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An efficient nonmonotone method for state-constrained elliptic optimal control problems. (English) Zbl 1447.49038

Summary: This paper presents a novel numerical strategy based on combination of an adaptive semismooth Newton (ASN) method and the Lavrentiev regularization technique for the solution of elliptic optimal control problems with state constraints. Using the global convergence proof for a nonmonotone semismooth Newton method, we will exploit an adaptive nonmonotone line search method such that the nonmonotonicity degree of this method can be increased when the results are far from the optimum solution and it can be reduced when they are close to the optimizer. In this strategy, the role of the Lavrentiev regularization technique is converting the original optimal control problem to a regularized optimal control problem. Using the finite difference discretization scheme and a Newton-Cotes rule, the regularized optimal control problem is converted to a bound constrained optimization problem (BCOP). Then the ASN method is implemented to solve the resulting BCOP. Numerical results show the efficiency of the proposed procedure.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
65N06 Finite difference methods for boundary value problems involving PDEs
90C30 Nonlinear programming
49M15 Newton-type methods
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