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Variational inequality

A variational inequality is a mathematical formulation that seeks to find an element u^* in a closed convex subset K of a vector space such that \langle Au^*, v - u^* \rangle \geq 0 for all v \in K, where A is a mapping (often nonlinear) from the space to its dual, and \langle \cdot, \cdot \rangle denotes the duality pairing.^[1] This inequality generalizes classical variational principles and equilibrium conditions by incorporating constraints directly into the formulation, allowing solutions to satisfy directional inequalities rather than equalities.^[2] The concept was introduced by Philip Hartman and Guido Stampacchia in their seminal 1966 paper, where it was developed as a tool to address nonlinear elliptic differential-functional equations arising in boundary value problems with obstacles or unilateral constraints.^[3] Building on earlier work in convex analysis and functional equations, variational inequalities provide a unified framework for studying a broad class of problems, including those where traditional equations fail due to discontinuities or non-smoothness.^[1] Key existence results, such as the Hartman-Stampacchia theorem, guarantee solutions under conditions like monotonicity (where \langle A(u) - A(v), u - v \rangle \geq 0), hemicontinuity, and coercivity of A, ensuring the problem is well-posed even in infinite-dimensional Hilbert or Banach spaces.^[2] In finite dimensions, the variational inequality problem VI(F, K)—finding x^* \in K \subseteq \mathbb{R}^n such that F(x^*)^T (x - x^*) \geq 0 for all x \in K, with F continuous and K closed convex—extends to applications in optimization, where it characterizes the optimality conditions for convex programs, and in game theory, modeling Nash equilibria in noncooperative games.^[2] Uniqueness often follows from strict monotonicity of F.^[2] Beyond pure mathematics, variational inequalities have profound impacts in applied fields: in mechanics, they model contact problems like Signorini's obstacle problem for elastic bodies; in economics, they describe spatial price equilibria and oligopolistic markets; and in engineering, they arise in traffic networks (via Wardrop equilibria), porous media flow, and financial modeling of option pricing under constraints.^[1]^[2] Numerical methods, including projection algorithms and fixed-point iterations, are well-developed for solving these problems efficiently, with extensions to stochastic and multivalued variants addressing uncertainty and set-valued mappings.^[4]

History

Origins and Early Concepts

The origins of variational inequalities can be traced to the classical calculus of variations, where the Euler-Lagrange equations served as key precursors by providing necessary conditions for extremal functions in optimization problems. Developed by Leonhard Euler in the 1740s through works such as his 1744 publication Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti, these equations addressed the minimization of functionals subject to constraints, revealing the need for modified conditions when inequalities restricted the admissible functions—such as in scenarios involving barriers or unilateral restrictions that prevent certain variations from being negative.^[5] In particular, problems like finding minimal surfaces above an obstacle highlighted how equality-based constraints in the Euler-Lagrange framework naturally extended to inequality forms to ensure the functional's variation remained non-negative on the feasible set.^[6] Building on Euler's foundations, Karl Weierstrass advanced the theory in 1879 by refining necessary conditions for extrema in variational problems under constraints, emphasizing sufficient conditions and the role of second variations to distinguish maxima, minima, and saddle points. His improvements addressed shortcomings in earlier approaches, particularly for constrained extrema where inequality barriers required careful analysis of the Legendre condition and corner conditions, laying groundwork for handling unilateral constraints without assuming smooth equality restrictions. David Hilbert's 1904 work on the Dirichlet principle further extended these ideas by rigorously justifying the existence of minimizers for variational integrals associated with elliptic partial differential equations. In his paper "Über das Dirichletsche Prinzip," Hilbert used compactness arguments in suitable function spaces to ensure the integral attains its minimum.^[7] Early 20th-century contributions, notably Richard Courant's analysis of elliptic partial differential equations with boundary constraints, applied variational methods in the context of physical problems like membrane vibrations and equilibrium states subject to rigid supports. Courant's inequality, derived for solutions bounded by obstacle-like conditions on the boundary, provided bounds on eigenvalues and energy functionals, illustrating how variational methods could incorporate unilateral constraints to model real-world barriers in elliptic boundary value problems.^[8] These developments bridged classical variational calculus with more general inequality formulations, paving the way for mid-20th-century formalizations.

