##
**Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis.**
*(English)*
Zbl 0826.49008

In the paper a new approach to the sensitivity analysis for parametric generalized equations of the form
\[
0\in f(p, z)+ Q(z)
\]
is presented where \(f(.,.)\) is a continuous vector-valued function and \(Q(.)\) is a multifunction with closed graph. The author describes sufficient and necessary conditions for the locally pseudo-Lipschitz dependence of the solution set on the parameter \(p\).

For this papers generalized differentiability notions are introduced using a new normal cone \(N(x, \Omega)\) to a set \(\Omega\) at a point \(x\in \Omega\). This normal cone is in general not convex, but it is smaller than the Clarke normal cone such that sharper stability assertions can be expected.

In the usual manner the coderivative of the multifunction \(\phi(.)\) at the point \((x, y)\in \text{gph } \phi\) is defined by \[ D^* \phi(x, y)(y^*):= \{x^*\mid (x^*, - y^*)\in N((x, y),\text{ gph } \phi)\}. \] Similarly, for a real-valued function the associated subdifferentials of first- and second-order are introduced which turn out to be smaller than the subdifferentials in sense of Clarke/Rockafellar.

Assuming special Lipschitz properties of the functions \(f\) and \(Q\) the author presents local stability conditions for the generalized equation near the point \((p, z)\) in such a form that the adjoint equation \[ 0\in D^*_z f(p, z)(y^*)+ D^* Q(z, -f(p, z))(y^*) \] admits only the trivial solution \(y^*= 0\).

First the theory is proved for the linear case, i.e., in case of \(f(p, z)= Ap+ z\). Then for the smooth case (i.e., \(f\) is strictly differentiable in \(z\)) the assertion is derived by linearizing the function \(f\). At least for the nonsmooth case the main result is proved using the Robinson strong approximation for the function \(f\) and a stability assertion of Dontchev/Hager.

In all cases some corollaries are formulated regarding a special structure of the multifunction \(Q(.)\), e.g., a) \(\text{gph } Q\) is convex; b) \(Q(z)= E\) for \(z\in \Omega\), \(=\emptyset\) otherwise, where \(\Omega\) and \(E\) are convex sets; c) \(Q(z)= \{y\mid \vartheta(z, y)= \Lambda\}\), where \(\vartheta\) is a (continuous or smooth, respectively) vector-valued function and \(\Lambda\) is a closed set; d) \(Q(z)= \{y\mid \vartheta_i(z, y)\leq 0\), \(i\in I\), \(\vartheta_i(z, y)= 0\), \(i\in J\}\), where \(\vartheta_i\) are smooth real-valued functions; e) \(Q(z)= \partial^-\varphi(z)\) if \(|\varphi(z)|< \infty\), \(=\emptyset\) otherwise using the introduced general subdifferential of a real-valued function \(\varphi\); e) \(Q(z)= \{0\}\).

In the last section the author points out that by assumption of additional monotony properties the solution set is locally single-valued and Lipschitz continuous.

For this papers generalized differentiability notions are introduced using a new normal cone \(N(x, \Omega)\) to a set \(\Omega\) at a point \(x\in \Omega\). This normal cone is in general not convex, but it is smaller than the Clarke normal cone such that sharper stability assertions can be expected.

In the usual manner the coderivative of the multifunction \(\phi(.)\) at the point \((x, y)\in \text{gph } \phi\) is defined by \[ D^* \phi(x, y)(y^*):= \{x^*\mid (x^*, - y^*)\in N((x, y),\text{ gph } \phi)\}. \] Similarly, for a real-valued function the associated subdifferentials of first- and second-order are introduced which turn out to be smaller than the subdifferentials in sense of Clarke/Rockafellar.

Assuming special Lipschitz properties of the functions \(f\) and \(Q\) the author presents local stability conditions for the generalized equation near the point \((p, z)\) in such a form that the adjoint equation \[ 0\in D^*_z f(p, z)(y^*)+ D^* Q(z, -f(p, z))(y^*) \] admits only the trivial solution \(y^*= 0\).

First the theory is proved for the linear case, i.e., in case of \(f(p, z)= Ap+ z\). Then for the smooth case (i.e., \(f\) is strictly differentiable in \(z\)) the assertion is derived by linearizing the function \(f\). At least for the nonsmooth case the main result is proved using the Robinson strong approximation for the function \(f\) and a stability assertion of Dontchev/Hager.

In all cases some corollaries are formulated regarding a special structure of the multifunction \(Q(.)\), e.g., a) \(\text{gph } Q\) is convex; b) \(Q(z)= E\) for \(z\in \Omega\), \(=\emptyset\) otherwise, where \(\Omega\) and \(E\) are convex sets; c) \(Q(z)= \{y\mid \vartheta(z, y)= \Lambda\}\), where \(\vartheta\) is a (continuous or smooth, respectively) vector-valued function and \(\Lambda\) is a closed set; d) \(Q(z)= \{y\mid \vartheta_i(z, y)\leq 0\), \(i\in I\), \(\vartheta_i(z, y)= 0\), \(i\in J\}\), where \(\vartheta_i\) are smooth real-valued functions; e) \(Q(z)= \partial^-\varphi(z)\) if \(|\varphi(z)|< \infty\), \(=\emptyset\) otherwise using the introduced general subdifferential of a real-valued function \(\varphi\); e) \(Q(z)= \{0\}\).

In the last section the author points out that by assumption of additional monotony properties the solution set is locally single-valued and Lipschitz continuous.

Reviewer: J.Thierfelder (Ilmenau)

### MSC:

49J52 | Nonsmooth analysis |

49J40 | Variational inequalities |

90C31 | Sensitivity, stability, parametric optimization |

58C15 | Implicit function theorems; global Newton methods on manifolds |

58C25 | Differentiable maps on manifolds |

58C07 | Continuity properties of mappings on manifolds |

26B10 | Implicit function theorems, Jacobians, transformations with several variables |

### Keywords:

coderivative; Lipschitz behaviour of multifunctions; sensitivity analysis; parametric generalized equations; subdifferentials
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\textit{B. Mordukhovich}, Trans. Am. Math. Soc. 343, No. 2, 609--657 (1994; Zbl 0826.49008)

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### References:

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