Quasi-Variational Inequalities

• Quasi-variational inequalities are mathematical problems that extend variational inequalities by allowing the feasible set to depend on the unknown solution.
• They employ advanced numerical methods such as projection iterations, augmented Lagrangian, and extragradient techniques to handle state-dependent constraints.
• Theoretical results on existence, uniqueness, and stability leverage monotonicity, coercivity, and fixed point frameworks, influencing applications in PDEs, control, and game theory.

A quasi-variational inequality (QVI) is a mathematical problem in which one seeks a point within a constraint set that itself depends on the unknown point and requires that a prescribed inequality (often involving a monotone or pseudomonotone operator) be satisfied for all points within this constraint set. QVIs generalize classical variational inequalities by allowing the feasible set to shift with the solution, thus capturing a much broader class of equilibrium and optimization phenomena, particularly in infinite-dimensional spaces, systems with feedback effects, and problems where the feasible region is state- or parameter-dependent. QVIs encompass frameworks ranging from convex analysis to dynamic equilibrium theory, and underpin modern developments in numerics, control theory, optimization under uncertainty, and game theory.

1. Mathematical Formulation and General Structure

A classical variational inequality (VI) on a Hilbert or Banach space seeks $x^* \in K$ such that

$$\langle F(x^*), y - x^* \rangle \geq 0, \quad \forall y \in K,$$

where $F$ is an operator (typically monotone or pseudomonotone) and $K$ is a fixed, closed, convex set.

A QVI relaxes the fixed-set assumption: for a set-valued mapping $K: X \to 2^X$, the QVI is to find $x^* \in K(x^*)$ such that

$$\langle F(x^*), y - x^* \rangle \geq 0, \quad \forall y \in K(x^*).$$

A variety of QVI formulations appear in the literature, including "split" QVIs where the solution must also satisfy a further inequality involving its image under a linear operator, hemivariational QVIs with additional nonsmooth terms, and inverse QVIs. Table 1 highlights several canonical models.

Model	Constraints	Operator
VI	$x \in K$	$F(x)$
QVI	$x \in K(x)$	$F(x)$
Split-GQVI (Kazmi, 2013)	$x \in K_1(x)$, $A x \in K_2(A x)$	$F_1(x)$, $F_2(A x)$
Hemivariational QVI (Migorski et al., 2023)	$u \in K(u)$	$A u + \partial j(Mu)$

Typical examples of $K(x)$ include sets defined by state-dependent constraints, obstacles in PDEs, or more generally sets derived from interdependent equilibrium or control constraints.

2. Existence, Uniqueness, and Structural Conditions

Existence and, when possible, uniqueness of solutions to QVIs require a blend of classical monotonicity, compactness, coercivity, and additional regularity of the solution set mapping. Several frameworks address these aspects:

1. Pseudomonotonicity and Mosco Continuity: On Banach spaces, assuming $F$ is pseudomonotone (in the sense of Brezis) and the set mapping $x \mapsto K(x)$ is weakly Mosco-continuous, one ensures existence under boundedness and closedness assumptions (Kanzow et al., 2018).
2. Strong Monotonicity and Lipschitz Continuity: In Hilbert spaces, uniqueness and Lipschitz stability are proved under smallness assumptions on the Lipschitz constant of $x \mapsto K(x)$ and strong monotonicity of $F$, e.g., $L_\Phi < 1/\gamma_A$ or its convex potential variant (Wachsmuth, 2019).
3. Order-Theoretic Fixed Point Theory: For monotone obstacle maps in lattice-ordered vector spaces, fixed point theorems such as the Birkhoff–Tartar theorem yield existence of minimal and maximal solutions (Alphonse et al., 2020, Alphonse et al., 2019). This is pivotal in applications where multiple solutions exist, as in obstacle-type QVIs.
4. Coercivity and Unbounded Constraints: For QVIs with possibly unbounded constraint maps, coercivity at infinity (forcing the sequence to "return" from infinity) enables existence results even for noncompact feasible regions (Sultana et al., 2023).
5. State-Dependent and Evolution Constraints: In evolution QVIs, where the constraint set depends on the process history or solution trajectory, existence follows from a combination of compactness results (parabolic analogues of the Aubin–Lions lemma), maximal monotonicity, and feedback (fixed-point) arguments (Gokieli et al., 2020, Zeng et al., 28 May 2024).

