For a feasible point \( x \in X(p)\) , the critical cone associated to the feasible set \( X(p)\) at \( x\) is defined as

Let \( (x^\star, y^\star, z^\star)\) be a KKT stationary solution. The critical set at the solution is the polyhedral cone defined as

Elements of \( \mathcal{K}_p(x^\star, z^\star)\) are called critical directions. The directional critical set in the direction \( h \in \mathbb{R}^{\ell}\) is defined as

LICQ holds if the gradients of active constraints

are linearly independent.

Suppose \( (x^\star, y^\star, z^\star)\) is a primal-dual solution of (4) satisfying LICQ. Then,

where \( \mathcal{C}_p(x)\) is the critical cone of the feasible set \( X(p)\) at \( x\) .

MFCQ holds if the gradients of the equality constraints

are linearly independent and if there exists a vector \( d \in \mathbb{R}^{n_x}\) such that

The set \( \mathcal{M}_p(x)\) is bounded if and only if MFCQ holds.

Using the theorem of alternatives, MFCQ is equivalent to \( 0\) being the unique solution of the linear system

SMFCQ holds if

are linearly independent and if there exists a vector \( d \in \mathbb{R}^{n_x}\) such that

Let \( w = (x, y, z)\) be a primal-dual solution. Then SMFCQ holds at \( x\) if and only if the multiplier vector \( (y, z)\) is unique.

CRCQ holds if there exists a neighborhood \( W\) of \( x^\star\) such that for any subsets \( I\) of \( \mathcal{I}_p(x^\star)\) and \( J\) of \( [m_e]\) the family of gradient vectors

has the same rank for all vectors \( x \in W\) .

SOSC holds at \( x\) if there exists \( (y, z) \in \mathcal{M}_p(x)\) such that

Suppose SOSC holds at a solution \( x^\star\) . Then, there exists \( \sigma > 0\) and a neighborhood \( V\) of \( x^\star\) such that for all \( x \in X(p) \cap V\) (with \( x \neq x^\star\) ),

SSOSC holds at \( x\) if there exists \( (y, z) \in \mathcal{M}_p(x)\) such that

If the condition (29) hold for all multipliers in \( \mathcal{M}_p(x)\) , then we say that the generalized second-order sufficiency condition (GSSOSC) hold.

Let \( F: \mathbb{R}^{n} \times \mathbb{R}^{\ell} \to \mathbb{R}^{n}\) be a \( C^1\) function and \( (x^\star, p^\star) \in \mathbb{R}^{n} \times \mathbb{R}^{\ell}\) such that \( F(x^\star, p^\star) = 0\) . If the Jacobian \( J_x F(x^\star, p^\star)\) is invertible, then there exists a neighborhood \( U \subset \mathbb{R}^{\ell}\) of \( p^\star\) and a differentiable function \( x(p)\) such that \( F(x(p), p)= 0\) for all \( p \in U\) , and \( x(p)\) is the unique solution in a neighborhood of \( x^\star\) . In addition, \( x(\cdot)\) is \( C^1\) differentiable on \( U\) , with

Suppose that at a KKT stationary solution \( x^\star\) of NLP\( (p^\star)\) , SOSC, LICQ and SCS hold. Then,

1. the primal-dual solution \( w^\star = (x^\star, y^\star, z^\star_\mathcal{I})\) is unique;

2. there exists a unique continuously differentiable function \( w(\cdot)\) defined in a neighborhood \( U\) of \( p^\star\) such that \( w(p^\star) =(x^\star, y^\star, z^\star_\mathcal{I})\) , with

   \[ \begin{equation} J_p w(p) = - M^{-1} N \; . \end{equation} \]

   (34)

