Nested Variational Inequalities

• Nested variational inequalities are hierarchical problems where the upper-level VI’s feasible set is defined by a lower-level VI, enabling complex equilibrium modeling.
• Tikhonov regularization and prox-penalization techniques enforce strong monotonicity and contraction, ensuring robust convergence of iterative schemes.
• These frameworks find practical applications in bilevel convex optimization, Nash equilibrium selection, and multi-follower game scenarios.

Nested variational inequalities refer to hierarchical problems where the solution set (or feasible region) of an upper-level variational inequality (VI) is itself defined as the solution set of a lower-level VI. This structure generalizes classical VI formulations, enabling the modeling of complex hierarchical equilibria, bilevel programming, and certain classes of equilibrium selection and multi-agent games. The field encompasses operator-theoretic, regularization, algorithmic, and complexity-theoretic developments across convex and monotone analysis, optimization, and game theory.

1. Mathematical Structure: Hierarchical VI Formalism

A nested VI problem consists of an upper-level VI: $$\text{Find } u \in H \text{ such that } \langle G(u), v-u\rangle \geq 0 \quad \forall v \in S_0$$ where the feasible set $S_0$ is characterized by: $$S_0 := \operatorname{zer}(A + F) = \{ x \in H : 0 \in A(x) + F(x) \}$$ with $G : H \to H$ and $F : H \to H$ monotone, Lipschitz continuous maps, and $A : H \to 2^H$ a maximally monotone operator (often the subdifferential of a convex function).

Nested VIs are pervasive in hierarchical convex bilevel optimization and multi-follower games, where the feasible region at the upper level is not directly accessible but implicitly specified through a lower-level VI. This structure generalizes quasi-variational inequalities (QVI), complementarity problems, and certain game-theoretic equilibrium and selection formulations (Kapron et al., 7 Nov 2024).

2. Tikhonov Regularized Exterior Penalty Methods

To compute solutions to nested VIs, a double loop prox-penalization approach leverages strong monotonicity through Tikhonov regularization. The key regularized operator is: $$\Phi_{\alpha,\beta}(v, w) = F(v) + \beta G(v) + \alpha (v - w)$$ where $\alpha > 0$ is the proximal penalty, $\beta > 0$ the Tikhonov regularization parameter, and $w$ the current anchor (reference) point. The extra term $\alpha (v-w)$ induces strong monotonicity and contractivity, ensuring unique solvability for fixed $w$.

The inner iteration computes the unique fixed point: $$0 \in A(\bar{u}_{\alpha,\beta}(w)) + \Phi_{\alpha,\beta}(\bar{u}_{\alpha,\beta}(w), w)$$ by (possibly relaxed and inertial) forward-backward splitting: $$v^{k+1} = J_{\gamma_k A}(v^k - \gamma_k \Phi_{\alpha,\beta}(v^k, w))$$ where $J_{\gamma_k A} = (I + \gamma_k A)^{-1}$ is the resolvent operator. The outer loop updates $w$ and $\beta$, with $\beta_t \downarrow 0$ to recover limiting equilibrium selection.

3. Convergence Guarantees and Algorithmic Properties

Under monotonicity and Lipschitz continuity assumptions for $F$ and $G$, and suitable step size choices ($\gamma_k < 2\alpha / L_{\alpha,\beta}^2$), the inner iteration is contractive: $$q_k(\alpha, \beta) = \sqrt{1 - \gamma_k (2\alpha - \gamma_k L_{\alpha,\beta}^2)} \in (0,1)$$ Strong convergence is guaranteed for the inner loop (all iterates $v^k$ converge to $\bar{u}_{\alpha,\beta}(w)$), provided the inexactness sequence ($\theta_k \delta_k$) is square summable, and the inertial and relaxation parameters ($\tau_k, \theta_k$) satisfy standard inequalities: $$v^{k+1} = (1-\theta_k)z^k + \theta_k \tilde{T}_k(z^k), \quad z^k = v^k + \tau_k (v^k - v^{k-1})$$ The outer loop sequence $(w^t)$ is bounded, with every weak cluster point belonging to the set: $S_1 = \operatorname{zer}(G + N_{S_0})$ where $N_{S_0}$ denotes the normal cone operator. This selects an equilibrium from the lower-level solution set according to the upper-level VI.

