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    The Convergent Generalized Central Paths for Linearly Constrained Convex Programming

    267699.pdf (313.7Kb)
    Access Status
    Open access
    Authors
    Qian, X.
    Liao, L.
    Sun, Jie
    Zhu, H.
    Date
    2018
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Qian, X. and Liao, L. and Sun, J. and Zhu, H. 2018. The Convergent Generalized Central Paths for Linearly Constrained Convex Programming. SIAM Journal on Optimization. 28 (2): pp. 1183-1204.
    Source Title
    SIAM Journal on Optimization
    DOI
    10.1137/16M1104172
    ISSN
    1052-6234
    School
    School of Electrical Engineering, Computing and Mathematical Science (EECMS)
    Funding and Sponsorship
    http://purl.org/au-research/grants/arc/DP160102819
    URI
    http://hdl.handle.net/20.500.11937/69781
    Collection
    • Curtin Research Publications
    Abstract

    The convergence of central paths has been a focal point of research on interior point methods. Quite detailed analyses have been made for the linear case. However, when it comes to the convex case, even if the constraints remain linear, the problem is unsettled. In [Math. Program., 103 (2005), pp. 63–94], Gilbert, Gonzaga, and Karas presented some examples in convex optimization, where the central path fails to converge. In this paper, we aim at finding some continuous trajectories which can converge for all linearly constrained convex optimization problems under some mild assumptions. We design and analyze a class of continuous trajectories, which are the solutions of certain ordinary differential equation (ODE) systems for solving linearly constrained smooth convex programming. The solutions of these ODE systems are named generalized central paths. By only assuming the existence of a finite optimal solution, we are able to show that, starting from any interior feasible point, (i) all of the generalized central paths are convergent, and (ii) the limit point(s) are indeed the optimal solution(s) of the original optimization problem. Furthermore, we illustrate that for the key example of Gilbert, Gonzaga, and Karas, our generalized central paths converge to the optimal solutions.

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