\`x^2+y_1+z_12^34\`
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Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control

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  • We consider the numerical solution of nonlinear and nonsmooth operator equations in Hilbert spaces. A semismooth Newton method is used for search direction generation. The operator equation is solved by a globalized semismooth Newton method that is equipped with an Armijo linesearch using a semismooth merit function. We prove that an accumulation point of the globalized algorithm is a solution and transition to fast local convergence under a directional Hadamard-like continuity assumption on the Newton matrix. In particular, no auxiliary descent directions or smoothing steps are required. Finally, we apply this method to a control-constrained and also to a regularized state-constrained optimal control problem subject to partial differential equations.

    Mathematics Subject Classification: Primary: 49J20, 49J52, 49M15; Secondary: 90C56, 65K10, 65N12.

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  • Figure 1.  Discrete solution of (P2) for $h=1/64$. Left-hand side: Optimal state $y^h(x_1, x_2)$ on $x_3$ axis vs. $x_1$ and $x_2$. Right-hand side: Optimal control $u^h(x_1, x_2)$ on $x_3$ axis vs. $x_1$ and $x_2$

    Figure 2.  Discrete solution of (P3) for $h=1/32$. Left-hand side: Optimal state $y^h(x_1, x_2)$ on $x_3$ axis vs. $x_1$ and $x_2$. Right-hand side: Optimal control $u^h(x_1, x_2)$ on $x_3$ axis vs. $x_1$ and $x_2$

    Table 1.  Iteration history for the solution of problem (P2) for $h=1/256$. Step size $\alpha_k$, norm $\Vert f(z_k)\Vert_{Z^*}$ and norm of the search direction $\Vert s_k\Vert_Z$ for the $k$-th iterate. These numerical results exhibit the superlinear convergence

    $k$ $\alpha_k$ $\left\Vert f(z_k)\right\Vert_{Z^*}$ $\left\Vert s_k\right\Vert_Z$
    0- 5.43111E-02 -
    1 9.76563E-04 5.43015E-02 5.00085E+00
    2 3.12500E-02 5.36304E-02 1.82556E+00
    3 5.00000E-01 2.91839E-02 1.55585E+00
    4 6.25000E-02 2.75202E-02 3.87423E-01
    16 0.25000E+00 1.65715E-02 2.48095E-02
    17 0.50000E+00 1.38976E-02 1.28644E-02
    18 1.00000E+00 1.24060E-02 6.81858E-03
    19 1.00000E+00 9.44693E-03 1.63072E-03
    20 1.00000E+00 5.60965E-06 4.47294E-05
    21 1.00000E+00 2.27743E-15 1.57318E-11
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    Table 2.  Iteration history for the solution of problem (P3) for $h=1/128$. Step size $\alpha_k$, norm $\Vert f(z_k)\Vert_{Z^*}$ and norm of the search direction $\Vert s_k\Vert_Z$ for the $k$-th iterate. We observe transition to local superlinear convergence

    $k$ $\alpha$ $\left\Vert f(z_k)\right\Vert_{Z^*}$ $\left\Vert s_k\right\Vert_{Z^*}$
    0 - 7.59736E+05 -
    1 1.00000E+00 1.14024E+05 1.93458E+03
    2 1.00000E+00 3.61620E+04 7.83427E+02
    3 1.00000E+00 1.59280E+04 1.62132E+03
    9 2.50000E-01 3.03640E-02 1.48894E-01
    10 1.00000E+00 9.69843E-03 3.23249E-02
    11 1.00000E+00 2.42234E-05 9.90030E-06
    12 1.00000E+00 3.15754E-06 2.56947E-07
    13 1.00000E+00 1.14583E-07 1.59876E-09
    14 1.00000E+00 1.70426E-13 5.17916e-13
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