Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind

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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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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Correspondence to C. Meyer.

Appendices

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

Proof of Lemma 3.3

We consider the mapping

$$\begin{aligned} g{:}\;L^1(\varOmega ) \ni y \mapsto \int _\varOmega |y(x)|\,\varphi (x)\, \mathrm{d}x \in \mathbb {R}. \end{aligned}$$

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

$$\begin{aligned} \frac{|y(x) + t_n \eta (x)| - |y(x)|}{t_n} \rightarrow {\text {abs}}'\big (y(x);\eta (x)\big ), \end{aligned}$$

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

$$\begin{aligned} - 2 |\eta (x)|\le \frac{|y(x) + t_n \eta (x)| - |y(x)|}{t_n} - {\text {abs}}'\big (y(x);\eta (x)\big ) \le 2 |\eta (x)| \quad \text {a.e. in }\, \varOmega , \end{aligned}$$

Lebesgue dominated convergence theorem thus yields

$$\begin{aligned} \frac{|y + t_n \eta | - |y|}{t_n}\rightarrow {\text {abs}}'(y;\eta ) \text { in } L^1(\varOmega ), \end{aligned}$$

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

$$\begin{aligned} g'(y;\eta ) = \int _\varOmega {\text {abs}}'(y(x);\eta (x))\,\varphi (x)\, \mathrm{d}x. \end{aligned}$$

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

$$\begin{aligned} \int _\varOmega \Big (\frac{|y_n| - |y|}{t_n} - {\text {abs}}'(y;\eta )\Big ) \varphi \, \mathrm{d}x = \frac{g(y + t_n\eta + r(t_n)) - g(y)}{t_n} - g'(y;\eta ) \rightarrow 0, \end{aligned}$$

since

$$\begin{aligned} r(t_n) := y_n - y - t_n \eta \end{aligned}$$

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

$$\begin{aligned} w_k(x) = w(x) - \min \big ((\max (w(x),-k),k\big ) \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \Vert w_k\Vert _{H^1(\varOmega )}^2 \ge c\, \Vert w_k\Vert _{L^m(\varOmega )}^2&= c\Big (\int _{A(k)} \big | |w| - k \big |^m \mathrm{d}x\Big )^{2/m}\\&\ge c\int _{A(h)}(h-k)^m \mathrm{d}x^{2/m} = c\,(h-k)^2 |A(h)|^{2/m}, \end{aligned} \end{aligned}$$
(86)

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

$$\begin{aligned} \alpha \,\Vert w_k\Vert ^2 \!\le \! \int _{A(k)} f\,w_k\, \mathrm{d}x \le \Vert f\Vert _{L^{m'}(A(k))} \, \Vert w_k\Vert _{L^m(A(k))} \le c\,\Vert f\Vert _{L^{m'}(A(k))} \, \Vert w_k\Vert _{H^1(\varOmega )}, \end{aligned}$$

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

$$\begin{aligned} m' = \frac{m}{m-1} = \frac{d}{d/2 + 1} \le \frac{d}{2} < p, \quad \text {if } d \ge 2, \end{aligned}$$

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

$$\begin{aligned} \Vert w_k\Vert ^2 \le c\Big (\int _{A(k)} |f|^{m'}\, \mathrm{d}x\Big )^{2/m'} \le c\,\Vert f\Vert _{L^p(\varOmega )}^2\,|A(k)|^{2r/m'} \end{aligned}$$
(87)

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

$$\begin{aligned} s = \frac{m}{m'}\, r = \frac{p}{(m'-1)(p-m')} \end{aligned}$$
(88)

we infer from (86) and (87) that

$$\begin{aligned} |A(h)|^{2/m} \le c\,\Vert f\Vert _{L^p(\varOmega )}^2\,\frac{1}{(h-k)^2} \, \big (|A(h)|^{2/m}\big )^s\quad \text {for all } h > k \ge 0. \end{aligned}$$
(89)

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

$$\begin{aligned} h^* = 2^{s/(s-1)} \sqrt{c \,|\varOmega |^{2(s-1)/m}}\, \Vert f\Vert _{L^p(\varOmega )}. \end{aligned}$$

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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Keywords

  • Variational inequalities
  • Optimality conditions
  • Mathematical programs with equilibrium constraints

Mathematics Subject Classification

  • 49K21
  • 90C33
  • 35R35
  • 49J40