4. Applications: Bilevel Optimization, Nash Equilibrium Selection, and Beyond

This methodology subsumes several important applications:

• Bilevel convex optimization: Problems of the form

$$\min_{u \in H} g(u) \text{ subject to } u \in \arg\min_{v \in H} \{ r(v) + f(v) \}$$

where $G = \nabla g, F = \nabla f, A = \partial r$.

• Structured convex programs: For

$$\min_{u \in H} g(u) \text{ subject to } u \in \arg\min_{v \in H} \{ f(v) + r(Lv) \}$$

Fenchel-Rockafellar duality reformulates the nested structure as a VI over coupled spaces.

• Equilibrium selection in Nash games: Select equilibrium $u \in S_0$ minimizing an additional criterion, e.g.,

$\min_{u \in S_0} \phi(u)$

with $S_0$ the equilibrium set.

• Multi-follower Stackelberg games: Nested VI captures the response mapping from followers characterized via a lower-level VI.

Additional applications cited include signal processing, inverse problems, and resource allocation in stochastic and deterministic regimes.

5. Connections to Set-Optimization, Multilevel Equilibria, and Complexity

Nested VI theory is tightly related to set-optimization via inf-translation operators (Crespi et al., 2013). For a convex set-valued objective $f : X \to \mathcal{P}(Z, C)$, the “inf-translation” over a solution set $M \subseteq X$ is: $\hat{f}(x; M) = \inf \{ f(m+x) : m \in M \}$ Directional derivatives and Minty/Stampacchia-type inequalities for the inf-translated function characterize optimality in set-optimization, vector optimization, and multilevel equilibrium problems.

In computational game theory, many complex solution concepts, including resilient Nash equilibria and multi-leader-follower settings, are recast as nested or quasi-variational inequalities where feasible sets themselves depend on other strategies or equilibrium mappings. Approximate solution algorithms using separation oracles, projection circuits, and fixed-point methods situate these problems within the PPAD-completeness class (Kapron et al., 7 Nov 2024).

6. Algorithmic and Practical Implications

Double-loop penalization and regularization provide robust strategies for hierarchical VIs, allowing proximal splitting and inertial/relaxation acceleration. Numerical experiments (e.g., on two-player zero-sum games) show strong convergence properties across different initializations and regularization parameters. Parameter selection (e.g., the decay schedule for $\beta_t$), step-size rules, and inner loop accuracy thresholds impact convergence rates and solution selection properties.

The framework accommodates inexact computation, strong monotonicity-induced contractivity, and regularization-based selection. These techniques are extensible to inf-translation set-optimization and complement classical approaches employing Stampacchia/Minty-type inequalities and scalarizations.

7. Outlook and Extensions

Nested variational inequalities, underpinned by regularization and exterior penalty approaches, have established computational, theoretical, and practical relevance for hierarchically constrained equilibrium, multilevel games, and bilevel optimization. Extensions include deep-nested multilevel equilibrium modeling, further analysis of sample complexity under strong monotonicity (Zhao et al., 28 Oct 2024), stochastic algorithms, and applications in distributed optimization, learning, and strategic networks.

Emerging directions involve modular algorithm design capable of stacking or composing forward-backward and extragradient schemes, analysis of variance reduction and batching for stochastic nested VIs (Pichugin et al., 15 Jan 2024), and unification with set-valued optimization, multilevel gap functions, and scalability in large-scale applications.

Nested VI frameworks thus offer a unified lens on hierarchical and multi-level optimization, equilibrium computation, and their algorithmic and complexity-theoretic properties.