3. Algorithmic Frameworks and Numerical Methods

Several algorithmic paradigms have emerged for QVIs, generalizing familiar methods from VI numerics:

1. Projection and Fixed Point Iterations: Iterative schemes combine projected evaluations of $F$ with sequential updates of the constraint set, as in the split general QVI algorithm (Kazmi, 2013), where at each step one projects a modified gradient onto $K(x^n)$.
2. Augmented Lagrangian and Penalty Schemes: For problems with complex parametric or coupling constraints, augmented Lagrangian or penalty methods iteratively approximate the QVI by a sequence of VIs with fixed, simplifying sets, updating multipliers and penalty parameters (Kanzow et al., 2018, Alphonse et al., 2020).
3. Extragradient and Inertial Methods: Extensions of extragradient and inertial projection methods provide robust convergence for monotone or strongly monotone (possibly stochastic) QVIs, with provable linear rates in the presence of strong monotonicity and appropriately regular set mappings (Yao et al., 22 Apr 2024, Alizadeh et al., 5 Jan 2024). Tseng-type methods and their continuous limits admit similar contraction properties with appropriately chosen parameters (Altangerel, 7 May 2025).
4. Dantzig-Wolfe Decomposition: For large-scale structured QVIs, decomposition into "master" and "subproblem" levels—solving VIs over decoupled sets and coordinating via gap functions—enables scalable computation and efficient parallelization (Jardim et al., 12 May 2025).
5. Value Function Reformulation and Semismooth Newton Methods: For nonconvex or nonmonotone QVIs, reformulation via the optimal value function enables coderivative-based robust stability results and global convergence of semismooth Newton algorithms (Dutta et al., 2022).
6. Neural Network Dynamical Systems: Continuous and discrete-time neural network flows are proposed for inverse QVIs, relying on carefully tuned contraction and dissipation estimates to guarantee asymptotic and exponential stability (Dey et al., 2022).

Implementation details for these algorithms typically hinge on computable projections onto the state-dependent feasible set, strong or pseudomonotonicity of underlying operators, Lipschitz regularity of projection mappings, and, in stochastic or splitting schemes, proper control of approximation and sampling errors.

4. Regularity, Sensitivity, and Control

Advanced QVI theory addresses not just existence, but also regularity (stability with respect to data), differentiability properties of the solution map, and control-theoretic implications:

• Directional Differentiability and Sensitivity: For QVIs with Hadamard directionally differentiable obstacle maps, a solution map is directionally differentiable, and its derivative solves a linearized QVI where the critical cone is determined by the active constraints and the obstacle's derivative (Alphonse et al., 2018, Alphonse et al., 2020, Wachsmuth, 2019).
• Stability of Extremal Solutions: For set-valued QVI solution mappings, the extremal (minimal, maximal) elements are shown to depend continuously on monotone data perturbations under suitable order and monotonicity conditions. Perturbation approaches crucially inform well-posedness and control (Alphonse et al., 2019).
• Optimal Control and Nash Equilibrium: Control problems with QVI constraints are tackled by deriving stationarity conditions (Bouligand, C-, and strong stationarity) that involve adjoint states and multipliers tied to the linearized QVI (Alphonse et al., 2020, Wachsmuth, 2019). These frameworks are directly applied to generalized Nash equilibrium problems, distributed control in PDEs with state-dependent constraints, and (for parabolic QVIs) feedback systems with dynamic admissible sets (Kanzow et al., 2018, Sultana et al., 2023, Zeng et al., 28 May 2024).
• Local Solution Concepts: For nonconvex or locally varying constraints, local concepts (local Stampacchia or Minty solutions) become essential, often characterized by the "local reproducibility" property of constraint mappings, reducing the analysis to VI problems on local sections (Aussel et al., 3 Feb 2024).