3. SCS and LICQ hold locally near \( p^\star\) .

If the problem (4) is a linear program (LP), we have \( \nabla_{xx}^2 \mathcal{L} = 0\) . The Goldman-Tucker theorem states there always exists a LP solution satisfying strict-complementarity. If the LP is not degenerate, LICQ holds and the solution is a vertex of the polyhedral feasibility set \( X(p)\) . This automatically implies SOCP (the critical cone reduces to \( \mathcal{C}_p(x^\star) = \{0 \}\) ) and there are exactly \( n\) binding constraints: \( m_e + | \mathcal{I}_p(x^\star) | = n\) . Hence Theorem 4 applies to the linear programming case. We denote the active Jacobian at the solution \( B = \begin{bmatrix} J_x g^\top & J_x h_\mathcal{I}^\top \end{bmatrix}^\top\) . The matrix \( B\) has dimension \( n \times n\) and is non-singular (LICQ implies it is full row-rank). We note that \( B\) is exactly the basis matrix used in the simplex algorithm. In that case, the inverse of the matrix \( M = \begin{bmatrix} 0 & B^\top \\ B & 0 \end{bmatrix}\) in (34) satisfies:

Hence, the computation of the sensitivities simplifies considerably if the problem is regular linear program. If the linear program is solved using the simplex algorithm, the solver usually returns a LU factorization for the basis matrix \( B\) .

Certain optimization problems have a graph structure: in that case the primal-dual variable can be reordered following a set of nodes \( \mathcal{V}\) such that \( w = \{w_i \}_{i \in \mathcal{V}}\) . As a consequence, the Jacobian matrix \( M\) appearing in (34) can also be reordered following the graph structure: in the language of linear algebra, there exists a permutation matrix \( P\) such that the matrix \( P^\top M P\) is banded. It is known that terms in the inverse of a banded matrix decay exponentially fast away from the diagonal [4]. Then, assuming SSOSC, SCS and LICQ the solution of (34) satisfies the exponential decay of sensitivity property [5], that is, for two parameters \( p, p'\) close enough,

with \( w_i(p)\) the primal-dual nodal solution at node \( i \in \mathcal{V}\) and \( C_{ij} = \Gamma \rho^{d_G(i, j)}\) (with \( \Gamma > 0\) and \( \rho > 0\) two positive constants) and \( d_G(i, j)\) the graph distance between node \( i\) and node \( j\) . The Equation (37) has important consequence for computing the sensitivities in a decentralized fashion [6] or building a preconditionner to solve (34) with an iterative method [7].

Let \( w = (x, y, z) \in \mathbb{R}^{n} \times \mathbb{R}^{m_e} \times \mathbb{R}^{m_i}\) be a primal-dual variable. A primal-dual solution \( w\) of the KKT system (9) is also solution of the generalized equation

with \( F(w, p) := \begin{bmatrix} \phantom{-}\nabla_x \mathcal{L}(x, y, z, p) \\ -\nabla_y \mathcal{L}(x, y, z, p) \\ -\nabla_z \mathcal{L}(x, y, z, p) \end{bmatrix}\) and \( C = \mathbb{R}^{n} \times \mathbb{R}^{m_e} \times \mathbb{R}^{m_i}_+\) .

Let \( w^\star = (x^\star, y^\star, z^\star)\) a primal-dual solution of the generalized equation (38). If SSOSC and LICQ hold at \( w^\star\) , then (38) is strongly regular at \( w^\star\) .

Suppose that at a KKT stationary solution \( x^\star\) of NLP\( (p^\star)\) , LICQ and SSOSC hold. Then

• there exists a local unique continuous primal-dual solution \( w(\cdot)\) which is directionally differentiable in every direction \( h\in \mathbb{R}^{\ell}\) ;

• for a given direction \( h \in \mathbb{R}^{\ell}\) , the directional derivative \( w'(p; h)\) is the primal-dual solution of the QP problem defined as

  \[ \begin{equation} \begin{aligned} \min_{d}\; & \frac{1}{2} d^\top (\nabla^2_{x x}\mathcal{L}) \, d + h^\top (\nabla^2_{x p}\mathcal{L}) \,d \\ \text{s.t.} ~ & \nabla_x g_i(x, p)^\top d + \nabla_p g_i(x, p)^\top h = 0 & (\forall i \in [m_e]) \,,\\ & h_j(x, p) + \nabla_x h_j(x, p)^\top d + \nabla_p h_j(x, p)^\top h \leq 0 & (\forall j \in [m_i])\,, \end{aligned} \end{equation} \]

  (39)

  with \( \nabla_{x x}^2 \mathcal{L} = \nabla_{x x}^2 \mathcal{L}(x, y, z, p)\) , \( \nabla_{x p}^2 \mathcal{L} = \nabla_{xp}^2 \mathcal{L}(x, y, z, p)\) .

