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Necessary and sufficient conditions of optimality for second order discrete and differential inequalities

Elimhan N. Mahmudov ORCID logo and Sevilay Demir Sağlam ORCID logo


The present paper studies the optimization of the Bolza problem with a system of convex and nonconvex, discrete and differential state variable second-order inequality constraints by deriving necessary and sufficient conditions of optimality. The problem with a system of discrete-approximation inequalities is investigated using the proposed method of discretization and equivalence theorems for subdifferential inclusions, which greatly contributes to the derivation of adjoint discrete inclusions generated by a given system of nonlinear inequality constraints. Furthermore, we formulate sufficient conditions of optimality for the continuous problem by passing to the case of limit. A numerical example is provided to illustrate the theoretical approach’s effectiveness.


The authors would like to thank the anonymous reviewers for their helpful comments that have improved the final manuscript.


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Received: 2021-04-10
Revised: 2021-06-09
Accepted: 2021-06-15
Published Online: 2022-01-23

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