Haraux Function: Theory & Applications

• Haraux function is a central analytical object in monotone operator theory, serving as a versatile tool to quantify discrepancies and energy properties in various applications.
• It underpins range formulae, duality approaches, and error estimates in splitting algorithms, linking monotonicity and convex representative functions.
• Recent advances sharpen its lower bounds and extend applications to Lyapunov functionals, stability analysis in PDEs, and asymptotic convergence in evolution equations.

The Haraux function is a central analytical object in monotone operator theory, variational analysis, infinite-dimensional dynamical systems, and stability theory. It takes diverse forms depending on context: as the Fitzpatrick (Haraux) function for monotone operators, as a Lyapunov or energy-type functional in PDE and control theory, as an orthogonal functional in the theory of almost periodic vibrations, and as a quantitative tool for error estimates in optimization algorithms. Its fundamental role is to quantify, in an explicit and often sharp way, the discrepancy, range, or energetic properties associated with operators, mappings, or function spaces.

1. Definition and Variants in Monotone Operator Theory

Central to monotone operator theory, the Haraux function is typically defined for a set-valued monotone operator $A : X \to 2^{X^*}$ on a reflexive Banach space as:

$$H_A(x, u^*) = \sup_{(y, y^*) \in \mathrm{gra}\,A} \langle x-y, y^*-u^* \rangle.$$

For maximally monotone operators, $H_A(x, u^*)$ takes values in $[0, +\infty]$ and vanishes only on the graph of $A$. This property is extensively utilized in error bounds, variational inequalities, and duality theory (Combettes et al., 21 Aug 2025). In convex analysis and optimization, the Fitzpatrick function,

$$F_A(x, x^*) = \sup_{(a, a^*) \in \mathrm{gra}\,A} [\langle x, a^* \rangle + \langle a, x^* \rangle - \langle a, a^* \rangle],$$

is often called the Haraux function, especially in connection with the work of Brezis and Haraux (Bauschke et al., 2012). It serves as a convex representative of the monotone operator, facilitating duality approaches and convergence proofs.

The Fenchel–Young function is closely related:

$$L_\varphi(x, u^*) = \varphi(x) + \varphi^*(u^*) - \langle x, u^* \rangle.$$

One has $L_\varphi(x,u^*) \geq H_{\partial \varphi}(x,u^*)$, allowing bounds for $L_\varphi$ to be deduced via the Haraux function (Combettes et al., 21 Aug 2025).

2. Range Formulae and Brezis–Haraux Theorem

The Brézis–Haraux theorem provides foundational range descriptions:

• For monotone operators $A,B$ (on Hilbert or Banach spaces), under mild conditions:

$$\operatorname{cl}(\operatorname{ran}(A+B)) = \operatorname{cl}(\operatorname{ran} A + \operatorname{ran} B), \quad \operatorname{int}\,\operatorname{ran}(A+B) = \operatorname{int}(\operatorname{ran} A + \operatorname{ran} B).$$

This result is pivotal for existence proofs in nonlinear equations and composite inclusions (Bùi, 20 Jan 2025, Yao, 2014, Bauschke et al., 2012).

Extensions in (Bùi, 20 Jan 2025) generalize this to composite operators:

$M = A + \sum_{k=1}^m L_k^* B_k L_k,$

yielding

$$\operatorname{ran} M \simeq A(D) + \sum_{k=1}^m L_k^* (\operatorname{ran} B_k),$$

where $$D = \bigcap_{k=1}^m L_k^{-1}(\operatorname{dom} B_k)$$ and $\simeq$ denotes equality of closure and interior.

For the Douglas–Rachford splitting operator $T$ associated with $A$ and $B$,

$$T = \mathrm{Id} - J_A + J_B R_A, \quad J_A = (\mathrm{Id} + A)^{-1}, \quad R_A = 2J_A - \mathrm{Id},$$

the range of $\mathrm{Id} - T$ is characterized:

$$\operatorname{ran}(\mathrm{Id} - T) \simeq \operatorname{ran} A + \operatorname{ran} B,$$

when $A$ is 3*-monotone and $\operatorname{dom} A = H$ (Bùi, 20 Jan 2025, Bauschke et al., 2014).

