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Path-following gradient-based decomposition algorithms for separable convex optimization

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  • Quoc Tran Dinh
  • Ion Necoara
  • Moritz Diehl

Abstract

A new decomposition optimization algorithm, called path-following gradient-based decomposition, is proposed to solve separable convex optimization problems. Unlike path-following Newton methods considered in the literature, this algorithm does not require any smoothness assumption on the objective function. This allows us to handle more general classes of problems arising in many real applications than in the path-following Newton methods. The new algorithm is a combination of three techniques, namely smoothing, Lagrangian decomposition and path-following gradient framework. The algorithm decomposes the original problem into smaller subproblems by using dual decomposition and smoothing via self-concordant barriers, updates the dual variables using a path-following gradient method and allows one to solve the subproblems in parallel. Moreover, compared to augmented Lagrangian approaches, our algorithmic parameters are updated automatically without any tuning strategy. We prove the global convergence of the new algorithm and analyze its convergence rate. Then, we modify the proposed algorithm by applying Nesterov’s accelerating scheme to get a new variant which has a better convergence rate than the first algorithm. Finally, we present preliminary numerical tests that confirm the theoretical development. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Quoc Tran Dinh & Ion Necoara & Moritz Diehl, 2014. "Path-following gradient-based decomposition algorithms for separable convex optimization," Journal of Global Optimization, Springer, vol. 59(1), pages 59-80, May.
    • Handle: RePEc:spr:jglopt:v:59:y:2014:i:1:p:59-80
    DOI: 10.1007/s10898-013-0085-7
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    References listed on IDEAS

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    1. I. Necoara & J. A. K. Suykens, 2009. "Interior-Point Lagrangian Decomposition Method for Separable Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 143(3), pages 567-588, December.
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    Cited by:

    1. Da Tian, 2015. "An exterior point polynomial-time algorithm for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 61(1), pages 51-78, May.
    2. Bingsheng He & Min Tao & Xiaoming Yuan, 2017. "Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 662-691, August.
    3. Frank E. Curtis & Arvind U. Raghunathan, 2017. "Solving nearly-separable quadratic optimization problems as nonsmooth equations," Computational Optimization and Applications, Springer, vol. 67(2), pages 317-360, June.

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