Key Developments in the Mid-20th Century

The mid-20th century marked the formal emergence of variational inequalities as a unified mathematical framework, building on prior work in functional analysis and partial differential equations. Independently, Gaetano Fichera around 1963–1964 developed variational inequality formulations for unilateral constraints in elasticity, notably addressing the Signorini problem of contact with frictionless rigid obstacles, establishing existence principles for such problems. In 1962, George J. Minty provided a pivotal generalization of Felix Browder's fixed-point theorem to monotone operators in Hilbert spaces, establishing the Minty-Browder formulation that ensured the surjectivity of the sum of the identity and a maximal monotone operator under appropriate conditions. This result laid essential groundwork for solving nonlinear problems via resolvent operators, influencing subsequent developments in operator theory.^[9] A landmark contribution came in 1964 from Guido Stampacchia, who introduced the classical variational inequality in the context of elliptic boundary value problems with discontinuous coefficients. In his seminal paper, Stampacchia extended the Lax-Milgram theorem to coercive bilinear forms on convex closed subsets of Hilbert spaces, proving existence and uniqueness for solutions to the problem: find u \in K such that a(u, v - u) \geq 0 for all v \in K, where K is a closed convex subset, a(\cdot, \cdot) is a continuous coercive bilinear form on the space, and the associated operator A satisfies a(u, v) = \langle Au, v \rangle. This formulation, often denoted as \langle Au, v - u \rangle \geq 0 for all v \in K, provided a powerful tool for handling inequalities arising in mechanics and potential theory. In 1966, Philip Hartman and Guido Stampacchia published a seminal paper on nonlinear elliptic differential-functional equations, further developing variational inequalities as a tool for boundary value problems with obstacles or unilateral constraints.^[3] By 1969, Jacques-Louis Lions significantly advanced the theory by extending variational inequalities to time-dependent evolution equations and nonlinear partial differential equations. In his monograph, Lions developed abstract frameworks for parabolic and hyperbolic problems, incorporating variational inequalities to address noncoercive cases and obstacle-type constraints in dynamic settings. These extensions enabled the treatment of complex phenomena such as flow through porous media and viscoelasticity, solidifying variational inequalities as a versatile method for nonlinear analysis.^[10]

Formal Definition

Finite-Dimensional Formulation

In finite-dimensional Euclidean space \mathbb{R}^n, the standard inner product is defined as \langle \mathbf{x}, \mathbf{y} \rangle = \mathbf{x}^T \mathbf{y} for \mathbf{x}, \mathbf{y} \in \mathbb{R}^n, inducing the Euclidean norm \|\mathbf{x}\| = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}.^[4] A nonempty subset K \subseteq \mathbb{R}^n is convex if, for all \mathbf{x}, \mathbf{y} \in K and \lambda \in [0, 1], the convex combination \lambda \mathbf{x} + (1 - \lambda) \mathbf{y} \in K.^[4] The finite-dimensional variational inequality problem, denoted \mathrm{VI}(F, K), is to find a vector \mathbf{x}^* \in K such that

\langle F(\mathbf{x}^*), \mathbf{y} - \mathbf{x}^* \rangle \geq 0 \quad \forall \, \mathbf{y} \in K,

where K \subseteq \mathbb{R}^n is a nonempty convex set and F: \mathbb{R}^n \to \mathbb{R}^n is a single-valued mapping.^[11] This formulation, often assuming F is continuous, provides an accessible framework for problems in optimization and equilibrium analysis.^[4] For set-valued mappings F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n, the problem generalizes to finding \mathbf{x}^* \in K and \mathbf{w} \in F(\mathbf{x}^*) such that

\langle \mathbf{w}, \mathbf{y} - \mathbf{x}^* \rangle \geq 0 \quad \forall \, \mathbf{y} \in K,

with continuity of the selection or upper semicontinuity of F typically imposed for theoretical analysis.^[11] A solution \mathbf{x}^* to \mathrm{VI}(F, K) equivalently satisfies the fixed-point equation

\mathbf{x}^* = \proj_K \bigl( \mathbf{x}^* - F(\mathbf{x}^*) \bigr),

where \proj_K: \mathbb{R}^n \to K is the orthogonal projection onto the closed convex set K, defined as \proj_K(\mathbf{z}) = \arg\min_{\mathbf{u} \in K} \|\mathbf{z} - \mathbf{u}\|^2.^[11]

General Formulation in Banach Spaces

The general formulation of a variational inequality in a Banach space setting extends the finite-dimensional case to infinite-dimensional spaces, leveraging tools from functional analysis such as duality pairings and weak topologies. Let V be a reflexive Banach space with dual space V^*, and let K \subseteq V be a nonempty, convex, and closed subset. Given a nonlinear operator A: V \to V^*, the variational inequality problem seeks an element u \in K satisfying

\langle A(u), v - u \rangle \geq 0 \quad \forall v \in K,

where \langle \cdot, \cdot \rangle denotes the duality pairing between V and V^*. This formulation captures equilibrium conditions in abstract spaces, where the operator A represents forces or gradients, and the constraint K models admissible configurations. The dual space V^* plays a crucial role, as it allows the inequality to be expressed in terms of linear functionals, facilitating analysis in spaces without inner products, unlike the Hilbert space case.^[12]^[4] Key assumptions underpin this setup to ensure well-posedness in the weak topology. The reflexivity of V guarantees the weak compactness of closed bounded sets via the Banach-Alaoglu theorem, which is essential for handling infinite-dimensional phenomena like weak convergence. The set K must be convex and closed to preserve the variational structure under weak limits. The operator A is typically assumed to be continuous, or more generally hemicontinuous—meaning that the map t \mapsto \langle A((1-t)u + tv), w \rangle is continuous on [0,1] for all u, v, w \in V—to align with weak convergence properties. These conditions enable the extension of finite-dimensional intuition to broader function spaces without relying on coordinate representations.^[12]^[4] A related, relaxed variant is the Minty variational inequality, which seeks u \in K such that

\langle A(v), v - u \rangle \geq 0 \quad \forall v \in K.

