## Improved Complexity for Maximum Volume Inscribed Ellipsoids (2001)

Venue: | SIAM Journal on Optimization |

Citations: | 10 - 0 self |

### BibTeX

@ARTICLE{Anstreicher01improvedcomplexity,

author = {Kurt Anstreicher},

title = {Improved Complexity for Maximum Volume Inscribed Ellipsoids},

journal = {SIAM Journal on Optimization},

year = {2001},

volume = {13},

pages = {309--320}

}

### OpenURL

### Abstract

Let P = fx j Ax bg, where A is an m \Theta n matrix. We assume that P contains a ball of radius one centered at the origin, and is contained in a ball of radius R centered at the origin. We consider the problem of approximating the maximum volume ellipsoid inscribed in P . Such ellipsoids have a number of interesting applications, including the inscribed ellipsoid method for convex optimization. We reduce the complexity of finding an ellipsoid whose volume is at least a factor e \Gammaffl of the maximum possible to O(m 3:5 ln(mR=ffl)) operations, improving on previous results of Nesterov and Nemirovskii, and Khachiyan and Todd. A further reduction in complexity is obtained by first computing an approximation of the analytic center of P . Keywords: Maximum volume inscribed ellipsoid, inscribed ellisoid method. 1 1

### Citations

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Citation Context ...um volume inscribed ellipsoid, inscribed ellisoid method. 1 1 Introduction Let P = fx j Axsbg, where A is an m \Theta n matrix. We assume that P is bounded, with a nonempty interior. It is then known =-=[5]-=- that there is a unique ellipsoid E ae P of maximum volume. We say that an ellipsoid E ae P is fl-maximal if Vol(E)sfl Vol(E ), where 0 ! fl ! 1 and Vol(\Delta) denotes n-dimensional volume. In this p... |

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Citation Context ...cube to relative accuracysin O(n ln(n=)) iterations, each requiring evaluation of the function and a subgradient. The order of this complexity, also achieved by the volumetric cutting plane algorithm =-=[1, 13], is optim-=-al [11]. Another application of fl-maximal ellipsoids is to provide a "rounding" of P. It is know that for the maximum volume inscribed ellipsoid (MVIE) E , E ae P ae nE ; where for an ellip... |

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Citation Context ...lated to that of computing an e \Gammaffl --maximal inscribed ellipsoid for P is that of computing an e ffl --minimal circumscribing ellipsoid for the convex hull of m given points in ! n . Khachiyan =-=[7] show-=-s that the latter problem can be solved in O ` m 3:5 ln ` m ffl " operations; note that this bound is independent of the parameter R. Notation: If A and B are symmetric matrices, AsB denotes that... |

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Citation Context ...amma fl fl ! E: Roundings of this type are required in several contexts, including Lenstra's algorithm for integer programming in fixed dimension [9], and randomized algorithms for volume computation =-=[6]-=-. Alternative methodologies for obtaining O(n)-roundings of P include the shallow cut ellipsoid algorithm [3, Section 4.6] and the volumetric cutting plane algorithm [2]. Assume that P contains a ball... |

28 |
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Citation Context ... of interesting applications of fl-maximal ellipsoids. For example, the computation of a fl-maximal ellipsoid, with fl ? 0:92, is required on each iteration of the inscribed ellipsoid algorithm (IEM) =-=[12]-=- for convex programming. The IEM minimizes a convex function over an n-dimensional cube to relative accuracysin O(n ln(n=)) iterations, each requiring evaluation of the function and a subgradient. The... |

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Citation Context ... bounds on R may be exponential in n [3, Lemma 3.1.25]. Several novel formulations of the MVIE problem are considered in [16]. Primaldual algorithms based on two of these formulations are analyzed in =-=[17]-=-, and their numerical performance is compared to original and modified versions of the algorithm from [9] on instances up to size n = 500, m = 1200. The performance of one of the primal-dual algorithm... |

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Citation Context ...Delta) obtained using ff = 1=(1 +st (x; x; X)); see Lemma 4.3 below. Our analysis of the main stage algorithm for MVIE is based on the well-known analysis of the barrier algorithm from [10] (see also =-=[4]-=-). The following result facilitates the use of directions based on the family of barrier functions F t (y; \Delta; \Delta). 7 Lemma 4.1 Suppose that x and y are interior points of P, and let f(X) = \G... |

21 |
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Citation Context ...cube to relative accuracysin O(n ln(n=)) iterations, each requiring evaluation of the function and a subgradient. The order of this complexity, also achieved by the volumetric cutting plane algorithm =-=[1, 13], is optim-=-al [11]. Another application of fl-maximal ellipsoids is to provide a "rounding" of P. It is know that for the maximum volume inscribed ellipsoid (MVIE) E , E ae P ae nE ; where for an ellip... |

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Citation Context ...a sequence of problems, each of which requires less work per iteration than the original problem considered by [10]. A primal-dual algorithm for computing an approximation of the MVIE is described in =-=[14]. In -=-this paper we show that an e \Gammaffl --maximal inscribed ellipsoid can be computed in O ` m 3:5 ln ` mR ffl " (3) operations. We also show that by first computing an approximation of the analyt... |

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Citation Context ...orithms for volume computation [6]. Alternative methodologies for obtaining O(n)-roundings of P include the shallow cut ellipsoid algorithm [3, Section 4.6] and the volumetric cutting plane algorithm =-=[2]-=-. Assume that P contains a ball of radius one centered at the origin, and is contained in a ball of radius R centered at the origin. Using the ellipsoid algorithm an e \Gammaffl --maximal inscribed el... |

2 |
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(Show Context)
Citation Context ... a specialized "rescaling" technique to lower the work required on each iteration. A further reduction to O / m 3:5 ln ` mR ffl ' ln ` n ln R ffl ' ! (2) operations was achieved by Khachiyan=-= and Todd [8]-=-, who apply an interior-point algorithm to a sequence of problems, each of which requires less work per iteration than the original problem considered by [10]. A primal-dual algorithm for computing an... |

1 |
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(Show Context)
Citation Context ...ed by Khachiyan and Todd [9], who apply an interior-point algorithm to a sequence of problems, each of which requires less work per iteration than the original problem considered by [12]. Nemirovskii =-=[11] lowe-=-rs the complexity of obtaining an e \Gammaffl --maximal inscribed ellipsoid to O ` m 3:5 ln ` mR ffl " (3) operations. The approach taken in [11] uses Lagrangian duality to reformulate the MVIE p... |