The Zero Duality Gap Property for an Optimal Control Problem Governed by a Multivalued Hemivariational Inequality
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The Zero Duality Gap Property for an Optimal Control Problem Governed by a Multivalued Hemivariational Inequality Fengzhen Long1 · Biao Zeng2
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We show in this work the zero duality gap property for an optimal control problem governed by a multivalued hemivariational inequality with unbounded constraint set. Based on the existence of solutions to the inequality, we establish several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem by using nonlinear Lagrangian methods. Moreover, we obtain a convergence result for the optimal control problem governed by a perturbed multivalued hemivariational inequality. Keywords Multivalued hemivariational inequality · Optimal control · Zero duality gap property · Nonlinear Lagrangian method · Convergence Mathematics Subject Classification 47J20 · 49N15 · 49J20
1 Introduction It is well known that the duality theory is an important topic in optimization. Various Lagrangian functions have been introduced to generalize the traditional Lagrangian duality theory, which play an important role in devising efficient schemes for solving
The work is supported by the Natural Science Foundation of Guangxi Province (No. 2019GXNSFBA185005), the Start-up Project of Scientific Research on Introducing talents at school level in Guangxi University for Nationalities (No. 2019KJQD04) and Xiangsihu Young Scholars Innovative Research Team of Guangxi University for Nationalities (No. 2019RSCXSHQN02).
B
Biao Zeng [email protected]
1
School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China
2
Faculty of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006, Guangxi, China
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Applied Mathematics & Optimization
constrained optimization problems. These problems are among the most important challenges in applied mathematics and industrial. Usually they can only be solved by employing numerical methods, namely computational optimization algorithms. The role of Lagrange type functions in the development of computational methods is to reduce the original constrained problem to a sequence of unconstrained subproblems, in such a way that information on the constraints is encapsulated in the Lagrangian. This reduction is carried out with the assumption that the subproblems are better behaved and simpler than the original one. Among the most well-known methods for reduction to unconstrained optimization, are the so-called augmented Lagrangian methods ([21] chapter 11, section K ∗ ), represented by the sum of the ordinary Lagrangian and an augmenting function. Methods incorporating Lagrangetype functions are successful when the optimal value of the dual problem coincides with that of the primal problem. This property, when holds, is referred as the zero duality gap property [2]. It follows from the Fenchel-Moreau theorem that the zero duality gap property implies the lower semicontinuity of the
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