Originally introduced for monotone operators in Hilbert spaces, this form extends naturally to reflexive Banach spaces and serves as an auxiliary problem. Under the assumptions of monotonicity—i.e., \langle A(u) - A(v), u - v \rangle \geq 0 for all u, v \in V—and hemicontinuity of A, solutions to the Minty inequality coincide with those of the standard (Stampacchia) formulation. This equivalence simplifies proofs involving surjectivity or maximality of monotone operators.^[9]^[4] In applications to nonlinear problems, the operator A is often the Gâteaux derivative of a functional J: V \to \mathbb{R}, defined as A(u) = J'(u) where the directional derivative satisfies \lim_{t \to 0^+} \frac{J(u + t h) - J(u)}{t} = \langle J'(u), h \rangle for h \in V. This perspective links variational inequalities to subdifferential inclusions for convex functionals, with the dual space V^* hosting the subgradients. The Gâteaux differentiability assumption accommodates nonlinearities beyond linear operators, such as those arising in elasticity or fluid dynamics, while maintaining the duality framework.^[12]^[4]

Theoretical Properties

Monotonicity Conditions

In the context of variational inequalities, monotonicity conditions on the operator A play a fundamental role in establishing the existence of solutions. A single-valued operator A: X \to X^*, where X is a reflexive Banach space with dual X^*, is said to be monotone if for all u, v \in X,

\langle A(u) - A(v), u - v \rangle \geq 0,

where \langle \cdot, \cdot \rangle denotes the duality pairing between X and X^*. This condition generalizes the increasing nature of functions to nonlinear operators in infinite-dimensional spaces. If the inequality is strict for u \neq v, then A is strictly monotone, which strengthens the separation properties between points. Furthermore, A is strongly monotone if there exists \mu > 0 such that

\langle A(u) - A(v), u - v \rangle \geq \mu \|u - v\|^2

for all u, v \in X, providing a quantitative measure of how much A "expands" differences, often leading to unique solutions in associated problems. Hemicontinuity is another key regularity condition that complements monotonicity. An operator A is hemicontinuous if it is locally bounded and continuous with respect to the weak topology along every line segment in its domain, meaning that for any u, v \in X with v \in D(A), the mapping t \mapsto \langle A((1-t)u + t v), w \rangle is continuous in t \in [0,1] for every w \in X. This property ensures that monotone operators behave sufficiently regularly to apply topological arguments, such as those in fixed-point theorems, without requiring full continuity. Maximal monotonicity extends the notion of monotonicity to set-valued operators A: X \rightrightarrows X^*, where the graph of A, defined as \{(u, u^*) \in X \times X^* \mid u^* \in A(u)\}, cannot be properly contained in the graph of any other monotone operator. Minty's characterization provides an equivalent condition for maximality: a monotone operator A is maximal if and only if, for every \lambda > 0, the range of I + \lambda A covers the entire dual space X^*, where I is the identity operator. This surjectivity criterion is particularly useful in Hilbert spaces, where maximal monotone operators are densely defined and single-valued on the interior of their domain. These properties underpin existence results for variational inequalities by ensuring that the operator cannot be extended further while preserving monotonicity, thus capturing the "full" behavior of the mapping.

Existence and Uniqueness Results

The existence of solutions to variational inequalities in reflexive Banach spaces is guaranteed under conditions of monotonicity, hemicontinuity, and coercivity for the operator involved. Specifically, the Browder-Minty theorem establishes that if A: V \to V^* is a monotone and hemicontinuous operator on a reflexive Banach space V, and coercive (i.e., \lim_{\|u\| \to \infty, u \in K} \frac{\langle A(u), u \rangle}{\|u\|} = +\infty), with K \subset V a nonempty, closed, and convex set, then the variational inequality \langle A(u), v - u \rangle \geq 0 for all v \in K admits at least one solution u \in K.^[9]^[13] This result extends Minty's earlier work in Hilbert spaces to the more general Banach setting, relying on the surjectivity of the operator I + A for maximal monotone extensions.^[9] Uniqueness of solutions follows from stricter monotonicity assumptions. If A is strictly monotone, meaning \langle A(u) - A(v), u - v \rangle > 0 for all u \neq v in K, then the variational inequality has at most one solution, as any two solutions would contradict the strict inequality. Furthermore, if A is strongly monotone with constant \mu > 0, i.e., \langle A(u) - A(v), u - v \rangle \geq \mu \|u - v\|^2, and Lipschitz continuous with constant L, then under the condition \mu > L/2, the proximal mapping associated with A acts as a contraction, ensuring a unique solution via fixed-point arguments.^[13] A key assumption for boundedness of solution sets is coercivity of the operator A, defined by \lim_{\|u\| \to \infty, u \in K} \frac{\langle A(u), u \rangle}{\|u\|} = +\infty. This condition ensures that the level sets \{u \in K : \langle A(u), u \rangle \leq C\} are bounded for any C > 0, preventing solutions from escaping to infinity and facilitating compactness arguments in existence proofs. For pseudomonotone operators, which generalize monotonicity: whenever u_n \to u strongly in V and \liminf_{n \to \infty} \langle A(u_n), u_n - v \rangle \geq 0 for all v \in V, then \langle A(u), u - v \rangle \geq 0 for all v \in V, the Lions-Stampacchia theorem provides existence in reflexive Banach spaces when A is continuous, pseudomonotone, and coercive, with K closed and convex. In the context of partial differential equations, Gårding's inequality ensures coercivity for bilinear forms associated with elliptic operators, stating that for a strongly elliptic operator of even order, there exists c > 0 such that \operatorname{Re} a(u, u) \geq c \|u\|_{H^s}^2 - C \|u\|_{H^{s-1}}^2 for u in a Sobolev space, which underpins existence for variational inequalities modeling obstacle problems or unilateral constraints. Proofs of these existence results often employ the Galerkin method, approximating the variational inequality on finite-dimensional subspaces of V (e.g., spanned by basis functions) to obtain a sequence of solutions that converges weakly to a solution of the original problem via monotonicity and compactness in reflexive spaces, as detailed in the Lions-Stampacchia framework. Alternatively, the proximal point method iteratively solves regularized inclusions u^{k+1} \in (I + \lambda_k A)^{-1}(u^k) for \lambda_k > 0, leveraging the maximal monotonicity of A to ensure convergence to a solution, with weak convergence in Hilbert spaces under the hemicontinuity assumption.

