The purpose of this paper is to provide the linearization technique to solve the multidimensional control optimization problem (MCOP) involving first-order partial differential equation (PDEs) constraints. Firstly, we...
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The purpose of this paper is to provide the linearization technique to solve the multidimensional control optimization problem (MCOP) involving first-order partial differential equation (PDEs) constraints. Firstly, we use the modified objective function approach for simplifying the aforesaid extremum problem (MCOP) and show that the solution sets of the original control optimization problem and its modified control optimization problem (MCOP) Omega$$ {}_{\Omega} $$ are equivalent under convexity assumptions. Further, we use the absolute value exact penalty function method to transform (MCOP) Omega$$ {}_{\Omega} $$ into a penalized control problem (MCOP) Omega & rhov;$$ {}_{\Omega \varrho } $$. Then, we establish the equivalence between a minimizer of the modified penalized optimization problem (MCOP) Omega & rhov;$$ {}_{\Omega \varrho } $$ and a saddle point of the Lagrangian defined for the modified optimization problem (MCOP) Omega$$ {}_{\Omega} $$ under appropriate convexity hypotheses. Moreover, the results established in the paper are illustrated by some examples of MCOPs involving first-order PDEs constraints.
In this paper, a nonconvex multitime control problem with first-order PDE constraints is considered. Then, we investigate the absolute value exact penalty function method which is used for solving the aforesaid contro...
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In this paper, a nonconvex multitime control problem with first-order PDE constraints is considered. Then, we investigate the absolute value exact penalty function method which is used for solving the aforesaid control problem. Namely, in order to ensure the effective use of the absolute value exact penalty function method in the considered case, the most important property of any exactpenaltyfunctionmethod, that is, exactness of the penalization, is analyzed in the case when the aforementioned method is applied for solving the considered multitime control problem with first-order PDE constraints in which the functionals involved are nonconvex. Thus, the equivalence between an optimal solution of the aforementioned control problem and a minimizer of its associated unconstrained multitime control problem constructed in the used absolute value exact penalty function method is proved under appropriate invexity hypotheses.
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