Non-Monotone Variational Inequality Problem

• Non-monotone VIP is defined as an operator inclusion where standard monotonicity is relaxed through conditions like pseudomonotonicity and local Minty criteria.
• Algorithmic schemes such as the extragradient, Popov, and proximal methods are adapted to handle the challenges posed by non-monotonicity.
• Error bounds and gap-function reformulations provide convergence guarantees, with practical implications in machine learning, game theory, and network economics.

A non-monotone variational inequality problem (VIP) is an operator-theoretic inclusion that generalizes the classic monotone VIP to settings where the monotonicity property fails or is replaced by weaker conditions such as pseudomonotonicity, locally Minty-type conditions, or the existence of error bounds on an associated gap function. These problems are of fundamental importance in optimization, equilibrium theory, and applications such as machine learning, game theory, and network economics, where the structure of the operator does not guarantee monotonicity yet solutions (often equilibria) must be found and computed algorithmically.

1. Definition and Mathematical Formulation

A variational inequality problem is defined as follows: Let $X \subseteq \mathbb{R}^n$ be a closed, convex set and $F : \mathbb{R}^n \to \mathbb{R}^n$ (or more generally, a set-valued mapping). The problem VIP$(F,X)$ asks to find $x^* \in X$ such that

$$\langle F(x^*), x - x^* \rangle \geq 0,\quad \forall x \in X.$$

A monotone VI is one where $F$ satisfies $(F(x) - F(y))^\top (x - y) \geq 0$ for all $x,y \in X$. By contrast, in a non-monotone VI, this property is weakened or absent, and $F$ may be merely continuous, pseudomonotone, or satisfy other generalized monotonicity conditions.

Key distinctions in the literature include:

• Pseudomonotonicity: $F$ is pseudomonotone if for all $x, y \in X$, $(x - y)^\top F(y) \geq 0$ implies $(x - y)^\top F(x) \geq 0$ (Kannan et al., 2014).
• Minty solution: A Minty solution $x^*$ satisfies $\langle F(x), x - x^* \rangle \geq 0$ for all $x \in X$; such a solution often plays a central role in the absence of monotonicity (Huang et al., 2023, Arefizadeh et al., 29 Aug 2024, Arefizadeh et al., 3 Oct 2025).
• Strong Minty/weak Minty solutions: Stronger or weaker versions as in (Alizadeh et al., 17 May 2024, Arefizadeh et al., 3 Oct 2025).
• Point-to-set operators: Non-monotone VIPs often generalize to set-valued $F$, further increasing the problem’s generality (Burachik et al., 2016).

Non-monotone VIPs can model Nash equilibria with nonconvexities, economic equilibrium with nonstandard preferences, or adversarial problems (e.g., GAN training) where the corresponding monotonicity fails.

2. Existence Theory: Sufficient Conditions for Non-Monotone VIPs

Unlike in the monotone setting, existence of a solution to a non-monotone VI cannot be guaranteed by Minty's lemma alone. Key sufficient conditions documented in the literature include:

• Inverse mapping theory: If $F$ is continuously differentiable, $F(\mathbb{R}^m)$ is closed, and $|\nabla F(x)| \neq 0$ wherever $F(x) \neq 0$ (i.e., the Jacobian is nonsingular except at potential solutions), then the unconstrained VI in $\mathbb{R}^m$ has a solution. An equivalent practical criterion is that the partial derivative in variable $x_i$ dominates the sum of the off-diagonal terms for each component, i.e.

$$| \nabla_{x_i} F_i(x) | > \sum_{j \neq i} | \nabla_{x_j} F_i(x) |,$$

for every $i$ (Arefizadeh et al., 29 Aug 2024, Arefizadeh et al., 3 Oct 2025).