Examples

Relation to Optimization Problems

Variational inequalities provide a unifying framework for classical optimization problems, particularly in the finite-dimensional setting. Specifically, for a continuously differentiable convex function f: \mathbb{R}^n \to \mathbb{R} and a nonempty closed convex set K \subseteq \mathbb{R}^n, a point x^* \in K minimizes f over K if and only if it solves the variational inequality \langle \nabla f(x^*), y - x^* \rangle \geq 0 for all y \in K.^[11] This equivalence establishes the variational inequality as the first-order necessary and sufficient optimality condition for convex minimization subject to convex constraints, extending the unconstrained case where \nabla f(x^*) = 0.^[2] A canonical example is the projection problem onto a convex set. Consider minimizing \frac{1}{2} \|x - b\|^2 over K, where b \in \mathbb{R}^n. The unique minimizer x^* = \mathrm{proj}_K(b) satisfies the variational inequality \langle x^* - b, y - x^* \rangle \geq 0 for all y \in K, which corresponds to the operator F(x) = x - b. This formulation is central in proximal algorithms and constrained optimization, highlighting how variational inequalities capture Euclidean distance minimization under constraints.^[11] In game theory, variational inequalities model Nash equilibria in noncooperative games with convex strategy sets. For an n-player game where player i chooses x_i \in K_i (convex compact) to maximize concave payoff f_i(x_1, \dots, x_n), a Nash equilibrium x^* = (x_1^*, \dots, x_n^*) \in \prod_{i=1}^n K_i satisfies \langle \nabla_{x_i} f_i(x^*), y_i - x_i^* \rangle \geq 0 for all y_i \in K_i and each i. This is precisely the variational inequality over the product set with operator F(x) whose i-th block is -\nabla_{x_i} f_i(x).^[11] Furthermore, convex optimization problems with inequality constraints can be reformulated as variational inequalities via the Karush-Kuhn-Tucker (KKT) conditions. For \min f(x) subject to g_j(x) \leq 0 (j=1,\dots,m), assuming convexity and differentiability, the KKT system—\nabla f(x^*) + \sum \lambda_j^* \nabla g_j(x^*) = 0, \lambda^* \geq 0, \lambda_j^* g_j(x^*) = 0, g(x^*) \leq 0—reduces to a variational inequality over the feasible set K = \{x : g(x) \leq 0\} with F(x) = \nabla f(x), or equivalently, a mixed variational inequality incorporating the multipliers. This reformulation enables unified treatment of constrained convex programs.^[2]

Complementarity and Obstacle Problems

The linear complementarity problem (LCP) provides a fundamental discrete example of a variational inequality arising in optimization and game theory. Given a vector q \in \mathbb{R}^n and a matrix M \in \mathbb{R}^{n \times n}, the LCP seeks vectors w, z \in \mathbb{R}^n satisfying w \geq 0, z \geq 0, w^T z = 0, and z = q + M w. This system enforces complementary slackness, where each pair (w_i, z_i) has at most one positive component. The LCP can be equivalently expressed as a variational inequality over the nonnegative orthant: find x \in \mathbb{R}_+^n such that

(q + M x)^T (y - x) \geq 0 \quad \forall y \in \mathbb{R}_+^n.

