## Abstract

We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal–dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.

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## References

Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

Mignot, F.: Controle dans les inéquations variationelles elliptiques. J. Funct. Anal.

**22**, 130–185 (1976)Mignot, F., Puel, J.-P.: Optimal control in some variational inequalities. SIAM J. Control Optim.

**22**, 466–476 (1984)Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, New York (1993)

Bergounioux, M.: Optimal control of problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim.

**36**, 273–289 (1998)Hintermüller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: C-and strong stationarity and a path-following algorithm. SIAM J. Optim.

**20**, 868–902 (2009)Outrata, J., Jarušek, J., Stará, J.: On optimality conditions in control of elliptic variational inequalities. Set-Valued Var. Anal.

**19**, 23–42 (2011)Kunisch, K., Wachsmuth, D.: Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities. ESAIM Control Optim. Calc. Var.

**180**, 520–547 (2012)Kunisch, K., Wachsmuth, D.: Path-following for optimal control of stationary variational inequalities. Comput. Optim. Appl.

**51**, 1345–1373 (2012)Herzog, R., Meyer, C., Wachsmuth, G.: B-and strong stationarity for optimal control of static plasticity with hardening. SIAM J. Optim.

**23**, 321–352 (2013)Schiela, A., Wachsmuth, D.: Convergence analysis of smoothing methods for optimal control of stationary variational inequalities. ESAIM Math. Model. Numer. Anal.

**47**, 771–787 (2013)Hintermüller, M., Mordukhovich, B., Surowiec, T.: Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Prog. A

**146**, 555–582 (2014)Bonnans, J.F., Casas, E.: An extension of Pontryagin’s principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities. SIAM J. Control Optim.

**33**, 274–298 (1995)Outrata, J.V.: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim.

**38**, 1623–1638 (2000)De los Reyes, J.C.: Optimal control of a class of variational inequalities of the second kind. SIAM J. Control Optim.

**49**, 1629–1658 (2011)De los Reyes, J.C.: Optimization of mixed variational inequalities arising in flows of viscoplastic materials. Comput. Optim. Appl.

**52**, 757–784 (2012)De los Reyes, J.C., Schönlieb, C.-B.: Image denoising: learning the noise model via nonsmooth PDE-constrained optimization. Inverse Probl. Imaging

**7**, 1183–1214 (2013)De los Reyes, J.C., Herzog, R., Meyer, C.: Optimal control of static elastoplasticity in primal formulation. Ergebnisberichte des Instituts für Angewandte Mathematik No. 474, TU Dortmund (2013)

Evans, L.C.: Partial Differential Equations: Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)

Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin (1978)

Hintermüller, M., Stadler, G.: An infeasible primal–dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput.

**28**, 1–23 (2006)Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (2000)

Scholtes, S., Stöhr, M.: Exact penalization of mathematical programs with equilibrium constraints. SIAM J. Control Optim.

**37**, 617–652 (1999)

## Acknowledgments

The authors would like to thank Gerd Wachsmuth (TU Chemnitz) for his hint concerning strong stationarity. This work was supported by a DFG grant within the Collaborative Research Center SFB 708 (*3D-Surface Engineering of Tools for Sheet Metal Forming Manufacturing, Modeling, Machining*), which is gratefully acknowledged. We also acknowledge support of MATHAmSud project “Sparse Optimal Control of Differential Equations” (SOCDE).

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## Appendices

### Appendix A: Directional Derivative of the \(L^1\)-Norm

###
*Proof of Lemma 3.3*

We consider the mapping

It is easily seen that \(g\) is Lipschitz continuous. Moreover, for arbitrary \(y,\eta \in L^1(\varOmega )\), the directional differentiability of \(\mathbb {R}\ni r \mapsto |r| \in \mathbb {R}\) yields

and, since almost all points in \(\varOmega \) are common Lebesgue points of \(y\) and \(\eta \), this pointwise convergence holds almost everywhere in \(\varOmega \). Due to

Lebesgue dominated convergence theorem thus yields

which in turn implies the directional differentiability of \(g\) with

Consequently \(g\) is Hadamard-differentiable and hence

since

so that \(\Vert r(t_n)\Vert _{L^1(\varOmega )} = o(t_n)\) thanks to (35) and the compact embedding \(V \hookrightarrow L^1(\varOmega )\). \(\square \)

### Appendix B: Boundedness for Functions in \(H^1(\varOmega )\)

For convenience of the reader, we prove Lemma 3.6. The arguments are classical and go back to [22].

###
*Proof of Lemma 3.6*

The truncated function defined in (43) is equivalent to

and therefore [22], Theorem A.1] implies \(w_k \in V\).

It remains to verify the \(L^\infty \)-bound in (45). If \(d=1\), then the assertion follows directly from (44) and the Sobolev embedding \(H^1(\varOmega ) \hookrightarrow L^\infty (\varOmega )\).

So assume that \(d\ge 2\). Then let \(k \ge 0\) be given and set \(A(k) :=\{x\in \varOmega \;|\;|w(x)|\ge k\}\). Note that \(w_k(x) = 0\) a.e. in \(\varOmega \!\setminus \! A(k)\). Next let \(h \ge k\) be arbitrary so that \(w(x) \ge h \ge k\) a.e. in \(A(h)\). Then Sobolev embeddings give that

where \(m = 2d/(d-2)\), see e.g., \(\ldots \) On the other hand, (44) implies

where \(m'\) is the conjugate exponent to \(m\), i.e., \(1/m + 1/m' = 1\). Note that

and thus \(f\in L^{m'}(\varOmega )\) by the assumption on \(f\) in Lemma 3.6. Together with Young’s inequality, then Hölder’s inequality yields

with \(r = p/(p-m') \ge 1\) so that \(r' = r/(r-1) = p/m'\). By setting

we infer from (86) and (87) that

Since \(m > 2\), we have \(m'< 2\) and therefore \((m'-1)(p-m') < p-m' < p\) such that (88) gives in turn \(s>1\). In this case, according to [22], Lemma B.1], it follows from (89) that the nonnegative and nonincreasing function \(\mathbb {R}\ni h \mapsto |A(h)|^{2/m} \in \mathbb {R}\) admits a zero at

By definition, \(|A(h^*)| = 0\) is equivalent to \(|w(x)| \le h^*\) a.e. in \(\varOmega \), which yields the assertion. \(\square \)

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De los Reyes, J.C., Meyer, C. Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind.
*J Optim Theory Appl* **168**, 375–409 (2016). https://doi.org/10.1007/s10957-015-0748-2

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DOI: https://doi.org/10.1007/s10957-015-0748-2