Let \( (x^\star, y^\star, z^\star)\) be a solution of KKT. If \( \Phi\) is coherently oriented with respect to \( (x, y, z)\) at \( (x^\star, y^\star, z^\star)\) then there exists a neighborhood \( U\) of \( p\) and \( V\) of \( x^\star\) and a \( PC^1\) mapping \( w: U \to V\) such that for each \( p \in U\) \( w(p)\) is the unique solution of (40). Moreover, for any \( k \in \mathbb{N}\) and any \( R \in \mathbb{R}^{p \times k}\) the LD-derivatives \( x'(p; R)\) , \( y'(p, R)\) and \( z'(p, R)\) are the unique solution \( X\) , \( Y\) and \( Z = [Z^+ Z^0 Z^-]\) of the following nonsmooth equation system

where \( \bf{LMmin}\) is the lexicographic matrix minimum function defined in (108), and

Let \( (x^\star, y^\star, z^\star)\) be a KKT stationary point. Suppose that LICQ and SSOSC holds at \( (x^\star, y^\star, z^\star)\) . Then \( \Phi\) is coherently oriented with respect to \( (x, y, z)\) .

Let \( (x^\star, y^\star, z^\star)\) be a solution of KKT. Suppose that LICQ and SSOSC hold. Then, for any \( k \in \mathbb{N}\) and \( R \in \mathbb{R}^{\ell \times k}\) , the LD-derivatives \( x'(p, R)\) , \( y'(p, R)\) and \( z'(p, R)\) in the direction \( R\) are given as

with, for \( j = 1, …, k\) ,

Suppose that at a KKT stationary solution \( x^\star\) of NLP\( (p^\star)\) , MFCQ and GSSOSC hold. Then, there are open neighborhoods \( U\) of \( p^\star\) and \( V\) of \( x^\star\) and a function \( x: U \to V\) such that \( x(\cdot)\) is continuous. The function \( x(p)\) is the unique local solution of NLP(\( p\) ) in \( V\) , and MFCQ holds at \( x(p)\) .

Suppose that at a solution \( x^\star\) of NLP\( (p^\star)\) , SMFCQ and SSOSC hold. Then, \( x(\cdot)\) is directionally differentiable at \( p\) for every \( h \in \mathbb{R}^{\ell}\) , and the directional derivative \( x'(p^\star, h)\) is the unique solution of the QP problem

with \( \mathcal{K}_{p^\star}\) the critical cone defined in (13) and \( \nabla_{x x}^2 \mathcal{L} = \nabla_{x x}^2 \mathcal{L}(x^\star, y^\star, z^\star, p^\star)\) , \( \nabla_{x p}^2 \mathcal{L} = \nabla_{x p}^2 \mathcal{L}(x^\star, y^\star, z^\star, p^\star)\) .

Suppose that at a KKT stationary solution \( x^\star\) of NLP\( (p^\star)\) , MFCQ, CRCQ and GSSOSC hold. Then, \( x(\cdot)\) is directionally differentiable at \( p^\star\) for every \( h \in \mathbb{R}^{\ell}\) , and the directional derivative \( x'(p^\star, h)\) is the unique solution of the QP problem, defined for a given \( (y, z) \in \mathcal{E}_{p^\star}(x^\star)\) ,

with \( \mathcal{K}_{p^\star}\) the critical cone defined in (13) and \( \nabla_{x x}^2 \mathcal{L} = \nabla_{x x}^2 \mathcal{L}(x^\star, y, z, p^\star)\) , \( \nabla_{x p}^2 \mathcal{L} = \nabla_{x p}^2 \mathcal{L}(x^\star, y, z, p^\star)\) .

The critical cone \( \mathcal{K}_p(x^\star, z; h)\) is nonempty if and only if \( (y, z) \in S_p(x^\star; h)\) .