This formulation highlights the LCP as a special case of the finite-dimensional variational inequality with the operator F(x) = q + M x and convex set K = \mathbb{R}_+^n. The LCP was originally introduced by Lemke in the context of bimatrix equilibrium points and linear programming. Its connection to variational inequalities was further elucidated in subsequent surveys.^[14] In continuum mechanics, the Signorini problem exemplifies variational inequalities modeling physical constraints due to unilateral contact between an elastic body and a rigid obstacle. Consider a linearly elastic body in equilibrium under given forces, with a portion of its boundary subject to nonpenetration against a rigid support defined by \psi. The key conditions on this contact boundary \Gamma are the nonpenetration u \geq \psi, the nonadhesion stress \sigma_n(u) \geq 0 (normal component of the stress tensor), and the complementarity (u - \psi) \sigma_n(u) = 0, ensuring no interpenetration or tensile forces across the contact set. These conditions lead to a variational inequality formulation in the appropriate Sobolev space, where the solution u satisfies the weak equilibrium equations interior to the domain and the boundary inequality on \Gamma. The problem was posed by Signorini in 1959 to describe ambiguous boundary conditions in elasticity, and its existence and uniqueness were established using variational methods by Fichera in 1964.^[15]^[16] The obstacle problem offers another canonical illustration of variational inequalities in the context of constrained minimization for elliptic partial differential equations. It models the deflection of an elastic membrane fixed at the boundary of a domain \Omega \subset \mathbb{R}^d and resting above a given obstacle function \psi, minimizing the Dirichlet energy subject to u \geq \psi in \Omega. Equivalently, the solution u solves the variational inequality: find u \in K = \{ v \in H^1_0(\Omega) : v \geq \psi \ \text{a.e. in } \Omega \} such that

a(u, v - u) \geq 0 \quad \forall v \in K,

where a(u,v) = \int_\Omega \nabla u \cdot \nabla v \, dx is the standard bilinear form associated with the Laplacian. A key property is that the solution u coincides with the least superharmonic majorant of \psi, meaning u is the smallest superharmonic function dominating \psi. This interpretation links the problem to classical potential theory while capturing the free boundary where u > \psi. The variational formulation of the obstacle problem was developed by Stampacchia in the mid-1960s as part of early work on elliptic variational inequalities.^[17]

Applications

In Mechanics and PDEs

Variational inequalities play a central role in modeling elastoplasticity, particularly in capturing the stress-strain relations under yield conditions. In the Prandtl-Reuss model for perfect plasticity, the evolution of plastic strain is governed by a monotone variational inequality that enforces the flow rule and the yield criterion, such as the von Mises condition. This formulation ensures that the stress tensor remains within the elastic domain while allowing irreversible deformation when the yield surface is reached, leading to a quasistatic rate-independent system in three dimensions.^[18] Unilateral contact problems in mechanics, such as the interaction between an elastic body and a rigid obstacle, are naturally expressed as variational inequalities in Sobolev spaces like H^1(\Omega). For frictionless contact, the displacement field u satisfies a variational inequality that incorporates the non-penetration condition \langle \sigma n, v - u \rangle \geq 0 on the potential contact boundary, where \sigma is the stress tensor and n the outward normal, ensuring unilateral constraints without interpenetration. This setup arises in boundary value problems for linear elasticity and has been foundational since the development of mixed formulations for such inequalities. Evolutionary variational inequalities extend these ideas to time-dependent phenomena, including phase transitions modeled by the Stefan problem. In the classical two-phase Stefan problem, the temperature evolution across a moving interface is reformulated as a parabolic variational inequality, where the phase change is captured by an obstacle-like condition on the enthalpy or temperature field, enforcing the latent heat release. This approach handles the free boundary dynamics through monotonicity and provides existence results for weak solutions in multidimensional settings.^[19] A specific application arises in the mixed formulation for Stokes flow past obstacles, where the velocity u and pressure p satisfy the variational inequality

a(u, v) + b(v, p) - b(u, q) \geq (f, v) \quad \forall (v, q) \in V \times Q,

with a(\cdot, \cdot) the viscous bilinear form, b(\cdot, \cdot) the divergence term, f the forcing, and the spaces V, Q incorporating no-slip and incompressibility, alongside obstacle constraints on u. This inequality models low-Reynolds-number flows with unilateral barriers, such as in glaciology or microfluidics.^[20] Regularity theory for solutions of variational inequalities in mechanics provides optimal Hölder estimates for the free boundary in obstacle problems. In two dimensions, Caffarelli established that the solution exhibits C^{1,\alpha} regularity near regular points of the coincidence set, with explicit exponents depending on the dimension, which is crucial for analyzing contact interfaces and phase boundaries in PDE models.^[21]

