## A New Self-Dual Embedding Method for Convex Programming (2001)

Venue: | Journal of Global Optimization |

Citations: | 9 - 2 self |

### BibTeX

@TECHREPORT{Zhang01anew,

author = {Shuzhong Zhang},

title = {A New Self-Dual Embedding Method for Convex Programming},

institution = {Journal of Global Optimization},

year = {2001}

}

### OpenURL

### Abstract

In this paper we introduce a conic optimization formulation for inequality-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. Keywords: Convex Programming, Convex Cones, Self-Dual Embedding, Self-Concordant Barrier Functions. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E. 1 1

### Citations

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(Show Context)
Citation Context ...int dom f and s # int dom f # the following three statements are equivalent s = -#f(x) (7) x = -#f # (s) (8) -x T s = f(x) + f # (s). (9) The famous bi-conjugate theorem asserts (see e.g. Rockafellar =-=[12]-=-) that f ## = cl f . In particular, for the convex barrier function F (x), we simply have F ## (x) = F (x). In addition to that, Nesterov and Nemirovski [8] showed that if F (x) is a self-concordant #... |

267 |
Interior-point polynomial methods in convex programming
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- 1994
(Show Context)
Citation Context ...lowing interior point method. In order to stay within the polynomial-time complexity realm, we will need to rely on the self-concordant barrier function theory developed by Nesterov and Nemirovski in =-=[8]-=-. For that purpose, as an example, we prove in Section 4 that if all the constraints in the original problem are convex quadratic functions, then the barrier function for the self-dual embedded proble... |

223 | A Mathematical View of Interior-Point Methods in Convex Optimization
- Renegar
- 2001
(Show Context)
Citation Context ...it will only a#ect C by a constant factor, where C is termed the parameter of the barrier by Nesterov and Nemirovski in [8], or the complexity value of the barrier function as suggested by Renegar in =-=[11]-=-. Just as the definition of ordinary convexity, self-concordancy is a line-property, i.e., the definition of a self-concordant function can be restricted to any line lying in the domain. To see this, ... |

95 |
An O( √ nL)-iteration homogeneous and self-dual linear programming algorithm
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- 1994
(Show Context)
Citation Context ...ction In this paper we propose to solve the constrained convex optimization problems by means of conic self-dual embedding. The original self-dual embedding method was proposed by Ye, Todd and Mizuno =-=[16]-=- for linear programming. The advantage of this method is twofold. First, it has a strong theoretical appeal, since it displays, and makes use of, the symmetricity of the primal-dual relationship in li... |

59 | A simplified homogeneous self-dual linear programming algorithm and its implementation
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- 1996
(Show Context)
Citation Context ...amming. For conventional convex programming with inequality constraints, Andersen and Ye [1, 2] developed a di#erent type of self-dual embedding model based on the simplified model of Xu, Hung and Ye =-=[14]-=- for linear programming. In fact, the method of Andersen and Ye is designed for nonlinear complementarity problems, thus more general. However, it is not exactly a self-dual embedding model due to the... |

37 | Initialization in semidefinite programming via a self-dual skew-symmetric embedding
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- 1997
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Citation Context ...al problem, or obtains a Farkas type certificate to conclude that the original problem is unsolvable. This technique was independently extended by Luo, Sturm and Zhang [7], De Klerk, Roos and Terlaky =-=[5]-=-, and Potra and Sheng [10] to solve semidefinite programming. In fact, the extension of Luo, Sturm and Zhang [7] allowed for a more general conic optimization framework. In this section we shall brief... |

36 | On homogeneous interior-point algorithms for semidefinite programming
- Potra, Sheng
- 1998
(Show Context)
Citation Context ...Farkas type certificate to conclude that the original problem is unsolvable. This technique was independently extended by Luo, Sturm and Zhang [7], De Klerk, Roos and Terlaky [5], and Potra and Sheng =-=[10]-=- to solve semidefinite programming. In fact, the extension of Luo, Sturm and Zhang [7] allowed for a more general conic optimization framework. In this section we shall briefly introduce this method. ... |

