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Exterior Minimum-Penalty Path-Following Methods in Semidefinite Programming

Author

Listed:
  • M. K. H. Fan

    (Georgia Institute of Technology)

  • Y. Gong

    (Georgia Institute of Technology)

Abstract

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable but convex. It covers several standard problems, such as linear and quadratic programming, and has many applications in engineering. In this paper, we introduce the notion of minimal-penalty path, which is defined as the collection of minimizers for a family of convex optimization problems, and propose two methods for solving the problem by following the minimal-penalty path from the exterior of the feasible set. Our first method, which is also a constraint-aggregation method, achieves the solution by solving a sequence of linear programs, but exhibits a zigzagging behavior around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centering and ascending steps. The global convergence of the methods is proved and their performance is illustrated by means of an example.

Suggested Citation

  • M. K. H. Fan & Y. Gong, 1999. "Exterior Minimum-Penalty Path-Following Methods in Semidefinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 100(2), pages 327-348, February.
    • Handle: RePEc:spr:joptap:v:100:y:1999:i:2:d:10.1023_a:1021782202975
    DOI: 10.1023/A:1021782202975
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    References listed on IDEAS

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    1. Y.M. Ermoliev & A.V. Kryazhimskii & A. Ruszczynski, 1995. "Constraint Aggregation Principle in Convex Optimization," Working Papers wp95015, International Institute for Applied Systems Analysis.
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