## Complexity of Convex Optimization Using Geometry-Based Measures and a Reference Point (2002)

Citations: | 16 - 7 self |

### BibTeX

@MISC{Freund02complexityof,

author = {Robert M. Freund},

title = {Complexity of Convex Optimization Using Geometry-Based Measures and a Reference Point},

year = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

Our concern lies in solving the following convex optimization problem: G P : minimize x c where P is a closed convex subset of the n-dimensional vector space X. We bound the complexity of computing an almost-optimal solution of G P in terms of natural geometry-based measures of the feasible region and the level-set of almost-optimal solutions, relative to a given that might be close to the feasible region and/or the almost-optimal level set. This contrasts with other complexity bounds for convex optimization that rely on data-based condition numbers or algebraic measures, and that do not take into account any a priori reference point information. AMS Subject Classification: 90C, 90C05, 90C60 Keywords: Convex Optimization, Complexity, Interior-Point Method, Barrier Method This research has been partially supported through the Singapore-MIT Alliance. Portions of this research were undertaken when the author was a Visiting Scientist at Delft University of Technology.

### Citations

1139 |
Geometric Algorithms and Combinatorial Optimization
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(Show Context)
Citation Context ...easy to see that a suitably designed version of the ellipsoid method would compute a feasible solution of GP in O (n 2 ln(˜g)) iterations, under the presumption that the norm �·� is ellipsoidal. (See =-=[4]-=- for an expository treatment of the ellipsoid algorithm.) In the more typical context in continuous optimization where we do not have an a priori bound on the distance from the feasible region to the ... |

137 | A Mathematical View of Interior-Point Methods in Convex Optimization. MPS/SIAMSeries on Optimization. Society forIndustrialand AppliedMathematics
- Renegar
- 2001
(Show Context)
Citation Context ...We employ the basic theoretical machinery of interior-point methods in our analysis using the theory of self-concordant barrier functions as articulatedsGEOMETRY-BASED COMPLEXITY 9 in Renegar [7] and =-=[8]-=-, based on the theory of self-concordant functions of Nesterov and Nemirovskii [5]. The barrier method is essentially designed to approximately solve a problem of the form OP : ˆz =min{¯c T w | w ∈ S}... |

86 | programming, complexity theory and elementary functional analysis
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- 1995
(Show Context)
Citation Context ...that we will use to solve GP is the barrier method based on the theory of self-concordant barriers, and we presume that the reader has a general familiarity with this topic as developed in [5] and/or =-=[7]-=-, for example. We therefore assume that we have a ϑP -self-concordant barrier FP (·) for P . We also assume that we have a ϑ� �-self-concordant barrier F� �(·) for the unit ball: B(0, 1) = {x |�x� ≤1}... |

71 | Some perturbation theory for linear programming - Renegar - 1994 |

47 | Sonie characterizations and properties of the distance to ill-posedness and the condition rrieasiire of a conic liriear sysytei~i." Tcchniçal Report. Sloan School of 'çlanagenient. 4IIT
- Freund, Vera
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(Show Context)
Citation Context ...� 1 − v T v � is a ϑ� �= 1-self-concordant barrier for the unit ball. Let us set x r := 0 and let x 0 ∈ intC be given, and assume that we have rescaled x 0 so that �x 0 � = 1. Then from Theorem 17 of =-=[3]-=-, it follows that g will satisfy: �x g ≤ 3C(d) 0� dist(x0 ,∂C) and from Theorem 1.1 and Lemma 3.2 of [6] it follows that Dɛ ≤C(d) 2 ɛ + C(d) �c�∗ where C(d) is defined here using (2). Then under the h... |

38 | Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm
- Freund, Vera
- 1999
(Show Context)
Citation Context ...the reference point, there is a natural projective transformation of the problem for which the ellipsoid algorithm will compute a feasible solution of GP in O (n 2 ln(g)) iterations, see Lemma 4.1 of =-=[2]-=-. Therefore g is a very relevant geometric measure for the Phase-I problem in the context of the ellipsoid algorithm. Herein, we will see that g is also relevant for the complexity of the Phase-I prob... |

38 | Warm-start strategies in interior-point methods for linear programming
- Yildirim, Wright
(Show Context)
Citation Context ... truly viable warm-start algorithm for GP whose iteration bound would be small to the extent that the starting point is close to the set of ɛ-optimal solutions. The recent work of Yildirim and Wright =-=[10]-=- is a promising step in this direction for the case of linear programming.sGEOMETRY-BASED COMPLEXITY 8 1.5.3 Relation to Condition-Number based Complexity Bounds In this subsection we indicate how the... |

27 |
Toward probabilistic analysis of interior-point algorithms for linear programming
- Ye
- 1994
(Show Context)
Citation Context ...exity of the Phase-I problem for a suitably constructed interior-point algorithm. (3) (4)sGEOMETRY-BASED COMPLEXITY 5 The geometric measure g is also a generalization of the condition measure σ of Ye =-=[9]-=- for linear programming, which has been used in linear programming complexity analysis. The linear programming feasibility problem in standard form is to find a solution x of the system: LF : Ax = b x... |

16 | Some characterizations and properties of the "distance to ill-posedness" and the condition measure of a conic linear system - Freund, Vera - 1999 |

9 | On the primal-dual geometry of level sets in linear and conic optimization
- Freund
- 2003
(Show Context)
Citation Context ...s the maximum distance to the ɛ-optimal solution set (which would be infinite in this case) rather than thesGEOMETRY-BASED COMPLEXITY 6 minimum distance (which would be finite in this case). Also, in =-=[1]-=- in the case of conic optimization with x r = 0, it is shown that Dɛ defined using (7) is inversely proportional to the size of the largest ball contained in the level sets of the dual problem, and so... |