Suppose that at a KKT stationary solution \( x^\star\) of NLP\( (p^\star)\) , MFCQ, CRCQ and GSSOSC hold. Then,

• There is open neighborhoods \( U\) of \( p^\star\) and \( V\) of \( x^\star\) and a function \( x:U \to V\) such that \( x(p)\) is the unique local solution of NLP(\( p\) in \( V\) and MFCQ holds at \( x(p)\) ;
• \( x(\cdot)\) is a PC\( ^1\) function (hence locally Lipschitz and B-differentiable);
• For each \( p \in U\) , the directional derivative \( x'(p; \cdot)\) is a piecewise linear function, and for \( h \in \mathbb{R}^{\ell}\) and \( (y, z) \in S_p(x^\star; h)\) , \( x'(p; h)\) is the unique solution of (46).

Suppose \( X\) is nonempty and compact, and \( f(x, \cdot)\) is continuously differentiable at \( p \in \mathbb{R}^{\ell}\) . Then \( \varphi\) is locally Lipschitz near \( p\) , and directionally differentiable in every direction \( h \in \mathbb{R}^{\ell}\) , with

with \( \Sigma(p)\) the solution set defined in (7).

Suppose \( M\) is a closed convex set, \( f(\cdot, p)\) and \( h_i(\cdot, p)\) are convex on \( M\) for all \( p\) , and continuously differentiable on \( M \times N\) , with \( N\) a neighborhood of \( p\) . If (i) the solution set \( \Sigma(p)\) is nonempty and bounded, (ii) \( \varphi(p)\) is finite, (iii) there is a point \( \hat{x} \in M\) such that \( h(\hat{x}, p) < 0\) (Slater), then \( \varphi\) is directionally differentiable, and for all \( h\in \mathbb{R}^{\ell}\) ,

with \( \mathcal{M}_p(x)\) the multiplier solution set.

Suppose that \( X(p)\) is nonempty and uniformly compact near \( p\) and MFCQ holds at each \( x \in \Sigma(p)\) . Then \( \varphi\) is locally Lipschitz near \( p\) , and for any \( h \in \mathbb{R}^{\ell}\) ,

Let \( f\) and \( h\) be convex functions in \( x\) , \( g\) be affine in \( x\) , such that \( f,g, h\) are all jointly \( C^1\) in \( (x, p)\) . Then, if \( X(p)\) is nonempty and uniformly compact near \( p\) and MFCQ holds at each \( x \in \Sigma(p)\) , then \( \varphi\) is directionally differentiable at \( p\) and

Suppose GSSOSC and MFCQ hold at a stationary solution \( x^\star\) . Then near \( p\) the function \( \varphi(\cdot)\) has a finite one-sided directional derivative, and for every \( h \in \mathbb{R}^{\ell}\) ,

Suppose SOSC, SCS and LICQ hold at a stationary KKT solution \( (x^\star, y^\star, z^\star)\) . Then the local value function \( \varphi_\ell\) is \( C^2\) near \( p\) , and

• \( \varphi_\ell(p) = \mathcal{L}(x^\star, y^\star, z^\star, p)\) .
• \( \nabla_p \varphi_\ell(p) = \nabla_p \mathcal{L}(x^\star, y^\star, z^\star, p)\) .

• and also

  \[ \begin{equation} \begin{aligned} \nabla^2_{pp} \varphi_\ell(p) = \nabla_p \big(\nabla_p \mathcal{L}(w^\star, p)\big) &= \nabla_{pp}^2 \mathcal{L} + \nabla_{wp}^2 \mathcal{L} \; J_p w(p) \;, \\ &= \nabla_{pp}^2 \mathcal{L} - \nabla_{wp}^2 \mathcal{L} \, (\nabla_{ww}^2 \mathcal{L})^{-1} \nabla_{pw}^2 \mathcal{L} \;. \end{aligned} \end{equation} \]

  (60)

Suppose the conditions of Theorem 11 hold at a stationary solution \( (x^\star, y^\star, z^\star)\) of problem (61). Then, in a neighborhood of \( p\) the local value function \( \varphi_\ell\) is \( C^2\) , with

• \( \varphi_\ell(p) = f(x^\star, p)\) .
• \( \nabla_p \varphi_\ell(p) = \nabla_p f(x^\star)\) .
• \( \nabla^2_{pp} \varphi_\ell(p) = \nabla_{pp}^2 f + (\nabla_{xp}^2 f) \, J_p x(p)\) .

