On the convergence of the conditional gradient method as applied to the optimization of an elliptic equation.

*(English. Russian original)*Zbl 1318.49055
Comput. Math. Math. Phys. 55, No. 2, 212-226 (2015); translation from Zh. Vychisl. Mat. Mat. Fiz. 55, No. 2, 213-228 (2015).

Summary: The optimal control of a second-order semilinear elliptic diffusion-reaction equation is considered. Sufficient conditions for the convergence of the conditional gradient method are obtained without using assumptions (traditional for optimization theory) that ensure the Lipschitz continuity of the objective functional derivative. The total (over the entire set of admissible controls) preservation of solvability, a pointwise estimate for solutions, and the uniqueness of a solution to the homogeneous Dirichlet problem for a controlled elliptic equation are proved as preliminary results, which are of interest on their own.

##### MSC:

49M30 | Other numerical methods in calculus of variations (MSC2010) |

49J20 | Existence theories for optimal control problems involving partial differential equations |

35J61 | Semilinear elliptic equations |

##### Keywords:

optimal control; semilinear elliptic diffusion-reaction equations; conditional gradient method; total preservation of solvability; solution estimate; solution uniqueness
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\textit{A. V. Chernov}, Comput. Math. Math. Phys. 55, No. 2, 212--226 (2015; Zbl 1318.49055); translation from Zh. Vychisl. Mat. Mat. Fiz. 55, No. 2, 213--228 (2015)

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##### References:

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