In Economics and Game Theory

Variational inequalities provide a powerful framework for modeling equilibrium states in economic systems and strategic interactions in game theory, where agents optimize their objectives subject to constraints, leading to conditions that capture market clearing, supply-demand balances, and best-response behaviors. In economics, they extend classical equilibrium concepts to scenarios involving transportation costs, congestion, and incomplete information, allowing for the analysis of decentralized decision-making without assuming full cooperation. In game theory, variational inequalities unify the formulation of Nash equilibria across continuous and discrete strategy spaces, facilitating the study of noncooperative outcomes where no player benefits from unilateral deviation. Recent applications include modeling equilibria in machine learning, such as in generative adversarial networks (GANs) and multi-agent reinforcement learning, where variational inequalities capture adversarial training dynamics and policy optimization under constraints as of 2024.^[22] Spatial price equilibrium problems, which determine commodity prices and flows across regions accounting for supply, demand, and transportation costs, can be formulated as a variational inequality. In this setting, the vector x includes supply quantities s, demand quantities d, and link flows f, with the feasible set K being the Cartesian product of nonnegative supply and demand sets and the feasible flow set. The operator F(x) is defined such that its components reflect effective prices minus marginal costs for supplies, marginal costs minus delivered prices for demands, and transportation costs minus price differentials for flows, often expressed as F(x) = c(f) + \nabla p(s) - \pi(d) where c denotes transportation costs, p supply prices, and \pi delivered prices, though variations like F(x) = c(x) - \nabla p appear in generalized forms. A solution x^* \in K satisfies \langle F(x^*), x - x^* \rangle \geq 0 for all x \in K, ensuring no arbitrage opportunities exist. This formulation, introduced in seminal work on sensitivity analysis, allows for stability studies under parameter perturbations like cost changes.^[23] Traffic network equilibrium, based on Wardrop's first principle that all used paths between an origin-destination pair have equal and minimal travel times, is equivalently expressed as a variational inequality over feasible flow patterns. Here, the operator F assigns to each edge e the travel cost c_e(f_e), which may depend on the flow f_e on that edge due to congestion, and the convex set K comprises all nonnegative flows satisfying conservation of flow and demand constraints. An equilibrium flow f^* \in K solves \sum_e c_e(f^*_e) (f_e - f^*_e) \geq 0 for all f \in K, implying that no user can reduce their cost by switching routes. This VI structure, established in foundational traffic models, supports existence proofs under monotonicity assumptions on costs and enables extensions to asymmetric or multi-class user behaviors.^[24] In noncooperative games, Nash equilibria correspond to solutions of variational inequalities where players select strategies from convex compact sets to maximize individual payoffs. For a game with n players, strategy profiles x = (x_1, \dots, x_n) \in K = K_1 \times \dots \times K_n, and payoff functions u_i(x_i, x_{-i}), the pseudogradient operator has components F_i(x) = \nabla_{x_i} (-u_i(x)), so the VI seeks x^* \in K such that \sum_i \langle F_i(x^*), x_i - x^*_i \rangle \geq 0 for all x \in K, equivalent to each player optimizing given others' strategies. This unification applies to both potential and general-sum games, with monotonicity of F ensuring uniqueness under strict conditions. Extensions of the Arrow-Debreu general equilibrium model to variational inequalities in the 1980s accommodated incomplete markets by incorporating asset trading constraints and uncertainty, where equilibrium prices clear both spot and futures markets without full spanning of states. These developments reformulated the excess demand function into a VI operator over price and allocation spaces, allowing existence results via fixed-point mappings even when securities do not complete the market structure. For finite bimatrix games, the variational inequality reduces to finding mixed strategies \sigma^* in the simplex such that

\sum_i (A \sigma^* - b)_i (\tau_i - \sigma^*_i) \geq 0 \quad \forall \tau \geq 0, \sum_i \tau_i = 1,

where A is the payoff matrix for one player, b is the vector of ones, and the condition ensures optimality against deviations; a symmetric VI holds for the opponent's strategy. This formulation highlights how VI captures mixed-strategy Nash equilibria in zero-sum or general bimatrix settings.

Numerical Methods

Projection and Iterative Algorithms

Projection-based methods play a central role in iterative algorithms for solving variational inequalities (VIs) in finite-dimensional Hilbert spaces, where the feasible set K is a nonempty, closed, and convex subset. The orthogonal projection onto K, denoted \proj_K(z), is defined as the unique point in K that minimizes the Euclidean distance to a given point z:

\proj_K(z) = \arg\min_{y \in K} \|y - z\|^2.

This operator is firmly nonexpansive and thus nonexpansive, making it a foundational building block for ensuring convergence in projection-based iterations. Assuming the VI operator F is monotone (i.e., \langle F(x) - F(y), x - y \rangle \geq 0 for all x, y \in K), many algorithms leverage projections to handle the constraint K while approximating solutions to \mathrm{VI}(F, K). The extragradient method, introduced by Korpelevich in 1976, is a seminal projection-based algorithm for solving VIs where F is monotone and Lipschitz continuous with constant L > 0. The iteration proceeds in two steps: first, compute an auxiliary point y^k = \proj_K(x^k - \alpha F(x^k)), where \alpha \in (0, 1/L); then, update x^{k+1} = \proj_K(x^k - \alpha F(y^k)). Under these assumptions on F, the method converges to a solution of the VI starting from any initial x^0 \in K. Recent improvements have established an O(1/k) last-iterate convergence rate for the extragradient method in the monotone case. This rate measures the gap function or duality gap, providing non-ergodic guarantees that enhance practical reliability.^[25] For the case of strong monotonicity (i.e., \langle F(x) - F(y), x - y \rangle \geq \mu \|x - y\|^2 for some \mu > 0), the method achieves linear convergence rates. The proximal point algorithm, developed by Rockafellar in 1976, extends the projection framework to handle maximal monotone operators F, which are monotone and have full domain with no proper enlargements. At each iteration, x^{k+1} solves the regularized VI: \mathrm{VI}(F + (1/\lambda) \mathrm{Id}, K), where \lambda > 0 is a stepsize and \mathrm{Id} is the identity operator; equivalently, x^{k+1} = \proj_K(x^k - \lambda F(x^{k+1})). For maximal monotone F, the algorithm converges weakly to a solution of the original VI from any starting point, with the regularization ensuring solvability of the subproblems. This method is particularly robust for ill-conditioned problems, as the added proximal term promotes stability. Douglas-Rachford splitting addresses VIs reformulated as finding zeros of the sum of two maximal monotone operators, such as A + B where A = F (monotone) and B = N_K (normal cone to K). The iteration involves reflections: compute \mathrm{refl}_B = 2 \prox_{\lambda B} - \mathrm{Id} and \mathrm{refl}_A = 2 \prox_{\lambda A} - \mathrm{Id}, then update via z^{k+1} = \frac{1}{2} ( \mathrm{Id} + \mathrm{refl}_A \circ \mathrm{refl}_B ) (z^k), with the projected iterate x^{k+1} = \prox_{\lambda B}(z^{k+1}). Lions and Mercier proved weak convergence to a zero of A + B in 1979 for maximal monotone operators in Hilbert spaces. This splitting technique is versatile for structured VIs, enabling efficient computation when individual proximals are available.

