Generalized Inverse Mixed VI Problems

• GIMVIPs are a generalized extension of variational inequalities, unifying inverse, mixed, and parametric identification problems over convex domains.
• They employ proximal-point and dynamical systems methods to guarantee finite and fixed-time convergence with rigorous error bounds.
• GIMVIPs enable precise modeling in mechanics, optimization, and PDEs, offering scalable, numerically robust solution schemes.

Generalized Inverse Mixed Variational Inequality Problems (GIMVIPs) are an extension of variational inequality (VI) and inverse VI frameworks that address a broad class of parametric and operator identification problems over convex domains. The mathematical structure of GIMVIPs enables formulation and solution of parameter estimation, control, and model inversion tasks subject to nonsmooth, nonlinear, and possibly constrained requirements, encompassing models arising in mechanics, optimization, and partial differential equations.

1. Formal Definition and Problem Setting

Let $\Omega \subset \mathbb{R}^d$ be a nonempty closed convex set. Given single-valued operators $F, h : \mathbb{R}^d \rightarrow \mathbb{R}^d$ and a proper, convex, lower semicontinuous functional $g: \Omega\rightarrow\mathbb{R}\cup\{+\infty\}$, the GIMVIP seeks $\overline{w}\in\mathbb{R}^d$ satisfying:

• $F(\overline{w})\in\Omega,$
• $$\langle h(\overline{w}), v - F(\overline{w})\rangle + g(v) - g(F(\overline{w})) \ge 0,\ \forall v\in\Omega.$$

A key equivalence (Proposition 2.3) shows that $\overline{w}$ solves the GIMVIP if and only if

$$\Xi(\overline{w}) := F(\overline{w}) - \operatorname{prox}_{\Omega}^{\gamma g}(F(\overline{w}) - h(\overline{w})) = 0,$$

where $\operatorname{prox}_{\Omega}^{\gamma g}$ denotes the proximal operator of $g$ over $\Omega$.

When particular choices are made, GIMVIPs recover classical inverse VIs, mixed VIs, or parameter identification problems as special cases, unifying a variety of well-studied frameworks (Tran, 13 Jan 2026, Gwinner, 2020).

2. Mathematical Assumptions and Existence/Uniqueness

Central to the theory of GIMVIPs are monotonicity and Lipschitz conditions on the operators:

• $h$ is $\alpha$-Lipschitz and $\lambda$-monotone: $\langle h(w)-h(v), w-v\rangle\ge\lambda\|w-v\|^2$,
• $(F,h)$ is a $\mu$-strongly monotone couple: $\langle F(w)-F(v), h(w)-h(v)\rangle\ge\mu\|w-v\|^2$,
• $F$ is $\beta$-Lipschitz (equivalently, $\rho$-cocoercive with $\rho=1/\beta$),
• $g$ is proper, convex, and lower semicontinuous.

The parameters $\Lambda := \sqrt{\beta^2+\alpha^2-2\mu}$ and $\Gamma := \Lambda+\alpha$ are introduced. The key condition

$$\sqrt{\beta^2+\alpha^2-2\mu} + \sqrt{1-2\lambda+\alpha^2} < 1, \quad \rho > \Lambda$$

guarantees the existence and uniqueness of the solution $\overline{w}$ (Theorem 2.8). Moreover, error bounds and coercivity are established:

• $$(\rho-\Lambda)\|w-\overline{w}\|\le\|\Xi(w)\|\le\Gamma\|w-\overline{w}\|,$$
• $$\langle w-\overline{w}, \Xi(w) \rangle\ge(\rho-\Lambda)\|w-\overline{w}\|^2.$$

These analytical properties facilitate both theoretical and algorithmic development (Tran, 13 Jan 2026).

3. Dynamical System Approaches: Finite-Time and Fixed-Time Stability

Two continuous-time dynamical systems are constructed for solving the operator equation $\Xi(w) = 0$:

Finite-Time Stable System:

Given $k>2$ and $\tau>0$,

$$\dot{w} = -\tau \cdot \frac{\Xi(w)}{\|\Xi(w)\|^{(k-2)/(k-1)}}.$$

A Lyapunov analysis with $V(w)=\frac12\|w-\overline{w}\|^2$ yields global finite-time stability, with

$\dot{V} \le -M V^r, \quad r\in(0,1),$

implying convergence to equilibrium in time $T(w(0))\le V(w(0))^{1-r}/[M(1-r)]$, which depends on the initial distance $w(0)-\overline{w}$.

