Douglas–Rachford Splitting for QVIs

• The paper presents a projection-based Douglas–Rachford splitting scheme that achieves global convergence and a linear rate for QVIs.
• It combines metric projection and resolvent operators to effectively manage non-self constraint maps in real Hilbert spaces.
• Contraction analysis via fixed-point theory guarantees robustness under strong monotonicity and uniform Lipschitz continuity.

The Douglas–Rachford splitting method for quasi-variational inequalities (QVIs) is a projection-based iterative scheme designed to find projected solutions to QVIs posed in real Hilbert spaces with non-self constraint maps. This approach leverages the interplay of the metric projection onto convex sets, the resolvent operator, and the reflected resolvent operator associated with a strongly monotone, Lipschitz continuous mapping. Under suitable regularity conditions—including uniform Lipschitz continuity of the moving-set projector and strong monotonicity of the underlying operator—the algorithm achieves global convergence with a provable linear rate, and it efficiently addresses the added complexity of QVIs versus standard variational inequalities (Ramazannejad, 2024).

1. Problem Formulation and Background

A quasi-variational inequality (QVI) posed in a real Hilbert space $H$ seeks a pair $(x^*,z^*)$ such that:

• $z^* \in \Phi(x^*)$,
• $x^* = P_C(z^*)$,
• $(T(z^*), y - z^*) \geq 0$ for all $y \in \Phi(x^*)$,

where $C \subset H$ is a nonempty, closed, convex "base set," $\Phi: C \rightrightarrows H$ is a set-valued "constraint map" with convex images, $T: H \to H$ is single-valued, and $P_{\Phi(x)}: H \to \Phi(x)$ denotes the metric projection onto $\Phi(x)$. Equivalently, the QVI can be written as:

$$0 \in T(z^*) + N_{\Phi(x^*)}(z^*), \quad z^* = P_{\Phi(x^*)}(x^*),$$

where $N_{\Phi(x^*)}$ is the normal-cone operator to $\Phi(x^*)$.

The standing assumptions are that $T$ is $L$-Lipschitz and $\mu$-strongly monotone ($L > \mu > 0$) and that the map $x \mapsto P_{\Phi(x)}(z)$ is uniformly Lipschitz in $x$, with Lipschitz constant $\ell \geq 0$. If $\Phi$ is constant, the problem reduces to a classical variational inequality (VI) and $\ell = 0$.

2. Key Operators: Resolvent and Reflected Resolvent

Given a maximal monotone operator $A: H \rightrightarrows H$ and scalar $\gamma > 0$, the resolvent and reflected resolvent are defined by:

• $J_{\gamma A} := (I + \gamma A)^{-1}$,
• $R_{\gamma A} := 2 J_{\gamma A} - I$.

For a single-valued $A = T$, these specialize to

• $J_{\gamma T}(u) = (I + \gamma T)^{-1}(u)$,
• $R_{\gamma T}(u) = 2 J_{\gamma T}(u) - u$.

These operators are central to the Douglas–Rachford scheme, where $R_{\gamma T}$ exhibits a key contraction property under strong monotonicity and appropriate choice of $\gamma$.

3. Algorithmic Structure: Reflection–Projection Sandwich

Algorithm 1 for QVIs using Douglas–Rachford splitting proceeds as follows:

1. Initialization: Select $x_0 \in C$, $y_0 \in H$, and a stepsize $\gamma > 0$.
2. For $k = 0, 1, 2, \dots$:
   • $z_{k+1} = P_{\Phi(x_k)}(y_k)$ // Project onto current constraint
   • $y_{k+1} = R_{\gamma T}(2 z_{k+1} - y_k)$
   • $x_{k+1} = P_C(z_{k+1})$
   • Stopping condition: If $y_{k+1} = y_k$ and $x_{k+1} = x_k$, halt.

The principal step is the so-called reflection–projection sandwich: given $y$, form $z = P_{\Phi(x)}(y)$, then apply $R_{\gamma T}(2z - y)$, after which the projected update $x = P_C(z)$ is performed. Each iterate can be interpreted in terms of a Douglas–Rachford step for the moving-set QVI (Ramazannejad, 2024).