Finite Element and Discretization Techniques

Discretization techniques for variational inequalities (VIs) often employ the Galerkin method to approximate solutions in finite-dimensional subspaces of the underlying function space V. In this approach, a VI defined over a convex set K \subset V is projected onto a discrete convex set K_h \subset V_h, where V_h is a finite element space, such as piecewise linear polynomials on a triangulation of the domain. For instance, in the obstacle problem, V_h consists of continuous functions that are linear on each element and satisfy the discrete obstacle constraint, leading to a finite-dimensional VI: find u_h \in K_h such that a(u_h, v_h - u_h) \geq \langle f, v_h - u_h \rangle for all v_h \in K_h, where a(\cdot, \cdot) is the bilinear form and f is the data functional. This formulation preserves the monotonicity and coercivity properties of the continuous problem when a is symmetric and positive definite, enabling efficient computation via standard finite element assembly. Error estimates for these Galerkin approximations rely on the quasi-best approximation property for monotone VIs. Under suitable regularity assumptions on the solution, the error \|u - u_h\|_V satisfies \|u - u_h\|_V \leq C \inf_{v_h \in K_h} \|u - v_h\|_V, where C is a constant independent of the mesh size h. For piecewise polynomial elements of degree r, this yields convergence rates of O(h^r) in the energy norm, with optimal rates achieved for linear elements (r=1) in problems like the obstacle or Signorini type, provided the exact solution is sufficiently smooth. These estimates have been rigorously established for both conforming and mixed formulations, highlighting the robustness of finite elements in handling the inequality constraints without loss of accuracy compared to elliptic PDEs. Solving the resulting nonlinear discrete VI requires specialized nonlinear solvers, such as the Newton method adapted to the variational structure. The method linearizes the discrete VI around an iterate by solving a linear saddle-point problem on the active set of constraints, with globalization achieved via line search or trust-region techniques to ensure descent in a merit function like the energy functional. This approach converges quadratically locally for strongly monotone operators and has been effectively combined with preconditioned conjugate gradient solvers for the linear steps, particularly in large-scale obstacle and contact simulations.^[26] For contact problems modeled as VIs, active set strategies provide an efficient alternative by iteratively identifying and updating the active (contact) and inactive sets based on Lagrange multipliers or gap functions. Starting from an initial guess, the algorithm solves a linear elasticity problem on the predicted active set, updates the sets according to Signorini conditions, and iterates until convergence, often requiring only a few outer iterations for moderate friction. These methods are particularly suited to finite element discretizations in three dimensions, offering superlinear convergence and integration with mortar techniques for non-matching meshes in multi-body contact.^[27] Nonconforming finite elements are valuable for mixed formulations of VIs, such as those arising in Stokes-flow with obstacles or plate contact, where standard conforming spaces may fail stability. These elements relax inter-element continuity, approximating the solution in a larger space while enforcing weak continuity via additional terms in the bilinear form. Stability is ensured by extending Brezzi's inf-sup condition to the VI setting, requiring that the discrete spaces satisfy \inf_{q_h \neq 0} \sup_{v_h \neq 0} b(v_h, q_h) / (\|v_h\| \|q_h\|) \geq \beta > 0 on the kernel of the constraint operator, which guarantees unique solvability and optimal error estimates akin to conforming methods. This extension, originally for saddle-point problems, has been adapted to VIs to handle incompressibility and contact without spurious modes.^[28]