• Norm-coercivity: If $\lim_{\|x\|\to\infty} \|F(x)\| = +\infty$, then a solution exists (Arefizadeh et al., 3 Oct 2025).
• Degree theory: If $F$ (or its natural mapping $F_K^{nat}(x) = x - \Pi_X[x - F(x)]$) is uniformly close to a $\xi$-monotone operator $\varphi$ with a unique solution, then solutions to the perturbed (non-monotone) VI exist via invariance of the degree (Arefizadeh et al., 3 Oct 2025).
• Existence of Minty solutions: When there is $x^* \in X$ such that $\langle F(x), x - x^* \rangle \geq 0$ for all $x \in X$, then $x^*$ is a Minty solution. A sufficient condition: if $F$ is close to a strongly monotone mapping $\varphi$ (i.e., $\|F(x) - \varphi(x)\| \leq d \|x - x^*\|$ with $d < \mu_\varphi$), then the unique solution of the strongly monotone VI for $\varphi$ is a Minty solution for $F$ (Arefizadeh et al., 29 Aug 2024, Arefizadeh et al., 3 Oct 2025).

3. Algorithmic Schemes and Convergence

Algorithmic treatment of non-monotone VIPs requires modifications of standard methods. The following summarizes key approaches:

Extragradient and Popov Methods

• Korpelevich’s extragradient method: The iteration

$$y^k = \Pi_X[x^k - \alpha F(x^k)],\quad x^{k+1} = \Pi_X[x^k - \alpha F(y^k)]$$

is shown to converge to a VI solution under mere Lipschitz continuity of $F$ and the existence of a Minty solution if $0 < \alpha < 1/L$ (Arefizadeh et al., 29 Aug 2024, Arefizadeh et al., 3 Oct 2025). Convergence is established via nonexpansiveness of $\Pi_X$ and recursive inequalities that leverage the Minty property.

• Popov algorithm: With updates

$$x^{k+1} = \Pi_X[x^k - \alpha F(y^k)],\quad y^{k+1} = \Pi_X[x^{k+1} - \alpha F(y^k)],$$

one obtains convergence under similar assumptions for $\alpha < 1/(3L)$ (Arefizadeh et al., 3 Oct 2025).

Projection-Type and Proximal Algorithms

• Projection-type methods: These generalize to point-to-set operators without monotonicity via carefully crafted linesearches (e.g., Algorithm F (Burachik et al., 2016)) and boundedness/semicontinuity requirements, with the dual solution set non-empty. Such algorithms project the initial iterate onto intersections of halfspaces determined by feasible directions, rather than relying on Fejér monotonicity.
• Proximal point algorithm: Finite convergence is guaranteed under weak sharpness without monotonicity, and explicit bounds on the number of steps can be derived (Al-Homidan et al., 2017).
• Adaptive methods: Recent results show that the projection, Korpelevich, and Popov methods can be made adaptive (with normalized or “clipped” stepsizes) and still guarantee convergence for non-monotone (but $\alpha$-symmetric and $p$-quasi sharp) operators (Vankov et al., 8 Feb 2024).

Gap-Function and Error Bound Reformulations

• Smooth gap functions: Non-monotone VIPs can be reformulated as the unconstrained minimization of a smooth gap function $$g_\lambda(x) = \max_{y \in X} \langle F(x), x - y \rangle - (1/2\lambda)\|x-y\|^2$$, with $x^*$ solving the VIP iff $g_\lambda(x^*) = 0$. Proximal-gradient algorithms on $g_\lambda$ converge locally linearly if level-set error bounds on the gap function are available and initialization is sufficiently close (Zhao et al., 14 Oct 2025). Homotopy continuation (starting from a strongly monotone perturbation) enables global convergence in the affine case.

Stochastic and Variance-Reduced Algorithms

• Stochastic extragradient-like methods: ESA (extragradient stochastic approximation) and mirror-prox generalizations are provably convergent under relaxed pseudomonotonicity or acute angle conditions, achieving $\mathcal{O}(1/k)$ mean-squared error rates (Kannan et al., 2014, Choudhury et al., 2023). Key regularity assumptions (e.g., expected residual or weak Minty conditions) replace classical monotonicity (Choudhury et al., 2023, Alizadeh et al., 17 May 2024).
• Variance reduction: Single-loop variance-reduced methods using Bregman distance prox-maps achieve state-of-the-art complexity under monotonicity and improved rates for weak Minty non-monotone VIs (Alizadeh et al., 17 May 2024).

