Quasi-Variational Inequalities (QVI)

• Quasi-variational inequalities are problems where the feasible set depends on the current solution, introducing nonmonotonicity and nonsmoothness.
• They unify models like classical variational inequalities, complementarity systems, and generalized Nash games to analyze state-dependent feasibility conditions.
• Numerical methods for QVIs include iterative, penalty, and splitting approaches designed to overcome challenges posed by variable constraints and potential nonconvexity.

A quasi-variational inequality (QVI) is a finite- or infinite-dimensional problem of finding a point $x^*$ lying in a set $K(x^*)$, where this feasible set itself depends on the variable, such that a variational inequality is satisfied with respect to $K(x^*)$. QVIs unify and extend classical variational inequalities, complementarity systems, generalized Nash games, and several domains of equilibrium, control, and PDE theory. QVIs are fundamentally more challenging than VIs due to the variable constraint set, which induces additional nonmonotonicity, nonsmoothness, and can break convexity or semicontinuity properties essential in VI theory.

1. General Formulation and Mathematical Setting

A canonical (Stampacchia-type) QVI in a vector space $X$ with dual $X^*$ is:

$$\text{Find } x^* \in K(x^*) \text{ such that } \langle F(x^*), y - x^* \rangle \geq 0\ \forall y \in K(x^*),$$

where

• $F: X \to X^*$ is typically monotone, pseudomonotone, or set-valued;
• $K: X \rightrightarrows X$ is a set-valued mapping, assigning to each $x$ a nonempty, closed, often convex set $K(x) \subset X$.

The QVI class covers several situations:

• When $K(x) = K_0$ (fixed), this reduces to a classical VI.
• If $F$ is the gradient of a functional and $K(x)$ is defined by constraints $g(x, y) \leq 0$, the QVI nests parametric feasibility and equilibrium.

The defining property of QVIs is that the admissible ("test") set depends explicitly and possibly nonsmoothly on $x^*$. The solution set may be highly nonconvex, potentially empty, or have multiple disconnected solutions.

2. Existence and Uniqueness: Core Theories

Several frameworks enable the analysis of existence and uniqueness, structured according to the properties of $F$ and $K(\cdot)$:

1. Order-theoretic and Birkhoff–Tartar Fixed-Point Results: For monotone or increasing constraint maps $K(\cdot)$ (or obstacle maps in PDEs), existence of minimal and maximal solutions follows from order-theoretic arguments if $K$ preserves order intervals and has monotonicity properties; see obstacle-type QVIs (Alphonse et al., 2020, Alphonse et al., 2019).
2. Contractive and Iterative Schemes: When $K(\cdot)$ is Lipschitz and sufficiently "nonexpansive" (specifically, when the Lipschitz constant is below a threshold depending on strong monotonicity or coercivity of $F$), Picard or Mann iteration converges to a unique solution (Alphonse et al., 2020, Kazmi, 2013).
3. Penalty/PDE Regularization: Approximating the QVI by solving a sequence of penalized (possibly PDE-constrained) VIs, controlling constraint violation by a monotone penalty, and passing to the limit yields existence under weak continuity and compactness (Alphonse et al., 2020, Alphonse et al., 2019).
4. Pseudomonotonicity + Mosco Continuity in Banach Spaces: In general (possibly infinite-dimensional) settings, the operator $F$ is assumed Brezis-pseudomonotone, and $K(x)$ satisfies weak Mosco continuity (i.e., stability under weak limits). Then QVIs have solutions under boundedness and compactness hypotheses (Kanzow et al., 2018).
5. Nonconvex, Nonmonotone, and Unbounded Cases: For QVIs with nonconvex $K(x)$ or unbounded feasible sets, existence may be rescued via value-function reformulations, coderivative techniques, and abstract coercivity or partial calmness conditions. The solution sets can be sharply characterized using Mordukhovich coderivatives and stability criteria, even with nonconvexity (Dutta et al., 2022, Sultana et al., 2023).

Uniqueness is typically not guaranteed unless $F$ is strongly monotone relative to the Lipschitzness of $K(\cdot)$, or, in set-valued cases, if suitable "strong regularity" or single-valuedness of solution correspondences is imposed.