References

[1]
[PDF] VARIATIONAL INEQUALITIES - HAL
The variational inequalities generalize the theory of equations ; on the other hand a variational inequality can be reduced to an equation with the following.Missing: sources | Show results with:sources
[2]
[PDF] Variational Inequalities
Variational inequality theory was introduced by Hart- man and Stampacchia (1966) as a tool for the study of partial differential equations with applications ...Missing: primary sources
[3]
On some non-linear elliptic differential-functional equations
1966 On some non-linear elliptic differential-functional equations. Philip Hartman, Guido Stampacchia ... DOI: 10.1007/BF02392210. ABOUT; FIRST PAGE; CITED BY ...
[4]
An Introduction to Variational Inequalities and Their Applications
Variational inequalities (equilibrium or evolution problems typically with convex constraints) are carefully explained in An Introduction to Variational ...Missing: primary sources
[5]
Leonhard Euler (1707 - 1783) - Biography - University of St Andrews
It applies variational principles to determine the optimal ship design and first established the principles of hydrostatics ... Euler here also begins ...
[6]
[PDF] The Calculus of Variations - College of Science and Engineering
Jan 7, 2022 · Some of these minimization problems played a key role in the historical development ... leading to a second order parametric variational problem, ...
[7]
Über das Dirichletsche Prinzip - EUDML
Hilbert, David. "Über das Dirichletsche Prinzip." Mathematische Annalen 59 (1904): 161-186. ... Dirichlet principle, Hilbert space, harmonic functions ...<|control11|><|separator|>
[8]
[PDF] An appreciation of R. Courant's `variational methods for the solution ...
Physically, rigid conditions correspond to rigid constraints of the system at the boundary C while natural conditions express equilib- rium of the system of C ...<|control11|><|separator|>
[9]
Monotone (nonlinear) operators in Hilbert space - Project Euclid
Monotone (nonlinear) operators in Hilbert space. George J. Minty. DOWNLOAD PDF + SAVE TO MY LIBRARY. Duke Math. J. 29(3): 341-346 (September 1962).
[10]
Lions, J.-L. (1969) Quelques méthodes de résolution des problèmes ...
Lions, J.-L. (1969) Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris.
[11]
Finite-dimensional variational inequality and nonlinear ...
Feb 17, 1989 · This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.
[12]
Variational inequalities - Wiley Online Library
Variational inequalities. J. L. Lions,. J. L. Lions. University of Paris. Search for more papers by this author · G. Stampacchia, ... Download PDF. back ...
[13]
Nonlinear monotone operators and convex sets in Banach spaces
September 1965 Nonlinear monotone operators and convex sets in Banach spaces. Felix E. Browder · DOWNLOAD PDF + SAVE TO MY LIBRARY. Bull. Amer. Math. Soc.Missing: original | Show results with:original
[14]
[PDF] A Theoretical Perspective of Convex Optimization and Variational ...
the relationship between convex optimization and variational inequality problems. ... The following result shows that the first-order condition for unconstrained ...
[15]
[PDF] Finite-Dimensional-Variational-Inequality-and-Nonlinear ...
P.T. Harker, J.S. Pang Variational inequality and complementarity problems. An important special case of VI(X, F') is the nonlinear complementarity problem.
[16]
Variational Inequalities and the Signorini Problem for Nonlinear ...
It is conceivable that the contact set could be especially cornplicated. Fichera (1964) was the first to study the existence and uniqueness of this problern ...
[17]
A Generalized Strange Term in Signorini\'s Type Problems
The general question which will make the object of this paper is the homogenization of Signorini's type-like problems in perforated domains. Their classical ...
[18]
The Obstacle Problem and Best Superharmonic Approximation
Least superharmonic majorants are fundamental in Potential Theory, where they are called réduites, and they occur in the the theory of elliptic PDEs as ...
[19]
A generalized Norton-Hoff model and the Prandtl-Reuss law of ...
The aim of this article is to study the quasistatic evolution of a three-dimensional elastic-perfectly plastic solid which satisfies the Prandtl-Reuss law.
[20]
Variational methods in the stefan problem | SpringerLink
Sep 12, 2006 · —The solution of a two-phase Stefan problem by a variational inequality, in: J.R. Ockendon, A.R. Hodgkins Eds., “Moving Boundary Problems in ...
[21]
[PDF] arXiv:2108.00046v3 [math.NA] 11 Oct 2022
Oct 11, 2022 · We propose a mixed formulation of the Stokes variational inequality where a La- grange multiplier is used to enforce the contact conditions ...
[22]
Sensitivity Analysis for the General Spatial Economic Equilibrium ...
Stella Dafermos, Anna Nagurney, (1984) Sensitivity Analysis for the General Spatial Economic Equilibrium Problem. Operations Research 32(5):1069-1086. https ...
[23]
Traffic Equilibrium and Variational Inequalities - PubsOnLine
Exchange rates and multicommodity international trade: insights from spatial price equilibrium modeling with policy instruments via variational inequalities.
[24]
A penalty/Newton/conjugate gradient method for the solution of ...
The numerical methodology combines penalty and Newton's method, the linearized problems being solved by a conjugate gradient algorithm.
[25]
A Primal-Dual Active Set Algorithm for Three-Dimensional Contact ...
In this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed.
[26]
Error estimates for the finite element solution of variational inequalities
We study the mixed finite element approximation of variational inequalities, taking as model problems the so called “obstacle problem” and “unilateral problem”.