For ultramaximally monotone operators (maximal in $X^{**} \times X^*$),

$$\operatorname{int}\,\operatorname{ran}(A+B) = \operatorname{int}(\operatorname{ran} A + \operatorname{ran} B), \quad \operatorname{ran}(A+B) = \operatorname{ran} A + \operatorname{ran} B,$$

providing “sum theorems” for monotone operators and underpinning composite splitting algorithms (Yao, 2014).

3. Lower Bound Theory and Algorithmic Error Estimates

Recent advances (Combettes et al., 21 Aug 2025) yield sharper lower bounds for the Haraux function, applicable to general monotone operators on reflexive Banach spaces. Central results include:

• Given $W : X \to 2^{X^*}$ (auxiliary operator) and $\gamma > 0$,

$z \in (W + \gamma A)^{-1}(Wx + \gamma u^*),$

one can select $x^* \in Wx$, $z^* \in Wz$ with

$$H_A(x, u^*) \geq \frac{\langle x - z, x^* - z^* \rangle}{\gamma}.$$

If $W$ is $\varphi$-uniformly monotone:

$$H_A(x, u^*) \geq \frac{\varphi(\|x - z\|)}{\gamma}.$$

If $W$ is $\alpha$-strongly monotone:

$$H_A(x, u^*) \geq \frac{\alpha \|x - z\|^2}{\gamma}.$$

If $W = \nabla f$ for a Legendre function $f$:

$$H_A(x, u^*) \geq \frac{D_f(x, z) + D_f(z, x)}{\gamma}, \quad D_f(x, z) = f(x) - f(z) - \langle x - z, \nabla f(z) \rangle.$$

These bounds are shown to be sharper (even in Hilbert space); explicit examples demonstrate improvements over previous resolvent-based estimates, e.g., for negative Burg entropy. Applications include error bounds for splitting algorithms in composite monotone inclusions:

$$\theta_{A,B}(x) \geq \frac{\|x - J_{\gamma A}(x - \gamma B(x))\|^2}{\gamma}, \quad \text{or} \quad \theta_{A,B}(x) \geq \frac{D_f(x, z) + D_f(z, x)}{\gamma},$$

with $z$ as above and $\theta_{A,B}$ a merit function constructed from the Haraux function (Combettes et al., 21 Aug 2025).

4. Fitzpatrick/Haraux Function in Optimization and Duality

The Fitzpatrick function, synonymous with the Haraux function in operator theory, is convex, lower semicontinuous, and yields a representative for maximally monotone operators. Its domain properties, particularly “rectangularity” (star-monotonicity, 3*-monotonicity), have substantive implications:

• $A$ is rectangular if $$\operatorname{dom} A \times \operatorname{ran} A \subset \operatorname{dom} F_A$$. This guarantees that the Fitzpatrick/Haraux function meaningfully captures the operator, which undergirds duality results and the analysis of variational inequalities (Bauschke et al., 2012, Yao, 2014).

Paramonotonicity is a related but independent property, enforcing stability when the monotonicity gap vanishes. In linear settings, rectangularity implies paramonotonicity; converse implications require additional structure.

In the setting of Douglas–Rachford splitting, the connections between the function–dual function pairing and the Haraux function are explicit:

• For subdifferentials A = $\partial f$, B = $\partial g$,

$$\operatorname{ran}(-T_{A,B}) \simeq (f - g) \cap (f^* + g^*),$$

revealing how primal and dual representations “interact” through the splitting operator and the Haraux function (Bauschke et al., 2014).