Suppose the conditions of Theorem 11 hold at a stationary solution \( (x^\star, y^\star, z^\star)\) of problem (62). Then, in a neighborhood of \( (p, q)\) the local value function \( \varphi_\ell\) is \( C^2\) , with

Let \( p^\star \in \mathbb{R}^{\ell}\) . Suppose that a stationary solution \( x^\star\) of NLP(\( p\) LICQ, SOSC and SCS hold. Then in a neighborhood of \( (0, p^\star)\) there exists a unique differentiable function \( w(r, p) = (x(r, p), y(r, p), z(r, p))\) satisfying

and such that \( w(0, p^\star) = (x^\star, y^\star, z^\star)\) . In addition, for any \( (r, p)\) near \( (0, p^\star)\) , \( x(r, p)\) is locally unique unconstrained local minimizing point of \( W(x, r, p)\) with \( h_i(x(r, p), p) < 0\) and \( \nabla^2_{x x} W(x(r, p), r, p)\) positive definite.

Let \( w^\star = (x^\star, y^\star, z^\star)\) be a local primal-dual stationary point of (78). Suppose that LICQ, SCS and SOSC hold at \( w^\star\) . We denote by \( x(\mu_k, p)\) the solution of the barrier problem (79) at iteration \( k\) . Then

• There is at least one subsequence of \( x(\mu_k, p)\) converging to \( x^\star\) and for every convergent subsequence, the corresponding barrier multiplier approximation is bounded and converges to \( (y^\star, z^\star)\) .

• There exists a unique continuously differentiable function \( w(\cdot, p)\) existing in a neighborhood of \( 0\) for \( \mu > 0\) and solution of (79), and such that

  \[ \begin{equation} \lim_{\mu\to 0} w(\mu, p) = w^\star\; . \end{equation} \]

  (81)

• For \( \mu\) small enough,

  \[ \begin{equation} J_p w(\mu, p) = - (J_w F_\mu)^{-1} J_p F_\mu \; . \end{equation} \]

  (82)

Let \( x \in \Omega\) . The function \( f\) has a directional derivative at \( x\) in the direction \( h \in E\) if \( x + th \in \Omega\) for \( t\) small enough, and if the following limit exists

The function \( f\) is Gâteaux differentiable (or \( G\) -differentiable) in \( x \in \Omega\) if there exists a directional derivatives for every direction \( h \in E\) , and if the application

is continuous and linear. We note \( f'(x)\) the operator such that \( f'(x; h) = f'(x) \cdot h\) .

The function \( f\) is Fréchet differentiable (\( F\) -differentiable, or simply differentiable) at \( x \in \Omega\) if there exists a bounded linear operator \( L\) such that

The function \( f\) is Hadamard differentiable at \( x\) if for any mapping \( \varphi: \mathbb{R}_+ \to X\) such that \( \varphi(0) = x\) and \( \frac{1}{t} (\varphi(t) - \varphi(0))\) converges to a vector \( h\) as \( t \downarrow 0\) , the limit

does exist.

Let \( f\) be Hadamard directionally differentiable at \( x\) , and \( g\) be Hadamard directionally differentiable at \( f(x)\) . Then the composite mapping \( g \circ f\) is Hadamard directionally differentiable at \( x\) , and satisfies the chain-rule \( d_H (g \circ f)(x; h) = d_H g (f(x); d_H f(x; y))\) .

Let \( f: \Omega \to F\) be a locally Lipschitz continuous function. Then \( f\) is differentiable almost everywhere in the sense of the Lebesgue measure.

Let \( x \in \Omega\) . The upper Dini derivative in the direction \( h\) is defined as

Accordingly, the lower Dini derivative is defined as

Let \( \mathcal{D}_f\) be the set of points at which \( f\) is (Fréchet) differentiable. The \( B\) -differential of \( f\) at \( x \in \Omega\) is the set defined as

The \( C\) -differential of \( f\) at \( x \in \Omega\) is the convex hull of the Bouligand-differential, that is

The Clarke-generalized directional derivative of \( f\) in the direction \( h\) is defined by

Suppose that \( f\) is \( L\) -Lipschitz near \( x \in \Omega\) . Then

1. \( \partial_C f(x)\) is nonempty compact and convex,
2. \( \partial_C f(x)\) is locally bounded and upper semi-continuous at \( x\) .