3. Key Methods and Algorithms for QVI Solution

Algorithmic progress on QVIs leverages either reduction to sequences of VIs or direct splitting/proximal approaches adapted to the variable-set regime.

(A) Iterative VI-based Approaches

• Fixed-Point Iteration: For certain obstacle maps or obstacle-type QVIs, iterative solution of $y_{n+1} = \mathrm{VI}\left(K(y_n); F, f\right)$ gives monotone (increasing or decreasing) sequences converging to extremal QVI solutions (Alphonse et al., 2019, Alphonse et al., 2020).
• Augmented Lagrangian and Penalization: At each outer iteration, solve a VI with augmented multipliers or penalty updates, e.g., penalizing constraint violation $G(x, x) \notin K$ (Kanzow et al., 2018).

(B) Projection-Based Splitting and Operator Methods

• Douglas–Rachford Splitting for QVIs: Extends classical DR methods by iterating between the resolvent and reflected resolvent of $F$ and projection onto the non-self constraint set $C(x)$; convergence is geometric under strong monotonicity and Lipschitz continuity of $F$, with Lipschitz-continuous constraint maps (Ramazannejad, 18 Jul 2024).
• Iterative Projection–Mann and Split QVI Schemes: For split or coupled QVIs in Hilbert spaces (e.g., networked or decentralized equilibrium models), three-step projection–Mann iterations with adaptive step sizes ensure strong convergence when the constraint sets and operators meet strong monotonicity and Lipschitz criteria (Kazmi, 2013).

(C) Stochastic and Game-theoretic QVI Algorithms

• Inexact Extra-Gradient and Stochastic QVI Solvers: For monotone QVIs with stochastic $F$, single- and extra-gradient schemes with inexact projections provide provable linear convergence in the quadratic-growth regime, with batch size, inexactness, and step-size finely tuned for contraction (Alizadeh et al., 5 Jan 2024). Applications include stochastic Nash equilibria and bilevel learning.
• Dantzig–Wolfe Decomposition: For large-scale QVIs, decompose the problem into a master QVI with small moving constraint set and subproblems that are classical VIs on larger (fixed) sets. Alternating algorithms converge globally under strong monotonicity or maximal monotonicity of $F$; subproblems are amenable to scalable solution (Jardim et al., 12 May 2025).

(D) First-Order, Semismooth, and Neural Approaches

• Semismooth Newton for Nonsmooth QVIs: For nonconvex or nonsmooth QVIs, transform to a value-function-based system of inequalities and apply semismooth Newton with generalized derivatives; rapid convergence is backed by CD-regularity (Dutta et al., 2022).
• Neural Network Flows for Inverse QVIs: A continuous ODE driven by projected mapped differences converges globally (exponentially under strong monotonicity) to the IQVI solution; discretizations with proper step-size design inherit these stability properties (Dey et al., 2022).

4. Directional Differentiability and Sensitivity Analysis

A hallmark advancement in QVI theory is the directional differentiability of the solution map $Q : f \mapsto y(f)$, relevant for sensitivity, optimization, and numerical methods.

Given the QVI with Lipschitz, Hadamard-differentiable obstacle map $\Phi$ and single-valued solution $y$, under local regularity:

• The map $f \mapsto y(f)$ is directionally (and locally Hadamard) differentiable. The derivative $\alpha = Q'(f; d)$ solves another obstacle-type QVI whose constraint cone involves both the tangent cone to $K(y)$ at $y$ and the differential of $\Phi$ at $y$ (Alphonse et al., 2020, Alphonse et al., 2018, Alphonse et al., 2019).
• The derivative is characterized as the solution to

$$\alpha \in K^y(\alpha), \quad \langle A\alpha - d, \alpha - v \rangle \leq 0 \ \forall v \in K^y(\alpha),$$

where $K^y(\alpha)$ mixes the tangent cone at $y$ and the image $\Phi'(y)[\alpha]$.

• This result enables the rigorous justification of "one-step linearization" (Newton-type) algorithms for QVIs and supplies analytic tools for sensitivity in control and optimization contexts.

