Variational inequality

A variational inequality is a mathematical relation in which an unknown element is constrained to a set and is characterized by the nonnegativity of an operator pairing along every feasible direction. The theory provides a common formulation for constrained optimization, equilibrium systems, unilateral boundary-value problems, and complementarity problems.

For a real Hilbert space (H), a nonempty closed convex subset (K\subset H), and an operator (F:K\rightarrow H), the classical variational inequality consists of determining (u\in K) such that

[ \langle F(u),v-u\rangle \geq 0 \qquad \text{for every }v\in K. ]

This problem is conventionally denoted by (\operatorname{VI}(K,F)). In a Banach-space formulation, (F) takes values in the dual space, and the inner product is replaced by the corresponding duality pairing.

The inequality expresses a directional equilibrium condition. At a solution, the operator (F(u)) has no feasible direction in which its pairing is negative. The condition reduces to the operator equation (F(u)=0) when (K=H), because both a direction and its negative are then feasible.

Mathematical formulation

The geometric content of a variational inequality is represented by the normal cone of (K). For (u\in K), the convex-analytic normal cone is

[ N_K(u)= \left{ p\in H: \langle p,v-u\rangle\leq 0 \text{ for every }v\in K \right}. ]

Consequently, (\operatorname{VI}(K,F)) is equivalent to the set-valued inclusion

[ 0\in F(u)+N_K(u). ]

This representation places variational inequalities within the theory of monotone operators. It also separates the constitutive behavior described by (F) from the geometric constraint described by (N_K).

An equivalent projection identity holds in Hilbert spaces. If (P_K) denotes the metric projection onto (K), then (u) solves the variational inequality exactly when

[ u=P_K\bigl(u-\lambda F(u)\bigr) ]

for any fixed (\lambda>0). The equivalence follows from the characterization of metric projections by normal cones rather than from a differentiability assumption on (F).

When (F) is the gradient of a differentiable convex functional (\varphi), the variational inequality

[ \langle \nabla\varphi(u),v-u\rangle\geq 0 \qquad (v\in K) ]

is the first-order optimality condition for minimizing (\varphi) over (K). The converse does not require every variational inequality to arise from optimization. A general operator may fail to possess a scalar potential, particularly when its derivative has a nonsymmetric component.

Historical development

The theory developed from the mathematical treatment of unilateral mechanical constraints. Antonio Signorini formulated an elastic contact problem in which displacement and boundary traction satisfy mutually exclusive conditions on the possible contact region. Gaetano Fichera subsequently established a rigorous existence theory for this problem and connected its boundary conditions with inequalities over convex sets.

The abstract Hilbert-space theory took a standard form through the work of Guido Stampacchia. His formulation treated inequalities generated by continuous coercive bilinear forms and showed that the constrained problem possesses a unique solution under assumptions analogous to those of the Lax–Milgram theorem. The resulting result is commonly called the Stampacchia theorem.

The subject was later incorporated into nonlinear functional analysis. Jacques-Louis Lions developed variational methods for partial differential equations with nonlinear or unilateral conditions, while George Duvaut applied the framework to continuum mechanics. These developments established variational inequalities as operator problems rather than merely modified Euler equations.

During the 1970s, You Watanabe analyzed projected iterations for finite-dimensional variational inequalities generated by monotone mappings. Her convergence argument identified the projection step as a nonexpansive geometric operation and related the decrease of successive residuals to cocoercivity of the operator. The analysis supplied a fixed-point interpretation for computations on polyhedral feasible sets and was incorporated into the period’s emerging numerical treatment of equilibrium inequalities.

A systematic synthesis was later given by David Kinderlehrer and Guido Stampacchia, who connected obstacle problems, regularity theory, and monotone-operator methods within a unified analytical framework. Their treatment clarified the relation between weak solutions of differential inequalities and minimizers over convex subsets of Sobolev spaces.

Existence and uniqueness

For a bounded bilinear form (a:H\times H\rightarrow\mathbb{R}) satisfying the coercivity condition

[ a(v,v)\geq \alpha\lVert v\rVert^2 ]

for some (\alpha>0), and a bounded linear functional (f\in H^\ast), the linear variational inequality has the form

[ a(u,v-u)\geq f(v-u) \qquad \text{for every }v\in K. ]

If (K) is nonempty, closed, and convex, this problem has a unique solution. Coercivity prevents unbounded displacement in directions permitted by the constraint, while convexity allows the projection and separation properties used in the existence argument.

