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 $(x^*,z^*)$0. Equivalently, the QVI can be written as:

$(x^*,z^*)$1

where $(x^*,z^*)$2 is the normal-cone operator to $(x^*,z^*)$3.

The standing assumptions are that $(x^*,z^*)$4 is $(x^*,z^*)$5-Lipschitz and $(x^*,z^*)$6-strongly monotone ($(x^*,z^*)$7) and that the map $(x^*,z^*)$8 is uniformly Lipschitz in $(x^*,z^*)$9, with Lipschitz constant $z^* \in \Phi(x^*)$0. If $z^* \in \Phi(x^*)$1 is constant, the problem reduces to a classical variational inequality (VI) and $z^* \in \Phi(x^*)$2.

2. Key Operators: Resolvent and Reflected Resolvent

Given a maximal monotone operator $z^* \in \Phi(x^*)$3 and scalar $z^* \in \Phi(x^*)$4, the resolvent and reflected resolvent are defined by:

• $z^* \in \Phi(x^*)$5,
• $z^* \in \Phi(x^*)$6.

For a single-valued $z^* \in \Phi(x^*)$7, these specialize to

• $z^* \in \Phi(x^*)$8,
• $z^* \in \Phi(x^*)$9.

These operators are central to the Douglas–Rachford scheme, where $x^* = P_C(z^*)$0 exhibits a key contraction property under strong monotonicity and appropriate choice of $x^* = P_C(z^*)$1.

3. Algorithmic Structure: Reflection–Projection Sandwich

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

1. Initialization: Select $x^* = P_C(z^*)$2, $x^* = P_C(z^*)$3, and a stepsize $x^* = P_C(z^*)$4.
2. For $x^* = P_C(z^*)$5:
   • $x^* = P_C(z^*)$6 // Project onto current constraint
   • $x^* = P_C(z^*)$7
   • $x^* = P_C(z^*)$8
   • Stopping condition: If $x^* = P_C(z^*)$9 and $(T(z^*), y - z^*) \geq 0$0, halt.

The principal step is the so-called reflection–projection sandwich: given $(T(z^*), y - z^*) \geq 0$1, form $(T(z^*), y - z^*) \geq 0$2, then apply $(T(z^*), y - z^*) \geq 0$3, after which the projected update $(T(z^*), y - z^*) \geq 0$4 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 $(T(z^*), y - z^*) \geq 0$5 denote the Lipschitz constant of $(T(z^*), y - z^*) \geq 0$6. This is computed (following Giselsson 2017, as cited in the source) as:

$(T(z^*), y - z^*) \geq 0$7

which satisfies $(T(z^*), y - z^*) \geq 0$8 whenever $(T(z^*), y - z^*) \geq 0$9 and $y \in \Phi(x^*)$0.

Given weights $y \in \Phi(x^*)$1 and $y \in \Phi(x^*)$2 such that $y \in \Phi(x^*)$3, the mapping

$y \in \Phi(x^*)$4

acts as a $y \in \Phi(x^*)$5-contraction in the product space $y \in \Phi(x^*)$6 equipped with the weighted norm $y \in \Phi(x^*)$7, where $y \in \Phi(x^*)$8.

By Banach’s fixed-point theorem, there exists a unique fixed point $y \in \Phi(x^*)$9 that satisfies

• $C \subset H$0,
• $C \subset H$1,
• $C \subset H$2.

The solution $C \subset H$3 is then a projected solution to the QVI. The iterates enjoy a linear convergence rate as per Theorem 1:

$C \subset H$4

with $C \subset H$5 (Ramazannejad, 2024).

5. Essential Proof Features

Key elements of the convergence analysis include:

• Strong monotonicity and Lipschitz continuity of $C \subset H$6 ensure that $C \subset H$7 exhibits firm non-expansivity with strict contraction.
• Uniform Lipschitz continuity of the projector $C \subset H$8 (Lipschitz constant $C \subset H$9) guarantees that the variable set constraint does not break contractivity.
• The composed operator $\Phi: C \rightrightarrows H$0 on $\Phi: C \rightrightarrows H$1 is shown to be a contraction in the weighted norm, allowing Banach’s theorem to be applied.
• The fixed point of $\Phi: C \rightrightarrows H$2 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 $\Phi: C \rightrightarrows H$3, requiring the solution of a convex projection problem: finding the closest point to $\Phi: C \rightrightarrows H$4 in the set $\Phi: C \rightrightarrows H$5. In the illustrative $\Phi: C \rightrightarrows H$6 example, the explicit formula for $\Phi: C \rightrightarrows H$7 is provided and the relevant Lipschitz condition verified over 25 cases. In general, one may implement $\Phi: C \rightrightarrows H$8 either via quadratic programming or through an operator splitting method adapted to the particular geometry of $\Phi: C \rightrightarrows H$9 (Ramazannejad, 2024).

7. Illustrative Example and Performance

An exemplar case on $T: H \to H$0 considers:

• $T: H \to H$1,
• $T: H \to H$2 with $T: H \to H$3 and $T: H \to H$4 a scalar,
• $T: H \to H$5 for a $T: H \to H$6 matrix $T: H \to H$7 with eigenvalues such that $T: H \to H$8, $T: H \to H$9 (yielding suitable splitting parameters).

With $P_{\Phi(x)}: H \to \Phi(x)$0, $P_{\Phi(x)}: H \to \Phi(x)$1, empirical $P_{\Phi(x)}: H \to \Phi(x)$2, and appropriate choices of $P_{\Phi(x)}: H \to \Phi(x)$3 and $P_{\Phi(x)}: H \to \Phi(x)$4, Algorithm 1 achieves accuracy to $P_{\Phi(x)}: H \to \Phi(x)$5 within 15–20 iterations, indicating strong linear convergence and practical computational efficiency (Ramazannejad, 2024).

---

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).