3. \( d_C f(x; \cdot)\) is the support function of \( \partial_C f(x)\) , that is,

   \[ \begin{equation} d_C f(x; h) = \max \{ s^\top h \; : \; \forall s \in \partial_C f(x) \} \;. \end{equation} \]

   (96)

If \( f\) is \( G\) -differentiable at \( x\) , then \( f'(x) \in \partial_C f(x)\) . If in addition \( f\) is \( C^1\) at \( x\) , then \( \partial_C f(x) = \{ f'(x) \}\) .

Suppose that \( f\) is \( L\) -Lipschitz continuous on \( \Omega\) . If the function \( g:F \to G\) is Lipschitz-continuous near \( f(x)\) , then

The function \( f:\Omega \to F\) is semi-smooth in \( x \in \Omega\) if:

1. \( f\) is \( L\) -Lipschitz near \( x\) ;
2. \( f\) has directional derivatives at \( x\) in all directions;
3. When \( h \to 0\) , \( \sup_{J \in \partial_C f} \|f(x+h) -f(x) - Jh \| = o(\|h\|)\) .

Suppose \( f\) is \( L\) -Lipschitz near \( x\) , and admits directional derivatives at \( x\) in all directions. Then the following statements are equivalent:

1. \( f\) is semi-smooth ;
2. for \( h \to 0\) , \( \sup_{J \in \partial_C f} \|Jh - f'(x;h)\| = o(\|h\|)\) .
3. for \( h \to 0\) such that \( x+h \in \mathcal{D}_f\) , \( f'(x+h) h - f'(x; h)= o(\|h\|)\) .

The function \( f:X \to F\) is piecewise-differentiable (PC\( ^1\) ) at \( x \in X\) if there exists a neighborhood \( N \subset X\) and a finite collection of \( C^1\) selection functions \( \mathcal{F}_f(x) = \{f_{(1)}, …, f_{(k)} \}\) defined on \( N\) such that \( f\) continuous on \( N\) and for all \( y \in N\) , \( f(y) \in \{f_{(i)}(y)\}_{i \in [k]}\) .

Let \( f:X \to F\) a PC\( ^1\) function at \( x \in X\) . Then

• \( \mathcal{I}_f^{ess}(x)\) is a non-empty set;

• \( f\) is Lipschitz continuous near \( x\) and directionally differentiable at \( x\) , with

  \[ \begin{equation} \partial_B f(x) = \{ \nabla f_{(i)}(x) \; : \; i \in \mathcal{I}_f^{ess}(x) \}. \end{equation} \]

  (99)

  In addition, the directional derivative \( f'(x; \cdot)\) is piecewise linear.

Let \( U \subset \mathbb{R}^n\) and \( V \subset \mathbb{R}^m\) be open. A \( PC^1\) function \( F:U \times W \to \mathbb{R}^n\) is said coherently oriented with respect to \( x\) at \( (x^\star, y^\star) \in U \times V\) if all matrices in \( \partial_{B,x} f(x^\star, y^\star)\) have the same non-vanishing determinant sign.

Let \( f:X \to \mathbb{R}^m\) a locally Lipschitz continuous function on \( X\) . \( f\) is lexicographically smooth at \( x\) if for any \( k \in \mathbb{N}\) and any \( M = [m_{(1)}, …, m_{(k)}] \in \mathbb{R}^{n \times k}\) the following higher-order derivatives are well defined

Let \( f:X \to \mathbb{R}^m\) a L-smooth function at \( x\) , and \( M \in \mathbb{R}^{n \times n}\) a nonsingular matrix. The lexicographic derivative of \( f\) at \( x\) in the direction \( M\) is defined as

The lexicographic subdifferential of \( f\) at \( x\) is defined as

For any \( k \in \mathbb{N}\) , \( M = [m_{(1)}, …, m_{(k)}] \in \mathbb{R}^{n \times k}\) the LD-derivative of \( f\) in the directions \( M\) is defined as

Let \( h:X \to Y\) and \( g:Y \to \mathbb{R}^q\) be L-smooth at \( x \in X\) and \( h(x) \in Y\) , respectively. Then the function \( g\circ h\) is L-smooth at \( x \in X\) , and for any \( k \in \mathbb{N}\) and \( M \in \mathbb{R}^{n \times k}\) ,

Let \( \Gamma: E \rightrightarrows F\) a point-to-set map.