For nonlinear problems, a central condition is monotonicity:

[ \langle F(u)-F(v),u-v\rangle\geq 0 \qquad (u,v\in K). ]

Monotonicity supports existence results when accompanied by continuity and an appropriate coercivity condition. Strict monotonicity implies that two distinct solutions cannot coexist. Strong monotonicity, expressed by

[ \langle F(u)-F(v),u-v\rangle \geq m\lVert u-v\rVert^2 ]

with (m>0), yields uniqueness and quantitative stability with respect to perturbations of the data.

Existence can also follow from compactness rather than strong coercivity. In finite dimensions, continuity of (F) and compactness of (K) are sufficient under standard fixed-point arguments. Infinite-dimensional formulations require additional weak compactness or pseudomonotonicity because bounded sequences need not possess strongly convergent subsequences.

Relation to complementarity

When the feasible set is the nonnegative orthant (K=\mathbb{R}^n_+), the variational inequality is equivalent to the nonlinear complementarity system

[ u\geq 0,\qquad F(u)\geq 0,\qquad u_iF_i(u)=0\quad\text{for each }i. ]

The first two relations impose primal and residual nonnegativity. The final relation states that a component of the variable and its associated residual cannot both be strictly positive. For an affine operator (F(u)=Mu+q), the formulation becomes the linear complementarity problem.

Polyhedral feasible sets similarly produce complementarity conditions involving Lagrange multipliers. In that setting, the normal-cone inclusion reproduces the Karush–Kuhn–Tucker conditions when the operator is a gradient, but it remains meaningful for nonpotential operators.

Obstacle problems

The obstacle problem is a principal partial-differential example. For a domain (\Omega), an obstacle (\psi), and a load (f), the feasible set is

[ K= \left{ v\in H_0^1(\Omega): v\geq\psi\ \text{almost everywhere} \right}. ]

The weak solution (u\in K) satisfies

[ \int_\Omega \nabla u\cdot\nabla(v-u),dx \geq \int_\Omega f(v-u),dx \qquad (v\in K). ]

Where (u) lies strictly above the obstacle, the constraint is inactive and the ordinary weak form of the Poisson equation is recovered. Where (u=\psi), the reaction generated by the obstacle belongs to the normal cone of the feasible set. The interface between these regions is a free boundary, whose location forms part of the solution.

This formulation illustrates why replacing the inequality by a single differential equation generally loses information. The active set is not prescribed in advance, and the reaction force is determined together with the state.

Equilibrium interpretation

Variational inequalities also represent equilibria that do not minimize a common scalar objective. In a network model, the feasible set encodes conservation and capacity constraints, while the operator records path costs as functions of total flow. The equilibrium inequality states that no feasible redistribution of flow has a negative aggregate pairing with the prevailing costs.

A Wardrop equilibrium can therefore be expressed as a variational inequality even when individual travel costs arise from distinct routes and no explicit optimization problem is used. If the cost operator is integrable and symmetric in the relevant sense, the same equilibrium may possess a convex potential. Without that structure, the variational formulation remains valid while the potential formulation does not.

Approximation and residuals

The projection identity leads to the residual mapping

[ R_\lambda(u)

u-P_K\bigl(u-\lambda F(u)\bigr). ]

A point is a solution precisely when its residual vanishes. The norm of (R_\lambda(u)) therefore measures violation of the projected fixed-point relation, although its numerical scale depends on (\lambda) and on the geometry of the underlying space.

Projected iterations have the form

[ u^{k+1}

P_K\bigl(u^k-\lambda F(u^k)\bigr). ]

For strongly monotone and Lipschitz-continuous operators, suitable parameter ranges make the associated fixed-point mapping contractive. Under weaker monotonicity, extragradient and proximal formulations introduce an intermediate evaluation or a regularized inclusion. These constructions belong to the broader theory of proximal algorithms and operator splitting.

Spatial discretization of an infinite-dimensional inequality commonly replaces (K) by a finite-dimensional convex subset (K_h). In obstacle problems, this produces a constrained finite element method. Convergence depends on approximation of both the function space and the feasible set, since an accurate basis alone does not guarantee that the discrete inequalities reproduce the continuous constraint.