizes the step into two half-resolvents. Its explicit VPTPD iteration computes

$f$6

forms reflected points $f$7 and $f$8, solves

$f$9

and then updates $$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$0, $$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$1 (Lorenz et al., 2023).

DualTPD for nonlinear PDEs uses a two-stage correction:

$$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$2

followed by

$$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$3

(Chen et al., 17 Oct 2025).

5. Convergence theory and stability results

The convergence theory attached to VPTPD depends on the specific formulation, but several recurring themes are visible: norm-control of the preconditioned coupling, Lyapunov decay, monotonicity under transformed dynamics, and variable-metric fixed-point arguments.

For OVDP in preconditioned primal-dual splitting, the critical quantity is

$$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$4

and the P–PDS convergence theorem requires

$$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$5

The paper proves that the OVDP choice of $$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$6 and $$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$7 satisfies the convergence condition and states: “Let $$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$8 be proper closed convex, and let $$\rho^{k+1} \in \mathrm{Arg}\,\min_\rho \left\{ \frac{1}{2\tau}d_{\mathcal W_m}^2(\rho,\rho^k)+\mathcal J(\rho)\right\},$$9, $\min_{u\in\mathcal V}F(Du)+G(u)$0 be chosen by OVDP as above. Then the sequence $\min_{u\in\mathcal V}F(Du)+G(u)$1 generated by the P–PDS iterations converges to a saddle point of the convex–concave problem” (Naganuma et al., 2023).

For TPDv in Hilbert spaces, the analysis is Lyapunov-based. The continuous flow is shown to be exponentially stable under assumptions including $\min_{u\in\mathcal V}F(Du)+G(u)$2 with $\min_{u\in\mathcal V}F(Du)+G(u)$3, and the discrete method has linear convergence. The paper states that there exists $\min_{u\in\mathcal V}F(Du)+G(u)$4 such that

$\min_{u\in\mathcal V}F(Du)+G(u)$5

and for the canonical choice of $\min_{u\in\mathcal V}F(Du)+G(u)$6 gives the explicit bound

$\min_{u\in\mathcal V}F(Du)+G(u)$7

(Chen et al., 2023).

For generalized Wasserstein gradient flows, the VPTPD iteration combines proximal splitting with Hessian-derived metrics. Under convexity of $\min_{u\in\mathcal V}F(Du)+G(u)$8, strong convexity of $\min_{u\in\mathcal V}F(Du)+G(u)$9, Lipschitz continuity of $y_j$0 with respect to the $y_j$1 metric, full-row-rank $y_j$2, and step-size bounds, the iteration “converges globally at a linear (geometric) rate”:

$y_j$3

for some $y_j$4 (Zeng et al., 18 Sep 2025). The same paper also proves “existence and uniqueness of bounded solutions for the resulting generalized proximal operator,” together with a “convergent and bound-preserving Newton solver” (Zeng et al., 18 Sep 2025).

DualTPD emphasizes strong monotonicity and mesh independence. Under the assumptions on $y_j$5 and the inf–sup stability of $y_j$6, the preconditioned nonlinear saddle map remains strongly monotone; a Lyapunov energy

$y_j$7

is shown to decrease, and the contraction factor is bounded away from $y_j$8 independently of mesh size, so “the iteration count is mesh-independent” (Chen et al., 17 Oct 2025).

The variable-metric Douglas–Rachford paper establishes weak convergence under bounded, self-adjoint, positive semidefinite, convergent preconditioners $y_j$9, summable metric variation, and single-valued resolvents; the proof uses quasi-Fejér arguments, Opial’s lemma, and the variable-metric proximal-point theorem (Lorenz et al., 2023). The continuous-time preconditioned dynamics paper, by contrast, derives “non-ergodic decay of the primal–dual gap” under asymptotically antisymmetric preconditioners, while warning that “in the symmetric case, the non-ergodic gap need not converge” (Apidopoulos et al., 31 May 2025).

