Abstract
In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.
Similar content being viewed by others
References
Anderson, E.D. and Ye, Y. (1997), A computational study of the homogeneous algorithm for large-scale convex optimization, Working Paper.
Anderson, E.D. and Ye, Y. (1999), On a homogeneous algorithm for monotone complementary problem, Mathematical Programming, 84, 375–399.
Berkelaar, A., Kouwenberg, R. and Zhang, S. (2000), A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming. Technical Report, SEEM2000-07, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, (Submitted for publication).
Brinkhius, J. (2001), private conversation, Erasmus University Rotterdam.
Glineur, F. (2001), Topics in Convex Optimization: Interior-Point Methods, Conic Duality and Approximations, Ph.D. Thesis, Faculté Polytechnique de Mons, Belgium.
De Klerk, E., Roos, C. and Terlaky, T. (1997), Initialization in semidefinite programming via a self-dual skew-symmetric embedding, Operations Research Letters, 20, 213–221.
Luo, Z.-Q., Sturm, J.F. and Zhang, S. (1997), Duality results for conic convex programming, Technical Report 9719/A, Econometric Institute, Eramus University Rotterdam, The Netherlands.
Luo, Z.-Q., Sturm, J.F. and Zhang, S. (2000), Conic convex programming and self-dual embedding, Optimization Methods and Software, 14, 169–218.
Nesterov, Yu. and Nemirovsky, A. (1994), Interior point polynomial methods in convex programming, Studies in Applied Mathematics 13, SIAM, Philadelphia, PA.
Nesterov, Yu., Todd, M.J. and Ye, Y. (1999), Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems, Mathematical Programming, 84, 227–267.
Potra, F.A. and Sheng, R. (1998), On homogeneous interior-point algorithms for semidefinite programming, Optimization Methods and Software, 9, 161–184.
Renegar, J. (2001), A mathematical view of interior-point methods in convex optimization, MPS-SIAM Series on Optimization.
Rockafellar, R.T. (1970), Convex Analysis, Princeton University Press, Princeton.
Sturm, J.F. (2000), Theory and algorithms for semidefinite programming, in Frenk et al. (eds.), High Performance Optimization, Kluwer Academic Publishers.
Xu, X., Hung, P.F. and Ye, Y. (1996), A simplified homogeneous self-dual linear programming algorithm and its implementation. Annals of Operations Research, 62, 151–171.
Ye, Y. (1987), Interior Algorithms for Linear, Quadratic and Linearly Constrained Convex Programming, Ph.D Thesis, Stanford University.
Ye, Y., Todd, M.J. and Mizuno, S. (1994), An \({\mathcal{O}}\left( {\sqrt n L} \right)\)-iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research, 19, 53–67.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhang, S. A New Self-Dual Embedding Method for Convex Programming. Journal of Global Optimization 29, 479–496 (2004). https://doi.org/10.1023/B:JOGO.0000047915.10754.a1
Issue Date:
DOI: https://doi.org/10.1023/B:JOGO.0000047915.10754.a1