• \( \Gamma\) is upper semicontinuous at \( x \in E\) if for each open set \( \Omega \subset F\) satisfying \( \Gamma(x) \subset \Omega\) there exists a neighborhood \( N(x)\) of \( x\) such that for all \( y \in N(x)\) , \( \Gamma(y) \subset \Omega\) .
• \( \Gamma\) is lower semicontinuous at \( x \in E\) if for each open set \( \Omega \subset F\) satisfying \( \Gamma(x) \cap \Omega \neq \varnothing\) there exists a neighborhood \( N(x)\) of \( x\) such that for all \( y \in N(x)\) , \( \Gamma(y) \cap \Omega \neq \varnothing\) .
• \( \Gamma\) is continuous at \( x\) if it is lower semicontinuous and upper semicontinuous at \( x\) .

The point-to-set map \( \Gamma:E \rightrightarrows F\) is closed at \( x\) if there exists a sequence \( x_n \in E\) , \( x_n\to x\) such that \( y_n \in \Gamma(x_n)\) and \( y_n \to y\) imply \( y \in \Gamma(x)\) .

The point-to-set map \( \Gamma:E \rightrightarrows F\) is open at \( x\) if \( y \in \Gamma(x)\) implies there exists a sequence \( x_n \in E\) , \( x_n\to x\) such that there exists \( m\) and \( \{y_n\}\) such that \( y_n \in \Gamma(x_n)\) for all \( n \geq m\) and \( y_n \to y\) .

The point-to-set map \( \Gamma:E \rightrightarrows F\) is uniformly compact near \( \hat{x}\) if the set \( \cup_{x \in N(\hat{x})} \Gamma(x)\) is bounded for some neighborhood \( N(\hat{x})\) of \( \hat{x}\) .

The point-to-set map \( \Gamma: E \rightrightarrows F\) is upper Lipschitzian with modulus \( L\) at \( \hat{x} \in E\) if there is a neighborhood \( N\) of \( \hat{x}\) such that for each \( x \in N\) , \( \Gamma(x) \subset \Gamma(\hat{x}) + L \|x - \hat{x} \|B\) , with \( B\) the unit-ball in \( F\) .

We define the normal cone operator associated to the set \( C\) as, for \( x \in X\)

Let \( p \in P\) be a parameter. For \( f: X \times P \to X^\star\) , we define the paramaterized variational inequality (also called generalized equation) the problem of finding \( x \in X\) such that

We say that (110) is strongly regular at a solution \( x_0\) with Lipschitz constant \( L\) if there exists a neighborhood \( U \subset X^\star\) of the origin and \( V \subset X\) of \( x_0\) such that the restriction to \( U\) of \( T^{-1} \cap V\) is a single-valued function from \( U\) to \( V\) and is \( L\) -Lipschitzian on \( U\) .

Let \( p \in P\) be a given parameter, \( x_0\) solution of the generalized equation (110). Suppose that \( f\) is Fréchet-differentiable w.r.t. \( x\) exists at \( p\) , and that both \( f(\cdot, \cdot)\) and \( \nabla_x f(\cdot, \cdot)\) are continuous at \( (x_0, p)\) . If (110) is strongly regular at \( x_0\) , then for any \( \varepsilon >0\) there exists neighborhood \( N\) of \( p\) and \( W\) of \( x_0\) as well as a single-valued function \( x:N \to W\) sucht that for any \( p \in N\) , \( x(p)\) is the unique solution of the inclusion

and, for all \( p, q \in N\) , one has

Suppose the hypotheses of Theorem 13 hold. Suppose in addition there exists a constant \( \kappa\) such that for each \( (p, q) \in N\) , for each \( x \in W\) , one has

Then \( x(\cdot)\) is Lipschitzian on \( N\) with modulus \( \kappa (L + \varepsilon)\) .