35 | Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems - Nesterov, Todd, et al. - 1999 |

32 | On a homogeneous algorithm for the monotone complementarity problem
- Andersen, Ye
- 1999
(Show Context)
Citation Context ...ction 2. The software package of Jos Sturm, SeDuMi, uses the self-dual embedding model for symmetric cone programming. For conventional convex programming with inequality constraints, Andersen and Ye =-=[1, 2]-=- developed a di#erent type of self-dual embedding model based on the simplified model of Xu, Hung and Ye [14] for linear programming. In fact, the method of Andersen and Ye is designed for nonlinear c... |

18 |
Interior algorithms for linear, quadratic, and linearly constrained non linear programming
- Ye
(Show Context)
Citation Context ...1s# m (11) and A := # # # # 1 0 0 T 0 1 0 T 0 0 A # # # # # # (m+2)(n+2) . (12) Let K = cl {x | p > 0, q - pf(x/p) # 0} # # n+2 , (13) which is a closed cone. The lemma below, which was used by Ye in =-=[15]-=-, shows that it is also convex. For completeness, we provide a proof here as well. 7 Lemma 3.1 The function -q + pf(x/p) is convex in # 1 ++s# 1s# n . Proof. We need only to show that pf(x/p) is conve... |

17 | S.: Duality results for conic convex programming
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- 1997
(Show Context)
Citation Context ...duality pair, (P ) and (D), enjoys a nice symmetric relationship, similar as in the case of linear programming where K = K # = # n + . For a detailed account on the subject, one is referred to either =-=[6]-=- or [13]. Take any x 0 # int K, s 0 # int K # , and y 0 # # m . Moreover, define r p = b - Ax 0 , r d = s 0 - c +A T y 0 , and r g = 1 + c T x 0 - b T y 0 . Consider the following embedded optimizatio... |

14 | A computational study of the homogeneous algorithm for large-scale convex optimization
- Andersen, Ye
- 1998
(Show Context)
Citation Context ...ction 2. The software package of Jos Sturm, SeDuMi, uses the self-dual embedding model for symmetric cone programming. For conventional convex programming with inequality constraints, Andersen and Ye =-=[1, 2]-=- developed a di#erent type of self-dual embedding model based on the simplified model of Xu, Hung and Ye [14] for linear programming. In fact, the method of Andersen and Ye is designed for nonlinear c... |

12 |
Theory and algorithms of semidefinite programming
- Sturm
- 2000
(Show Context)
Citation Context ... pair, (P ) and (D), enjoys a nice symmetric relationship, similar as in the case of linear programming where K = K # = # n + . For a detailed account on the subject, one is referred to either [6] or =-=[13]-=-. Take any x 0 # int K, s 0 # int K # , and y 0 # # m . Moreover, define r p = b - Ax 0 , r d = s 0 - c +A T y 0 , and r g = 1 + c T x 0 - b T y 0 . Consider the following embedded optimization model ... |

8 | A Primal-Dual Decomposition Algorithm for Multistage Stochastic - Berkelaar, Gromicho, et al. - 2005 |

1 |
private conversation
- Brinkhuis
- 2001
(Show Context)
Citation Context ... log(u - vf # (s/v)). Q.E.D. (I like to thank Jan Brinkhuis of Erasmus University for pointing out to me the form of the dual cone K # as described in the above theorem, during a private conversation =-=[4]-=-.) Now we consider the conic form self-dual embedding path following scheme as stipulated by Equation (10) for (CCP ), where the data of the problem, ( A, b,sc), is given by (11) and (12), and the bar... |

1 |
Conic convex programming and self-dual
- Luo, Sturm, et al.
- 2000
(Show Context)
Citation Context ...s extended to solve more general constrained convex optimization problems in two di#erent ways. For conically constrained convex optimization, including semidefinite programming, Luo, Sturm and Zhang =-=[7]-=- proposed a self-dual embedding model; for more details and an overview, see Section 2. The software package of Jos Sturm, SeDuMi, uses the self-dual embedding model for symmetric cone programming. Fo... |