5. Functional Analytic Role in PDE, Lyapunov, and Stability Theory

In PDE analysis, especially for the 1D wave equation with nonlinear/nonmonotone dampings, the Haraux function manifests as an energy-type or Lyapunov functional, vital for establishing well-posedness in nonstandard $L^p$ spaces and guaranteeing exponential decay:

• Define, for $z_t$, $z_x$,

$$\Phi(t) = \int_0^1 [F(z_t + z_x) + F(z_t - z_x)]\,dx,$$

where $F$ is even and convex. The functional is monotonic along trajectories:

$$\frac{d}{dt} \Phi(t) \leq 0 \implies \Phi(t) \leq \Phi(0).$$

Careful selections of $F$ (e.g., $F(s) = [\text{Pos}(|s| - 2M)]^2$) yield uniform bounds:

$$\max (\|z_x(t)\|_{L^\infty}, \|z_t(t)\|_{L^\infty}) \leq 2 \max (\|z_0'\|_{L^\infty}, \|z_1\|_{L^\infty}),$$

facilitating stability analysis and interpolation to all $L^p$ spaces (Marx et al., 2019, Chitour et al., 2019). The Haraux function thus enters directly into the Lyapunov framework of infinite-dimensional stability.

6. Almost Periodic Functions, Orthogonality, and Vibration Theory

In studies of almost periodic functions and oscillatory phenomena (Delage, 2015), the Haraux function refers to a unique (up to scaling) orthogonal function $h_n$ such that, for subspaces $X$ defined by sums of periodic components,

$$X = \{ f \in L^2(0,T) : \langle f, h_n \rangle = 0 \},$$

and for $n=3$ (three periods),

$$T_{\text{crit}} = T_1 + T_2 + T_3 - (T_1, T_2) - (T_1, T_3) - (T_2, T_3) + (T_1, T_2, T_3),$$

with $h_3$ constructed as a convolution of $h_2$ via an explicit generating polynomial. The sign and support of $h_3$ determine oscillatory and vibration properties. The existence, nonnegativity, and construction of these “Haraux functions” address subtle questions in periodic vibration theory.

7. Role in Asymptotic Analysis of Evolution Equations

In the convergence theory of nonautonomous monotone evolution equations,

$\dot{x}(t) + A_t(x(t)) \ni 0,$

the Brezis–Haraux function (often denoted $G_{A_t}$) quantifies the excess of $\mathrm{gph} A_\infty$ over $\mathrm{gph} A_t$:

• For $A_t = \partial \varphi_t$, $G_{A_t}(z,0) = \varphi_t(z) - \inf \varphi_t$. Summability conditions,

$\int_0^{+\infty} G_{A_t}(z, p)\,dt < +\infty,$

yield ergodic and sometimes strong convergence of trajectories—enabling sharp asymptotic analysis in infinite-dimensional dynamical systems, hierarchical minimization, viscosity selection, and sweeping processes (Attouch et al., 2016).

Table: Core Haraux Function Variants

Context	Haraux/Fitzpatrick Expression	Principal Application
Monotone Operator Theory, Duality	$F_A(x, x^*)$, $H_A(x,u^*)$	Range, error, duality, splitting bounds
PDE/Control/Lyapunov	$\Phi(t) = \int F(z_t \pm z_x)\,dx$	Boundedness, stability, decay estimates
Vibration/Oscillation, Almost Periodic Functions	$h_n$, convolution/generating formula	Orthogonality, oscillation, decomposition
Evolution Equations, Asymptotic Theory	$G_{A_t}(z,p)$	Convergence rate, equilibrium selection

Conclusion

The Haraux function, in its various incarnations, is a powerful analytical device that encapsulates monotonicity, convexity, and invariance properties crucial to optimization theory, nonlinear analysis, operator splitting, PDE stability analysis, spectral theory, and almost periodic function theory. Recent developments extend lower bound estimates, unify range computations for linearly composite operators, clarify error bounds for algorithms, and enhance energy-based Lyapunov analysis. Its fundamental property—quantifying deviation from equilibrium or optimality—provides a quantitative linkage between deep structural features of operators and the convergence, stability, and solvability of modern mathematical models.