On implementation of a self-dual embedding method for convex programming

Johnny Tak Wai Cheng, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we implement Zhang's method [Zhang, S., 2004, Journal of Global Optimization , 29, 479-496], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of self-dual embedding and central path-following for solving the resulting conic optimization model. A crucial advantage of the approach is that no initial solution is required and the method is particularly suitable when the feasibility status of the problem is unknown. In our implementation, we use a merit function approach proposed by Andersen and Ye [Andersen, E.D. and Ye, Y., 1998, Computational Optimization and Applications , 10, 243-269] to determine the step-size along the search direction. We evaluate the efficiency of the proposed algorithm by observing its performance on some test problems, which include logarithmic functions, exponential functions and quadratic functions in the constraints. Furthermore, we consider in particular the geometric programming and L p -programming problems. Numerical results of our algorithm on these classes of optimization problems are reported. We conclude that the algorithm is stable, efficient and easy-to-use in general. As the method allows the user to freely select the initial solution, it is natural to take advantage of this and apply the so-called warm-start strategy, whenever the data of a new problem is not too much different from a previously solved problem. This strategy turns out to be effective, according to our numerical experience.

Original languageEnglish (US)
Pages (from-to)75-103
Number of pages29
JournalOptimization Methods and Software
Volume21
Issue number1
DOIs
StatePublished - Feb 2006

Bibliographical note

Funding Information:
The research was supported by Hong Kong RGC Earmarked Grant CUHK4233/01E.

Keywords

  • Conic optimization
  • Convex programs
  • Path-following
  • Self-dual embedding

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