4. Contraction Analysis and Convergence Guarantees

Let $L_{\gamma T}$ denote the Lipschitz constant of $R_{\gamma T}$. This is computed (following Giselsson 2017, as cited in the source) as:

$$L_{\gamma T} = \sqrt{1 - \frac{4\gamma\mu}{(1 + \gamma\mu)^2} + \gamma^2 L^2}$$

which satisfies $L_{\gamma T} < 1$ whenever $\mu > 0$ and $0 < \gamma < 2\mu / L^2$.

Given weights $\alpha > 0$ and $\beta \geq 2\ell$ such that $\alpha L_{\gamma T} + 2\ell < 1$, the mapping

$$p: (z, y, x) \mapsto (J_{\gamma T}(y),\; R_{\gamma T}(2 J_{\gamma T}(y) - y),\; P_C(z))$$

acts as a $\theta$-contraction in the product space $H^3$ equipped with the weighted norm $$\|(z, y, x)\|_1 = \alpha \|z\| + \|y\| + \beta \|x\|$$, where $\theta = \alpha + 2\ell < 1$.

By Banach’s fixed-point theorem, there exists a unique fixed point $(z^*, y^*, x^*)$ that satisfies

• $z^* = J_{\gamma T}(y^*)$,
• $y^* = R_{\gamma T}(2z^* - y^*)$,
• $x^* = P_C(z^*)$.

The solution $(x^*, z^*)$ is then a projected solution to the QVI. The iterates enjoy a linear convergence rate as per Theorem 1:

$$\|z_k - z^*\| + \|y_k - y^*\| + \beta\|x_k - x^*\| \leq \theta^{k-1} [\alpha\|z_1 - z^*\| + \|y_1 - y^*\| + \beta\|x_1 - x^*\|]$$

with $0 < \theta < 1$ (Ramazannejad, 2024).

5. Essential Proof Features

Key elements of the convergence analysis include:

• Strong monotonicity and Lipschitz continuity of $T$ ensure that $R_{\gamma T}$ exhibits firm non-expansivity with strict contraction.
• Uniform Lipschitz continuity of the projector $x \mapsto P_{\Phi(x)}(z)$ (Lipschitz constant $\ell$) guarantees that the variable set constraint does not break contractivity.
• The composed operator $p$ on $(z, y, x)$ is shown to be a contraction in the weighted norm, allowing Banach’s theorem to be applied.
• The fixed point of $p$ is proved to yield a projected solution to the original QVI via the resolvent–normal–cone and projection relationships (Ramazannejad, 2024).

6. Practical Implementation Considerations

For each iteration, the computational bottleneck lies in $z_{k+1} = P_{\Phi(x_k)}(y_k)$, requiring the solution of a convex projection problem: finding the closest point to $y_k$ in the set $\Phi(x_k)$. In the illustrative $ℝ^2$ example, the explicit formula for $P_{\Phi(x)}$ is provided and the relevant Lipschitz condition verified over 25 cases. In general, one may implement $P_{\Phi(x)}$ either via quadratic programming or through an operator splitting method adapted to the particular geometry of $\Phi(x)$ (Ramazannejad, 2024).

7. Illustrative Example and Performance

An exemplar case on $H = ℝ^2$ considers:

• $$C = \{x \in ℝ^2\;|\; 0 < x_1 < 1,\, 0 < x_2 < 1,\, x_1 + x_2 > 1\}$$,
• $\Phi(x) = Q + 2\delta x$ with $Q = [0,1]^2$ and $\delta$ a scalar,
• $T(x) = Mx$ for a $2 \times 2$ matrix $M$ with eigenvalues such that $L = 0.25$, $\mu = 0.22$ (yielding suitable splitting parameters).

With $\gamma = 4$, $L_{\gamma T} < 1$, empirical $\ell \approx 0.390$, and appropriate choices of $\alpha$ and $\beta$, Algorithm 1 achieves accuracy to $10^{-8}$ within 15–20 iterations, indicating strong linear convergence and practical computational efficiency (Ramazannejad, 2024).

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Citation: The content in all sections directly references the results and discussion in "Douglas-Rachford splitting algorithm for projected solution of quasi variational inequality with non-self constraint map" (Ramazannejad, 2024).