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Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space

Abstract

This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on the Moreau–Yosida approximation, we prove the global existence and uniqueness of the solution of the proposed evolution differential inclusion with projection and the unique solution of the proposed system is just its “slow solution” when the constrained set is defined by the affine equalities. When the outer layer objective function ψ is strongly convex, any solution of the proposed system is strongly convergent to the unique minimizer of the constrained optimization problem, while, the strongly convergence is also given when the inner layer objective function ϕ is strongly convex. Furthermore, we present some other optimization problem models, which can be solved by the proposed system. All the results obtained are new not only in the infinite dimensional Hilbert space framework but also in the finite dimensional space.

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Author information

Correspondence to Wei Bian.

Additional information

This work is supported the NSF foundation (11101107, 10971043) of China, Heilongjiang Province foundation for distinguished young scholars (CJ200810), the program of excellent team in Harbin Institute of Technology, the project (HIT. NSRIF. 2009048), the Hong Kong Polytechnic University Postdoctoral Fellowship Scheme and The Education Department of Heilongjiang Province (1251112).

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Bian, W., Xue, X. Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space. Set-Valued Anal 20, 203–227 (2012). https://doi.org/10.1007/s11228-011-0194-8

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Keywords

  • Hilbert space
  • Constrained optimization
  • Project operator
  • Strong convergence
  • Subgradient

Mathematics Subject Classifications (2010)

  • 65K05
  • 49M37
  • 90C30
  • 93B40