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Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 )

Author

Listed:
  • Masoud Ahookhosh

    (University of Vienna)

  • Arnold Neumaier

    (University of Vienna)

Abstract

This paper describes an algorithm for solving structured nonsmooth convex optimization problems using the optimal subgradient algorithm (OSGA), which is a first-order method with the complexity $$\mathcal {O}(\varepsilon ^{-2})$$ O ( ε - 2 ) for Lipschitz continuous nonsmooth problems and $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 ) for smooth problems with Lipschitz continuous gradient. If the nonsmoothness of the problem is manifested in a structured way, we reformulate the problem so that it can be solved efficiently by a new setup of OSGA (called OSGA-V) with the complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 ) . Further, to solve the reformulated problem, we equip OSGA-O with an appropriate prox-function for which the OSGA-O subproblem can be solved either in a closed form or by a simple iterative scheme, which decreases the computational cost of applying the algorithm for large-scale problems. We show that applying the new scheme is feasible for many problems arising in applications. Some numerical results are reported confirming the theoretical foundations.

Suggested Citation

  • Masoud Ahookhosh & Arnold Neumaier, 2018. "Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 )," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 110-145, April.
    • Handle: RePEc:spr:topjnl:v:26:y:2018:i:1:d:10.1007_s11750-017-0462-3
    DOI: 10.1007/s11750-017-0462-3
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    References listed on IDEAS

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