6. Applications, empirical behavior, and practical interpretation

The application range of VPTPD is unusually broad. In the OVDP paper, the three demonstration problems are “mixed noise removal of hyperspectral images, hyperspectral unmixing, and graph signal recovery” (Naganuma et al., 2023). For hyperspectral mixed-noise removal with SSTV regularization, “OVDP2 typically cut the required iterations/time by 3×–10× compared to unpreconditioned PDS.” For hyperspectral unmixing in CLSUnSAL, “OVDP1/2 reduced runtime roughly 5× over scalar PDS and 10×–20× over ASP/PDP.” For graph-signal recovery with GTV sampling, “OVDP2 gave the fastest convergence in both iterations and CPU time” (Naganuma et al., 2023).

In generalized Wasserstein gradient flows, the tested simulations range “from 1D to 3D settings” and include “saturation equation, Cahn–Hilliard, droplet wetting.” The reported gains are: “Reduced iteration counts by up to 99 % over basic PDJKO and 70–90 % over PrePDJKO,” “Achieved 4–20× CPU–time speedups, especially in higher dimensions,” “Preserved energy dissipation, mass conservation and bounds exactly,” and “Adaptive step-size further accelerated convergence by 20–30 %” (Zeng et al., 18 Sep 2025).

In the TPDv paper, two nonlinear PDEs are used: Darcy–Forchheimer flow and a nonlinear Maxwell problem. For Darcy–Forchheimer, the excerpted table for refinement levels $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$0–$$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$1 states that “iteration counts remain constant; CPU time grows $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$2.” For the nonlinear Maxwell problem, “VPTPD requires 30–50 outer iterations, each costing 2–3 V-cycles, vs. 70–100+ iterations in the other methods. CPU times are 2–5× faster in large-scale tests” (Chen et al., 2023).

DualTPD extends this PDE performance picture. For the $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$3-Laplacian with $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$4, the reported behavior is “iteration counts $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$5–$$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$6 independent of $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$7, CPU time $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$8,” and “for $$\min_{x}\max_{y}\;\sum_{i=1}^N f_i(x_i)\;+\;\sum_{j=1}^M\langle L_{j,i}x_i,y_j\rangle \;-\;\sum_{j=1}^M g_j^*(y_j),$$9, DualTPD-J exhibits superlinear (Newton-like) behavior from zero initial data.” For nonlinear $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$00 problems, including ferromagnetism and the $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$01-curl analogue, the reported results are “15–18 DualTPD iterations, $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$02–10 PCG inner steps, iteration count independent of mesh $$\min_{x\in\mathbb R^n}\;\sum_{i=1}^Nf_i(x_i)\;+\;\sum_{j=1}^M g_j\Bigl(\sum_{i=1}^NL_{j,i}x_i\Bigr),$$03, CPU time growing linearly with DoF.” The paper further states that, compared with “a ‘plain’ primal–dual (Chambolle–Pock) solver or the primal TPD method, DualTPD is 2–5× faster and has far fewer iterations” (Chen et al., 17 Oct 2025).

The continuous-time preconditioned dynamics experiments support the asymptotically antisymmetric design principle. In quadratic minimization, linearly constrained quadratic programs, and LASSO, “the antisymmetric variant decays orders of magnitude faster than either the symmetric preconditioning or the un-preconditioned flow,” and “antisymmetric again gives the fastest primal–dual gap decay” (Apidopoulos et al., 31 May 2025).

A common misconception is to equate VPTPD with one specific update formula. The literature instead shows a stable conceptual core—transformed primal-dual structure, variable preconditioning, and preserved tractability of the primal and dual subproblems—but multiple realizations adapted to split convex models, JKO schemes, Douglas–Rachford factorizations, and nonlinear PDE discretizations (Naganuma et al., 2023, Chen et al., 2023, Zeng et al., 18 Sep 2025, Chen et al., 17 Oct 2025). A plausible implication is that future developments will continue to differ mainly in how they choose the metric operators and in which structural invariants—such as bound-preservation, energy dissipation, or mesh-independence—they enforce.