Fixed-Time Stable System:

A nonlinear scaling $\tau(w)$ is used:

$$\dot{w} = -\tau(w)\Xi(w), \qquad \tau(w)=\frac{a_1}{\|\Xi(w)\|^{1-k_1}}+\frac{a_2}{\|\Xi(w)\|^{1-k_2}}+a_3/\|\Xi(w)\|,$$

with $a_1,a_2>0$, $a_3\ge 0$, $k_1\in(0,1)$, $k_2>1$. The system admits a uniform upper bound $T_{\mathrm{max}}$ on the settling time, independent of initial conditions, determined via the Lyapunov method:

$$T_{\mathrm{max}}\le \frac{1}{A_1(1-s_1)}+\frac{1}{A_2(s_2-1)},$$

where $s_1=(1+k_1)/2$, $s_2=(1+k_2)/2$.

These approaches provide accelerated solution schemes with explicit convergence guarantees (Tran, 13 Jan 2026).

4. Discretization and Algorithmic Realizations

The explicit forward-Euler discretization of the fixed-time system leads to a proximal point-type algorithm:

$w_{n+1} = w_n - \theta \cdot \tau_n \cdot \Xi_n,$

where $\Xi_n = \Xi(w_n)$ and $\tau_n = \tau(w_n)$. Under suitable small step sizes ($\theta\in(0,\theta^*]$), the iteration converges globally to $\overline{w}$ in a uniform finite number of steps,

$$n^* = \left\lceil \frac{\chi\pi}{2\lambda\sqrt{A_1A_2}} \right\rceil,$$

and the error's explicit dependence on iteration is provided. This discrete framework enables efficient numerical solution of GIMVIPs and generalizes classical proximal point and gradient-type schemes to nonlinear, nonsmooth, and inverse settings. Allowable step size ranges are dictated by Lyapunov-based conditions and operator parameters.

An explicit instance with $\Omega = [0, \infty)$; $h(w) = w/2$; $F(w) = 3w/4$; $g(w) = w^2 + 2w + 1$ verifies theoretical predictions: the method achieves machine-precision solution within a finite and uniformly bounded number of steps (Tran, 13 Jan 2026).

5. Relation to Generalized Inverse and Mixed VI Frameworks

GIMVIPs encompass and extend a variety of canonical problems:

• Inverse mixed VI: $h = \mathrm{Id}$,
• Inverse VI: $h = \mathrm{Id},\ g \equiv 0$,
• Mixed VI/classical VI: $F = \mathrm{Id}$.

In parameter identification scenarios, as detailed in (Gwinner, 2020), the setup pertains to spaces $V$ (Hilbert), convex sets $K\subset V$ and parameter-dependent operators such as bilinear forms $a(u, v; \theta)$ and nonsmooth functionals $j(v; \theta)$. The inverse problem is cast as \emph{output-least-squares} with a variational inequality constraint, regularized by smoothing and Tikhonov techniques. The direct link to the GIMVIP structure arises via reformulation and regularization, allowing for differentiability of the parameter-to-solution map and enabling adjoint-based optimization algorithms.

The regularization approach smooths nonsmooth contributions, and convergence analysis establishes rates for recovery both at the state and parameter levels, leveraging coercivity, compactness, and monotonicity (Gwinner, 2020).

6. Analytical Properties, Convergence, and Algorithmic Implications

Table: Key Analytical Properties of GIMVIPs

Property	Continuous-Time	Discretized Algorithm
Well-posedness	Unique solution ($\overline{w}$)	Guaranteed by coercivity/monotonicity
Finite-time convergence	Depends on initial state	Initial-state-dependent step bound
Fixed-time convergence	Uniform, independent of initial	Uniform iteration count $n^*$
Parameter dependence	Explicit via operator norms	Explicit via step size and model constants

The differentiability of the regularized parameter-to-solution map in the inverse setting permits efficient gradient-based schemes and adjoint methodologies, with stationarity and optimality conditions readily formulated. A continuation approach with vanishing regularization parameter $\epsilon$ systematically recovers the original constrained problem. Stability and convergence are characterized explicitly in terms of the regularization error, operator coercivity, and the geometry of the feasible set (Gwinner, 2020).

7. Extensions, Special Cases, and Connections

GIMVIPs provide a unifying structure for identification, calibration, and optimal control in settings where underlying constraints are expressed as variational inequalities with mixed or inverse structure. When the operators and functionals specialize, classic monotone VI, mixed VI, and classical inverse VI problems are recovered, and solution theory as well as algorithmic convergence enjoy parallel results. The Lyapunov-based dynamical systems perspective facilitates both theoretical insight and the practical construction of discretization-robust algorithms with verifiable finite- or fixed-time guarantees (Tran, 13 Jan 2026).

Ongoing research targets further extensions to infinite-dimensional spaces, nonsmooth and set-valued operator regimes, and integration with data-driven and PDE-constrained frameworks (Gwinner, 2020).