4. Generalizations and Error Bound Structures

• Error bounds: Uniform and level-set error bounds on gap functions play a pivotal role in error analysis and local linear convergence of first-order algorithms (Zhao et al., 14 Oct 2025). The uniform Kurdyka–Łojasiewicz (u-KL), subdifferential error bounds, and H\"older error bounds provide a hierarchy of sufficient conditions that can be verified under polyhedrality or specific problem structure.
• Algorithm-based sufficient conditions: Beyond global Minty conditions, localized properties (e.g., local Minty, GP, GP+, and GP* as in (Huang et al., 2023))—formulated in terms of the iterates and evaluated solutions via projection/extra-gradient mappings—guarantee convergence even without monotonicity or existence of Minty solutions.

5. Applications and Practical Significance

Non-monotone VIPs arise in many domains:

• Game theory: Nash and quasi-Nash equilibria can be cast as non-monotone VIs; sufficient "weak coupling" of player objectives or certain Jacobian non-singularity conditions guarantee pure equilibrium existence (Arefizadeh et al., 29 Aug 2024).
• Economic equilibrium: Revenue-maximization, product pricing, and certain fractional optimization are naturally non-monotone, motivating the use of pseudomonotonicity and Minty conditions (Kannan et al., 2014).
• Learning and optimization: In GAN training, robust optimization, distributionally robust optimization, traffic control, and machine learning, non-monotonicity often arises due to adversarial structure, non-convexities, or coupled constraints (Huang et al., 2023, Zhao et al., 14 Oct 2025, Alizadeh et al., 17 May 2024).
• Polynomial VIPs: For VIPs defined by polynomial data and constraints, solution or infeasibility can be computed via reformulation to polynomial systems and semidefinite relaxations, even in the absence of monotonicity (Nie et al., 2023).

6. Summary Table: Sufficient Conditions and Principal Algorithms

Existence Condition	Main Algorithmic Guarantee	Reference
Inverse mapping/Nonsingularity	Solution exists (unconstrained/constrained)	(Arefizadeh et al., 29 Aug 2024, Arefizadeh et al., 3 Oct 2025)
Weak coupling via Jacobian	Solution exists (unconstrained case)	(Arefizadeh et al., 29 Aug 2024)
Minty solution present	Extragradient / Popov / PG converge	(Arefizadeh et al., 29 Aug 2024, Arefizadeh et al., 3 Oct 2025, Zhao et al., 14 Oct 2025)
Strongly monotone approximation	Minty solution exists, gap error bound holds	(Zhao et al., 14 Oct 2025)
Uniform/level-set error bound	Local linear convergence of PG	(Zhao et al., 14 Oct 2025)
Pseudomonotonicity / Weak Minty	Stochastic/extragradient rate guarantees	(Kannan et al., 2014, Choudhury et al., 2023, Alizadeh et al., 17 May 2024)

7. Open Research Directions and Outlook

Substantial progress has been made in extending core methodology (extragradient, Popov, adaptive projection, stochastic, gap-function methods) from monotone to non-monotone VIs by exploiting weaker regularity properties, Minty-type solutions, error bounds, and degree-theoretic arguments (Huang et al., 2023, Arefizadeh et al., 3 Oct 2025, Zhao et al., 14 Oct 2025). Notable ongoing questions include:

• How to characterize intrinsic error-bound properties in complex non-monotone structured VIs and guarantee global convergence without relying on monotonicity or compactness.
• Designing adaptive algorithms that balance regularization, scaling, and variance reduction specifically for high-dimensional and non-monotone saddle-point settings.
• Developing practical verification tools for Minty solutions or error bound conditions in large-scale games and complex machine learning problems, enabling robust computational implementation.

These advances mark a significant broadening of the variational inequalities paradigm, actively connecting theory and computation in non-monotone environments previously considered intractable without global monotonicity.