## On the Riemannian geometry defined by self-concordant barriers and interior-point methods

Venue: | Found. Comput. Math |

Citations: | 34 - 0 self |

### BibTeX

@ARTICLE{Nesterov_onthe,

author = {Yu. E. Nesterov and M. J. Todd},

title = {On the Riemannian geometry defined by self-concordant barriers and interior-point methods},

journal = {Found. Comput. Math},

year = {},

volume = {2},

pages = {333--361}

}

### OpenURL

### Abstract

We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primal-dual central path are in some sense close to optimal. The same is true for methods that follow the shifted primal-dual central path among certain infeasible-interior-point methods. We also compute the geodesics in several simple sets. ∗ Copyright (C) by Springer-Verlag. Foundations of Computational Mathewmatics 2 (2002), 333–361.

### Citations

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(Show Context)
Citation Context ...ne of the most intensely studied methods for both linear programming and certain convex programming problems, because of their excellent computational and theoretical properties; see, e.g., Karmarkar =-=[4]-=-, Nesterov and Nemirovskii [8], M. Wright [13], and S. Wright [14]. Perhaps their greatest successes have been for linear and semidefinite programming (in the latter, a linear function is optimized ov... |

512 | Primal-Dual Interior-Point Methods
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(Show Context)
Citation Context ...ing and certain convex programming problems, because of their excellent computational and theoretical properties; see, e.g., Karmarkar [4], Nesterov and Nemirovskii [8], M. Wright [13], and S. Wright =-=[14]-=-. Perhaps their greatest successes have been for linear and semidefinite programming (in the latter, a linear function is optimized over the set of symmetric matrices satisfying linear equations and t... |

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Citation Context ...mplexity of optimization algorithms. Also, differential and Riemannian geometry has been studied in optimization in relation to the need to remain feasible – see, e.g., Tanabe [11] and Edelman et al. =-=[1]-=-. In this paper, however (at least in feasible methods), the algorithms move in the intersection of an affine manifold and the interior of the constraint set, and thus maintaining feasibility is not a... |

376 |
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Citation Context ...is a cone K and F is logarithmically homogeneous, then the function 〈F ′ ( ˆ ξ(t)), ˆ ξ ′ (t)〉 is constant in t. Proof: That ρ defines a metric on int Q is a standard result; see for example Helgason =-=[3]-=-, p. 51. The completeness is a consequence of Lemma 3.1 below, which relates ρ(y, x) to �y − x�x when either is small (see Corollary 3.1). Then part (b) follows; see Theorem 10.4 of [3]. Part (c) is a... |

87 | Interior methods for constrained optimization
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(Show Context)
Citation Context ...both linear programming and certain convex programming problems, because of their excellent computational and theoretical properties; see, e.g., Karmarkar [4], Nesterov and Nemirovskii [8], M. Wright =-=[13]-=-, and S. Wright [14]. Perhaps their greatest successes have been for linear and semidefinite programming (in the latter, a linear function is optimized over the set of symmetric matrices satisfying li... |

51 | Barrier functions in interior-point methods
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(Show Context)
Citation Context ... self-dual cones has been studied extensively in the mathematics literature, as it relates to the classification of these cones via Jordan algebras – see Koecher [6] and Rothaus [10] as well as Güler =-=[2]-=-, who brought this line of research to the attention of optimizers. But their concerns were far removed from the complexity of optimization algorithms. Also, differential and Riemannian geometry has b... |

35 | Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems
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- 1999
(Show Context)
Citation Context ...for the primal central path with respect to the cone K. 16s5.2 Shifted primal-dual central path This path arises in infeasible-start interior-point primal-dual schemes. The idea of this approach (see =-=[9]-=-) is to solve the following minimization problem: (x, y, s, τ, κ) ∈ F := min F (x) + F∗(s) − ln τ − ln κ, ⎧ ⎪⎨ ⎪⎩ Ax = rx + τb, A T y + s = rs + τc, 〈c, x〉 − 〈b, y〉 + κ = rc, x ∈ K, s ∈ K ∗ , y ∈ R m ... |

27 |
Long-step strategies in interior-point primal-dual methods
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(Show Context)
Citation Context ...barrier for the cone ˆ K is ˆF (z) := F (x) + F∗(s), z := (x, s) ∈ int ˆ K. Note that the value of parameter of this barrier is ˆν = 2ν. Let us present a well-known duality theorem (see, for example, =-=[7]-=-). Theorem 5.1 (a) If the primal-dual feasible set in (5.15) has a strictly feasible primaldual point (i.e., x ∈ int K and s ∈ int K ∗ ), then the optimal value of this problem is zero and the primal-... |

14 |
Geometric method in nonlinear programming
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(Show Context)
Citation Context ... far removed from the complexity of optimization algorithms. Also, differential and Riemannian geometry has been studied in optimization in relation to the need to remain feasible – see, e.g., Tanabe =-=[11]-=- and Edelman et al. [1]. In this paper, however (at least in feasible methods), the algorithms move in the intersection of an affine manifold and the interior of the constraint set, and thus maintaini... |

12 |
Riemannian Geometry Underlying Interior-Point Methods for
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(Show Context)
Citation Context ... suggested by short-step methods are likely to be good for taking longer steps also.) The first discussion of these ideas in relation to the complexity of optimization algorithms appears in Karmarkar =-=[5]-=- for linear programming. A different view, relating to homotopy algorithms in general, is discussed in Todd [12]. Here we consider arbitrary selfconcordant barriers and the resulting Riemannian geomet... |

9 |
Domains of positivity, Abh
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- 1960
(Show Context)
Citation Context ...eometry of homogeneous self-dual cones has been studied extensively in the mathematics literature, as it relates to the classification of these cones via Jordan algebras – see Koecher [6] and Rothaus =-=[10]-=- as well as Güler [2], who brought this line of research to the attention of optimizers. But their concerns were far removed from the complexity of optimization algorithms. Also, differential and Riem... |

8 | On adjusting parameters in homotopy methods for linear programming, Approximation Theory and Optimization
- Todd
- 1997
(Show Context)
Citation Context ...hese ideas in relation to the complexity of optimization algorithms appears in Karmarkar [5] for linear programming. A different view, relating to homotopy algorithms in general, is discussed in Todd =-=[12]-=-. Here we consider arbitrary selfconcordant barriers and the resulting Riemannian geometry, concentrating on geodesics and their relation to interior-point methods. 2sWe note that the Riemannian geome... |

4 |
Positivitätsbereiche im R n
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- 1957
(Show Context)
Citation Context ...the Riemannian geometry of homogeneous self-dual cones has been studied extensively in the mathematics literature, as it relates to the classification of these cones via Jordan algebras – see Koecher =-=[6]-=- and Rothaus [10] as well as Güler [2], who brought this line of research to the attention of optimizers. But their concerns were far removed from the complexity of optimization algorithms. Also, diff... |