5. Applications in Analysis, Economics, and Engineering

QVIs have been successfully applied and numerically treated in a range of scientific and engineering domains:

• Nonlinear PDEs and Free Boundary Problems: Fractional elliptic QVIs, modeled with nonlocal diffusion operators and state-dependent obstacles, require advanced analytic and numerical methodology (extension problems, Mosco convergence, semismooth Newton solvers; see (Antil et al., 2017)). Quasi-hemivariational inequalities provide frameworks for semipermeability and contact phenomena in mechanics (Migorski et al., 2023).
• Fluid Dynamics and Phase Change: Stefan/Navier-Stokes models with temperature-dependent velocity or gradient constraints (modeling, for example, freezing/melting interfaces) are cast as parabolic QVIs and solved using compactness-based time-discretization and variational formulations (Gokieli et al., 2018, Gokieli et al., 2020).
• Optimal Control, Market, and Game Theory: Generalized Nash equilibrium problems, optimal distributed and boundary control with QVI constraints, and pure exchange Walrasian equilibrium systems are modeled and solved using QVI theory with advanced decomposition schemes and convergence properties (Kanzow et al., 2018, Jardim et al., 12 May 2025, Sultana et al., 2023).
• Nonsmooth Parameter Identification: Evolution QVIs with nonsmooth and multivalued nonlinearities (e.g., governed by Clarke subdifferentials) support robust models for parameter identification in complex control systems (Zeng et al., 28 May 2024).

6. Advanced Topics: Nonconvexity, Stochasticity, Evolution, and Hemivariational Structures

Extensions to nonconvex QVIs, stochastic problems, and hybrid structures enrich the scope and complexity of the field:

• Nonconvex QVIs: Transformation via value function reformulation provides coderivative-based robust stability and enables solver design even in the absence of convexity (Dutta et al., 2022).
• Stochastic QVIs: Monotone stochastic QVIs admit gradient and extragradient-type methods with rigorous rates under quadratic growth conditions, accommodating inexact constraint projections and sample-average approximations (Alizadeh et al., 5 Jan 2024).
• Evolution QVIs: Strongly time-dependent feasible sets and feedback-coupled parabolic structures are addressed via variational time-derivative operators, compactness theorems, and fixed-point strategies (Gokieli et al., 2020, Zeng et al., 28 May 2024).
• Hemivariational Components: The inclusion of locally Lipschitz (possibly nonconvex) superpotentials, as handled via Clarke subdifferentials, broadens the QVI framework to encompass contact mechanics, friction, and nonmonotone transport (Migorski et al., 2023).

7. Future Directions and Open Problems

Key avenues for further research include:

• Infinite-dimensional, unbounded, and time-dependent constraint maps: Further extension of existence, regularity, and numerical methodology, particularly under relaxed compactness or coercivity assumptions (Sultana et al., 2023, Sultana et al., 2023).
• Data-driven and neural network approaches: Neural network flows for QVIs and their control-theoretic properties have only begun to be explored (Dey et al., 2022).
• Numerical Scalability: Parallel and decomposed solution methods, including Dantzig–Wolfe and block-separable strategies, will be critical for large-scale and high-dimensional systems (Jardim et al., 12 May 2025).
• Local and approximate solution frameworks: Analytical techniques and algorithms based on local solution concepts, enabling robust computation in nonconvex or practically constrained domains (Aussel et al., 3 Feb 2024).
• Coupled/multiphysics QVIs: Handling QVIs arising from coupled systems (e.g., electrochemical, rheological, or coupled PDE-ODE networks) remains an open challenge in terms of both theory and large-scale simulation.

Quasi-variational inequalities thus provide a unifying, extensible framework for a wide spectrum of problems in analysis, PDEs, control, games, and optimization, with ongoing developments at the intersection of mathematical theory, computational algorithms, and domain-driven modeling.