Mixed Variational Inequality Problem

• MVI is a framework combining a monotone operator with a convex function, generalizing classical variational inequalities to include composite optimization and equilibrium problems.
• Recent algorithms leverage adaptive step-sizes, Bregman prox mappings, and inertial corrections to achieve provable convergence rates in challenging monotone and non-monotone settings.
• Applications span matrix games, saddle-point problems, and regularized regression, demonstrating MVI's versatility in modeling complex real-world systems.

A mixed variational inequality problem (commonly abbreviated MVI or MVIP) generalizes the classical variational inequality (VI) by incorporating both a monotone operator and a convex (potentially nonsmooth) function, thereby subsuming optimization, monotone inclusions, and equilibrium models. The problem is formulated on a real Hilbert space $\mathcal{H}$ with a monotone “cost operator” $A: \mathcal{H} \to \mathcal{H}$ and an extended-real-valued proper, convex, lower-semicontinuous function $g: \mathcal{H} \to (-\infty, +\infty]$. The goal is to find $w^* \in \mathcal{H}$ such that

$$\langle A(w^*), w - w^* \rangle + g(w) - g(w^*) \ge 0, \quad \forall w \in \mathcal{H}.$$

For $g \equiv 0$, this reduces to the classical VI, whereas for nonsmooth or highly structured functions $g$, MVI models encapsulate composite optimization and monotone inclusions. These formulations arise in matrix games, saddle-point problems, regularized regression, conic constraints in non-monotone VIs, electrical circuit analysis, and hierarchical fixed-point problems.

1. Formal Statement and Fundamental Properties

Let $\mathcal{H}$ be a finite-dimensional real Hilbert space, equipped with inner product $\langle \cdot, \cdot \rangle$ and norm $\|\cdot\|$. The mixed variational inequality seeks $w^* \in \mathcal{H}$ such that

$$\langle A(w^*), w - w^* \rangle + g(w) - g(w^*) \geq 0\quad \forall w \in \mathcal{H}, \tag{MVIP}$$

where $A: \mathcal{H} \to \mathcal{H}$ is monotone and $g: \mathcal{H} \to (-\infty, +\infty]$ is proper, convex, lower-semicontinuous. The solution set is denoted by $S = \{w^*\,|\,w^*\text{ satisfies (MVIP)} \}$.

In the generalization to constrained settings, MVI is frequently expressed over a convex set $\mathcal{C} \subset \mathcal{H}$ as

$$\langle T(x^*), y - x^* \rangle + h(y) - h(x^*) \geq 0, \quad \forall y \in \mathcal{C},$$

with a monotone Lipschitz operator $T$ and convex function $h$.

2. Algorithmic Approaches and Adaptive Schemes

A large array of first-order and proximal-point algorithms has emerged for MVI, exploiting Bregman distances, inertial extrapolation, contraction steps, and adaptive line-search for step-size selection. Key approaches include:

• Modified Bregman Golden Ratio Algorithm (Kumar et al., 8 Mar 2025): Employs a Bregman-proximal mapping using a Legendre distance-generating function $h$, with an adaptively nondecreasing step-size $\lambda_k$ that avoids prior knowledge of the global Lipschitz constant of $A$. The update sequence is:
  • Compute $$\bar{w}_k = (\nabla h)^{-1}\Big(\frac{(\phi-1)\nabla h(w_k)+\nabla h(\bar{w}_{k-1})}{\phi}\Big)$$, $\phi = (1+\sqrt{5})/2$.
  • Update $w_{k+1}$ by minimizing $$\langle A(w_k), w-w_k \rangle + g(w) + \frac{1}{\lambda_k} B_h(w, \bar{w}_k)$$.
  • Locally adapt $\lambda_k$ as a function of the local Lipschitz quotient $\|A(w_k)-A(w_{k-1})\|/\|w_k-w_{k-1}\|$.
• Forward–Backward–Forward Dynamical System (Nwakpa et al., 23 Nov 2025): Models the solution trajectory by a continuous-time ODE system where the proximal mapping returns $y(t)$, and the main system evolves as:

$$\dot{x}(t) + x(t) = y(t) + \lambda[T(x(t)) - T(y(t))], \quad y(t) = \operatorname{prox}_{\lambda h}(x(t)-\lambda T(x(t))).$$

Weak convergence is achieved if the operator is monotone and Lipschitz; exponential stability arises under h-strong pseudomonotonicity.

• Proximal–Contraction Algorithm with Inertia and Corrections (Nwakpa et al., 23 Nov 2025): Incorporates inertial extrapolation, two sequential correction terms, a self-adaptive step-size rule, relaxation, and contraction directions for accelerated convergence. The update steps involve
  • Multi-term inertial correction: $$w_n = x_n + \alpha(x_n-x_{n-1}) + \delta(1+\alpha)(w_{n-1}-x_n) - \alpha\delta(w_{n-2}-x_{n-1})$$.
  • Adaptive proximal and contraction steps, with self-adaptive update of $\lambda_n$ based on local operator differences.
• Adaptive Proximal Methods with Abstract Model Inequality (Stonyakin, 2019): Universal algorithms under only local model smoothness and monotonicity assumptions, adapting the regularization parameter using Bregman divergence mismatch; achieves complexity $O(L\Omega/\epsilon)$.

