Inverse Quasi-Variational Inequalities

• Inverse quasi-variational inequalities are defined by reversing traditional variational inequality roles, where the operator and constraint set depend on the unknown solution.
• They enable parameter recovery in applications like traffic assignment, inverse mechanics, and PDE constraints through robust projection-based formulations.
• Advanced dynamical systems and neural network approaches ensure accelerated convergence under strong monotonicity, Lipschitz conditions, and fixed-time stabilization.

An inverse quasi-variational inequality problem (IQVI) is a class of operator and set-valued problems motivated by models where the admissible constraint set itself depends on the yet-unknown solution, and one seeks to determine not only a variable constrained by such sets, but an input or parameter mapping a given outcome or data to an underlying structure. IQVIs generalize classical quasi-variational inequalities (QVIs) by reversing or inverting the roles of operator and test-point variables, and further extend parameter-identification inverse problems for differential and variational models. Such problems arise in function identification, traffic assignment, inverse mechanics, and parameter recovery in constrained partial differential equations.

1. Mathematical Formulation and Relationship to Standard QVIs

Let $H$ denote a finite-dimensional real Hilbert space, typically $H = \mathbb{R}^n$, equipped with the standard inner product $\langle \cdot, \cdot \rangle$ and norm $\|\cdot\|$. The data of an IQVI includes:

• A set-valued constraint mapping $\Phi: H \to 2^H$ with $\Phi(x)$ nonempty, closed, and convex for all $x \in H$.
• A single-valued operator $f: H \to H$.

The IQVI seeks $u^* \in H$ such that: $$\begin{cases} f(u^*) \in \Phi(u^*) \ \langle u^*, y - f(u^*) \rangle \geq 0 \quad \forall\, y \in \Phi(u^*) \end{cases}$$ This formulation inverts the standard QVI, which generally takes the form: "Find $x^* \in C(x^*)$ such that $$\langle F(x^*), y - x^* \rangle \geq 0, \forall y \in C(x^*)$$." For IQVI, the variable $u^*$ appears in the role of the test point, while $f(\cdot)$ assumes the place of the operator's image.

The IQVI is equivalently characterized in projection form by introducing a parameter $\alpha > 0$ and the metric projection $P_{\Phi(\cdot)}$: $f(u^*) = P_{\Phi(u^*)} (f(u^*) - \alpha u^*)$ This projection-based formulation is pivotal for algorithmic design and theoretical analysis (Tran et al., 6 Mar 2025, Hai et al., 16 Jan 2026, Dey et al., 2022).

A special case of IQVI recovers the classical variational inequality when $\Phi(x) = \Omega$ (constant), and $f = F^{-1}$, revealing a unifying structure.

2. Existence, Uniqueness, and Regularization Frameworks

Rigorous existence and uniqueness results for IQVI require monotonicity, Lipschitz continuity, and regularity assumptions on both the operator $f$ and the set mapping $\Phi$. Sufficient conditions include:

• $f$ is $L$-Lipschitz and $\beta$-strongly monotone: $\|f(x) - f(y)\| \leq L \|x - y\|$, $$\langle f(x) - f(y), x - y\rangle \geq \beta \|x - y\|^2$$.
• The projection $P_{\Phi(x)}(z)$ is Lipschitz in $x$: $$\| P_{\Phi(u)}(w) - P_{\Phi(v)}(w) \| \leq \mu \| u - v \|$$ for all $u,v,w \in H$.

Under the compatibility condition

$$\sqrt{L^2 + \alpha^2 - 2\alpha\beta} + \mu < \alpha$$

uniqueness and existence of an IQVI solution are guaranteed (Tran et al., 6 Mar 2025, Dey et al., 2022, Hai et al., 16 Jan 2026).

In Banach space settings, particularly for inverse parameter identification, existence is established via fixed-point arguments and Tikhonov-type regularization. For the inverse problem of recovering a parameter $\theta$ (e.g., in a state-dependent constraint), the objective is to minimize

$$J_\alpha(\theta) := \inf_{u \in I(\theta)} \| u - z^\delta \|_Z^2 + \alpha R(\theta)$$

over an admissible set $A \subset B$, with $R$ a convex regularizer and $z^\delta$ observed (possibly noisy) data, guaranteeing existence under standard coercivity and lower semicontinuity conditions (Migorski et al., 2019).

3. Dynamical System and Neural Network Approaches

Several continuous and discrete dynamical system frameworks have been developed to solve IQVIs:

Finite-Time and Fixed-Time Dynamical Systems

Define the residual operator

$T(u) = f(u) - P_{\Phi(u)} (f(u) - \alpha u)$

Two continuous-time dynamical systems are prominent:

1. Finite-Time Stable System:

$$\dot{u}(t) = -\sigma \frac{T(u(t))}{\|T(u(t))\|^{\frac{\gamma-2}{\gamma-1}}}$$

for $\sigma>0$, $\gamma>2$, guarantees global finite-time convergence to $u^*$ with settling time depending on initialization.

