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    A strategy of global convergence for the affine scaling algorithm for convex semidefinite programming

    Access Status
    Fulltext not available
    Authors
    Qian, X.
    Liao, L.
    Sun, Jie
    Date
    2018
    Type
    Journal Article
    
    Metadata
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    Citation
    Qian, X. and Liao, L. and Sun, J. 2018. A strategy of global convergence for the affine scaling algorithm for convex semidefinite programming. Mathematical Programming.
    Source Title
    Mathematical Programming
    DOI
    10.1007/s10107-018-1314-0
    ISSN
    0025-5610
    School
    School of Electrical Engineering, Computing and Mathematical Science (EECMS)
    URI
    http://hdl.handle.net/20.500.11937/72064
    Collection
    • Curtin Research Publications
    Abstract

    The affine scaling algorithm is one of the earliest interior point methods developed for linear programming. This algorithm is simple and elegant in terms of its geometric interpretation, but it is notoriously difficult to prove its convergence. It often requires additional restrictive conditions such as nondegeneracy, specific initial solutions, and/or small step length to guarantee its global convergence. This situation is made worse when it comes to applying the affine scaling idea to the solution of semidefinite optimization problems or more general convex optimization problems. In (Math Program 83(1–3):393–406, 1998), Muramatsu presented an example of linear semidefinite programming, for which the affine scaling algorithm with either short or long step converges to a non-optimal point. This paper aims at developing a strategy that guarantees the global convergence for the affine scaling algorithm in the context of linearly constrained convex semidefinite optimization in a least restrictive manner. We propose a new rule of step size, which is similar to the Armijo rule, and prove that such an affine scaling algorithm is globally convergent in the sense that each accumulation point of the sequence generated by the algorithm is an optimal solution as long as the optimal solution set is nonempty and bounded. The algorithm is least restrictive in the sense that it allows the problem to be degenerate and it may start from any interior feasible point.

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