5. QVI-Constrained Optimal Control and Game Theoretic Applications

QVIs serve as constraints in a diverse spectrum of equilibrium and optimal control problems:

(A) Optimal Control with QVI State Constraints

• The prototypical problem is: minimize $J(y, u)$ subject to $u \in U_\mathrm{ad}$, $y \in Q(u)$, with $Q(u)$ the QVI solution set for control $u$. Under existence and differentiability of $Q$, a hierarchy of first-order conditions is available, including Bouligand, C-stationarity, and strong stationarity (Alphonse et al., 2020).
• PDE-regularization and penalization techniques enable the derivation of multipliers and stationarity systems for potentially nonunique or nonsmooth settings, directly supporting efficient numerical solvers.

(B) Generalized Nash Equilibria, Bilevel, and Coupled Games

• Generalized Nash Equilibrium Problems (GNEP) with competitor-dependent feasible sets are precisely QVIs, and existence/algorithmic theory for QVIs directly applies (Alizadeh et al., 5 Jan 2024, Stupia et al., 2013, Jardim et al., 12 May 2025).
• Bilevel optimization, where lower-level optimality induces state-dependent constraints, is a canonical QVI, broadly motivating solution regularity and directional differentiability results.

(C) Other Applications

• Dynamic programming for impulse control, HJBI QVIs in deterministic games, sandpile evolution, semipermeability models, and problems in mechanics and economics are several fields where QVI structures are fundamental (Asri et al., 2021, Barrett et al., 2012, Migorski et al., 2023).

6. Generalizations, Special Structures, and Challenges

QVIs span numerous generalizations:

• Nonconvex QVIs: Via value-function and coderivative reformulations, stability and optimality theory can be established without convexity or even differentiability (Dutta et al., 2022).
• Hemivariational and nonsmooth QVIs: Models that combine nonmonotone superpotentials and solution-dependent constraints have existence and compactness theory via fixed-point arguments and Mosco continuity (Migorski et al., 2023).
• Fractional and Parabolic QVIs: Extensions to fractional diffusion operators, parabolic PDEs, and time-dependent state constraints are handled by spectral extensions, monotone iteration, and compactness arguments (Antil et al., 2017, Alphonse et al., 2019, Gokieli et al., 2020).
• Unbounded and Locally-Defined QVIs: Existence in unbounded feasible regions, and the introduction of local (rather than global) solutions based on local reproducibility and int-dual semicontinuity (Aussel et al., 3 Feb 2024, Sultana et al., 2023).

The main obstacles to tractable QVI theory and computation are:

• Set-dependence and potential nonconvexity of $K(x)$;
• Loss of standard VI regularity (upper semicontinuity, convexity);
• Failure of standard fixed-point/monotonicity arguments without additional structure (Mosco continuity, pseudomonotonicity, coercivity, incremental boundedness).

7. Numerical and Computational Aspects

QVIs pose significant numerical challenges:

• Algorithms must handle the moving set structure, nonsmoothness, and lack of "easy" projection.
• In practice, schemes based on penalization/semi-smooth Newton, Dantzig–Wolfe, splitting, primal-dual active-set, and neural network flows are deployed. These reduce QVIs to successive VIs, LPs, or penalized systems, or approximate solution trajectories via ODEs.
• Strong performance gains arise for large-scale, structured QVIs—e.g., economic equilibria, games—via decomposition and splitting (notably in (Jardim et al., 12 May 2025)).
• Sensitivity and stability theory underpins robust optimization under data uncertainty, as parametric and coderivative formulas become accessible (Dutta et al., 2022, Alphonse et al., 2019).

Convergence rates depend on the regularity and monotonicity conditions, with linear convergence achievable under strong monotonicity and quadratic-growth assumptions, and superlinear convergence for semismooth Newton approaches when suitable regularity holds.

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Quasi-variational inequalities continue to drive the analysis of interacting equilibrium systems with state-dependent feasibility, from PDE-constrained control to game theory and stochastic optimization. Advances in regularity, sensitivity, algorithm design, and variational analysis have significantly broadened their applicability and the robustness of their computational solution.