2. Globally Fixed-Time Stable System:

$$\dot{u}(t) = - \left[ a_1 \| T(u) \|^{r_1 -1} + a_2 \| T(u) \|^{r_2 -1} \right] T(u)$$

for $a_1, a_2 > 0$, $0 < r_1 < 1 < r_2$, ensures convergence to $u^*$ within a fixed time bound, independent of initialization (Tran et al., 6 Mar 2025).

Second-Order and Inertial Algorithms

Second-order continuous dynamical systems, and their discretizations, yield acceleration and inertial effects: $$\ddot{x}(t) + \sigma(t)\dot{x}(t) + \tau(t) [V(x(t)) - P_{\psi(x(t))}(V(x(t)) - \mu x(t)) ] = 0$$ With step size $h$ and constants $\sigma, \tau$: $$x_{n+1} = x_n + (1-\sigma)(x_n - x_{n-1}) + \tau \left[ P_{\psi(x_n)} (V(x_n) - \mu x_n) - V(x_n) \right]$$ Such inertial projection algorithms converge linearly to the unique IQVI solution under technical bounds on the operator and projection regularity (Hai et al., 16 Jan 2026).

Neural Network Flows

A continuous-time neural network inspired ODE for IQVI is given by

$$\dot{x}(t) = \lambda(t) \left[ P_{\Phi(x(t))}(f(x(t)) - \alpha x(t)) - f(x(t)) \right]$$

Global asymptotic or exponential stability can be proved using Lyapunov methods under monotonicity, Lipschitz, and spectral gap constraints. Forward-Euler discretization yields linearly convergent iterative solvers under analogous parameter conditions (Dey et al., 2022).

4. Inverse IQVI in Banach Spaces and Regularized Identification

Beyond finite dimensions, IQVIs are formulated in reflexive Banach spaces $X$ with dual $X^*$, particularly in the context of inverse parameter identification. For an operator $A: X \to X^*$ (possibly nonlinear, e.g., $p$-Laplacian), a family of set-valued constraint mappings $K=K(\theta, \cdot): X \to 2^X$ parameterized by $\theta$ in a Banach space $B$, and given noisy data $z^\delta \in Z$, the problem is to determine $\theta$ such that some solution $u(\theta)$ of the QVI matches the data. Under coercivity, Mosco-continuity of the constraint mapping, and weak continuity in the parameter, the Tikhonov-regularized minimization problem admits solutions. Uniqueness may remain open except under strong monotonicity and injectivity of the parameter-to-state map (Migorski et al., 2019).

A notable application is the identification of spatial material parameters for an implicit obstacle $p$-Laplacian problem, where variational selection and regularity properties facilitate existence of solutions with bounded variation and total variation regularization.

5. Numerical Algorithms and Applications

Numerical approaches for IQVI leverage explicit discretizations of the underlying continuous dynamical flows or neural network ODEs. Explicit Euler schemes, under appropriate step size constraints, maintain the fixed-time/finite-time convergence characteristics in practice. Iteration-complexity estimates and error bounds can be provided using discrete Lyapunov theory (Tran et al., 6 Mar 2025, Dey et al., 2022).

Benchmark examples include:

• 2D test problem: Linear $f(u) = Au$, box-type constraint set $\Phi(u)$, with rapid convergence to machine precision in under 50 iterations (finite-time flow) or fewer than 30 (fixed-time flow) (Tran et al., 6 Mar 2025).
• Traffic Assignment: User-equilibrium flows in constrained road network models, with capacity and toll bounds encoded as IQVI constraints. Fixed-time algorithms reach feasible solutions in $\sim$40 iterations, and inertial algorithms reduce iteration counts by 30–40% compared to first-order methods (Tran et al., 6 Mar 2025, Hai et al., 16 Jan 2026).

6. Synthesis, Limitations, and Future Extensions

IQVI theory and algorithms enable accelerated convergence to global solutions in variational models with implicit, state-dependent constraints and inversion requirements. Fixed-time stabilization is particularly advantageous for applications with unknown or unbounded initial conditions, and contrasts with the merely asymptotic or exponential rates possible with classical projection or forward-backward methods.

Extensions include:

1. Infinite-dimensional Hilbert or Banach formulations, relevant for PDE-constrained inverse problems or optimal control.
2. Relaxation from strong monotonicity to pseudomonotonicity with suitable regularization.
3. Stochastic noise models and distributed multi-agent IQVIs.
4. Adaptive or parameter-free step-size selection and extensions of the inertial and fixed-time theory.

Limitations reside in the intricacy of parameter feasibility regions (for step-size, damping, and projection regularity), open uniqueness issues in non-strongly monotone or mixed QVI settings, and the need for more general error and rate estimates in stochastic or nonconvex environments (Tran et al., 6 Mar 2025, Hai et al., 16 Jan 2026, Dey et al., 2022, Migorski et al., 2019).