3. Convergence Theory and Rate Results

Rigorous convergence guarantees exist for most contemporary MVI schemes, under monotonicity, strong convexity, and coherence-type assumptions:

Algorithm	Convergence Rate	Requirements
Modified B-GRAAL (Kumar et al., 8 Mar 2025)	R-linear under Bregman-strong monotonicity	Monotonicity, strong convexity of $h$
Forward–Backward–Forward DS (Nwakpa et al., 23 Nov 2025)	Weak convergence (general monotonicity), exponential under h-strong pseudomonotonicity	Lipschitz, h-strong pseudomonotonicity
Proximal–Contraction w/ Inertia (Nwakpa et al., 23 Nov 2025)	Weak convergence	Lipschitz, monotone, convex function
Inertial PPA (Chen et al., 2014)	$o(1/k)$ residual rate	$H$-monotonicity, inertia $< 1/3$
ALAVI (augmented Lagrangian) (Zhao et al., 2023)	$o(1/\sqrt{k})$ global, $O(1/k)$ ergodic; linear under metric subregularity	Variational coherence, metric subregularity
Adaptive Proximal UVI (Stonyakin, 2019)	$O(1/\epsilon)$ complexity	Model inequality, convexity, monotonicity

• The “Modified B-GRAAL” algorithm provides both global convergence (Lemma 3.3, Theorem 3.1) and R-linear rate (Theorem 4.2) under strong monotonicity in the Bregman sense.
• The ALAVI method achieves global convergence and sublinear rates in the absence of monotonicity, with local linear rate under metric subregularity (Zhao et al., 2023).
• Proximal-type methods with inertial steps, as in (Chen et al., 2014), guarantee $o(1/k)$ non-ergodic rate under mild $H$-monotonicity and inertia bounds.

4. Extensions and Generalizations

MVI naturally extends to structured and non-monotone settings:

• Conically constrained, non-monotone VIs (Zhao et al., 2023): Handles constraints of the form $\Theta(u) \in -\mathcal{C}$ for convex cones, and non-monotone mappings via coherence conditions. Solutions are characterized as saddle points of suitably augmented Lagrangian systems.
• Generalized Mixed Equilibrium Problems (Karahan, 2014): Further includes bifunctions, equilibrium constraints, and hierarchical fixed-point problems.
• Composite Saddle-Point Problems (Stonyakin, 2019): MVI subsumes primal-dual saddle formulations, with the bifunction representation and strong convex/concave structure supporting complex constraints.

5. Numerical Performance and Benchmark Results

Empirical evaluations across recent works demonstrate MVI algorithms' superior performance over traditional methods:

• Matrix Game and Sparse Logistic Regression (Kumar et al., 8 Mar 2025): The locally adaptive Modified B-GRAAL outperforms fixed-step and non-backtracking B-GRAAL variants in both iteration count and wall-clock time, while eliminating the dependence on the global Lipschitz constant.
• Constrained Non-monotone VI Examples (Zhao et al., 2023): ALAVI reliably solves highly nonlinear/non-monotone VIs up to dimension $n = 2000$, achieving KKT residuals $<10^{-6}$.
• Comparison against contemporary approaches (Nwakpa et al., 23 Nov 2025): The contraction-proximal algorithm with inertia and corrections requires fewer iterations and less CPU time than alternatives by Kim, Maingé, Dong–Cho, and Jolaoso–Shehu–Yao.

6. Connections to Broader Research Directions

Mixed variational inequalities interface with multiple core research areas:

• First-order splitting algorithms: MVI is the unifying abstraction for mirror descent, operator splitting, extragradient, and ADMM variants (including inertial linearized ADMM (Chen et al., 2014)).
• Monotone inclusions and equilibrium problems: The bifunction representation and general monotonicity in MVI spans monotone operator inclusions, Nash equilibria, and saddle-point models.
• Adaptive methods and line-search: State-of-the-art MVI solvers employ locally adaptive step-sizes, often bypassing oracle access to global smoothness parameters, and robustly handle inexact or noisy computation.

7. Notable Extensions, Special Cases, and Limitations

• When $g \equiv 0$, MVI reduces to classical VI and all convergence results for monotone VIs apply.
• For non-convex or non-monotone problems, primal-dual variational coherence and metric subregularity afford convergence and rate guarantees (Zhao et al., 2023).
• Incorporation of inertial (heavy-ball) terms provides empirical acceleration, contingent on careful parameter control to avoid destabilization (Chen et al., 2014).
• Further generalization to Banach spaces and broader classes of bifunctions broadens the applicability, though some strong convergence guarantees weaken to weak convergence or only ergodic rates.

In summary, the mixed variational inequality framework subsumes numerous models in optimization and equilibrium theory. Recent advances in adaptive step-size, Bregman-proximal schemes, inertial acceleration, and augmented Lagrangian techniques have yielded provably and empirically fast algorithms capable of handling monotone, non-monotone, and non-smooth cases, as validated on matrix games, logistic regression, structured saddle-point, and large-scale constrained VIs (Kumar et al., 8 Mar 2025, Nwakpa et al., 23 Nov 2025, Nwakpa et al., 23 Nov 2025, Zhao et al., 2023, Karahan, 2014, Stonyakin, 2019, Chen et al., 2014).