## On the behavior of the homogeneous self-dual model for conic convex optimization (2004)

Venue: | MIT Operations Research |

Citations: | 3 - 1 self |

### BibTeX

@TECHREPORT{Freund04onthe,

author = {Robert M. Freund},

title = {On the behavior of the homogeneous self-dual model for conic convex optimization},

institution = {MIT Operations Research},

year = {2004}

}

### OpenURL

### Abstract

There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of ε-optimal solutions, and (ii) the maximum distance of ε-optimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stopping-rule theory for HSD-based interior-point methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the ε-optimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous self-dual model that might improve the resulting solution time in practice.

### Citations

471 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
(Show Context)
Citation Context ...e second assertion, let (x ∗ ,y ∗ ,z ∗ ,τ ∗ ,κ ∗ ,θ ∗ ) be an optimal solution of H and recall from Lemma 3.1 that (x 0 ,y 0 ,z 0 ,τ 0 ,κ 0 ,θ 0 )isfeasibleforH. Let λ = ε/¯αθ 0 , and notice that λ ∈ =-=[0, 1]-=- for 0 ≤ ε ≤ (x 0 ) T z 0 + κ 0 τ 0 =¯αθ 0 , whereby (x, y, z, τ, κ, θ) :=(1− λ)(x ∗ ,y ∗ ,z ∗ ,τ ∗ ,κ ∗ ,θ ∗ )+λ(x 0 ,y 0 ,z 0 ,τ 0 ,κ 0 ,θ 0 ) is a feasible solution of H with objective value ¯αθ = ... |

163 |
Self-scaled barriers and interior-point methods for convex programming
- Nesterov, Todd
- 1997
(Show Context)
Citation Context ...ms with common objective function value �¯v� =max{|v1|, �¯v�}, proving the result in this case. Now consider an arbitrary given w 0 ∈ intQ n . Let v =(v1, ¯v) ∈ Q n ,and consider the self-scaled (see =-=[6]-=-) barrier function f(v) :=− ln(v 2 1 − ¯v T ¯v) for v ∈ intQ n . The Hessian of f(·) isgivenby: H(v) = 1 (v2 1 − ¯vT ¯v) 2 � 2v2 1 +2¯v T ¯v −4v1¯v T −4v1¯v 2(v2 1 − ¯vT ¯v)I +4¯v¯v T � and it follows... |

85 |
An O(nL)-iteration homogenous and self-dual linear programming algorithm
- Ye, Todd, et al.
- 1994
(Show Context)
Citation Context ..., pointed, and has nonempty interior), whereby C ∗ is also a regular cone. We say that P (D) is strictly feasible if there exists ¯x ∈ intC (¯y and ¯z ∈ intC ∗ ) that is feasible for P (D). Following =-=[11]-=- (also see [10]) we consider the following homogeneous selfdual (HSD) embedding of P and D. Given initial values (x 0 ,y 0 ,z 0 ) satisfying x 0 ∈ intC, z 0 ∈ intC ∗ , as well as initial constants τ 0... |

71 | Some perturbation theory for linear programming
- Renegar
- 1994
(Show Context)
Citation Context ...h similar remarks about R D ε . Indeed, Renegar’s data-perturbation condition measure C(d) mustsatisfy C 2 (d)+C(d) ε �c�∗ ≥ R P ε for ε ≤�c�∗; this follows directly from Theorem 1.1 and Lemma 3.2 of =-=[7]-=-. A closely related measure of the behavior of P /D is the maximum distance of ε-optimal solutions from the boundary of the cone variables: r P ε := maxx dist(x, ∂C) r D ε := maxz dist∗(z,∂C ∗ ) s.t. ... |

56 | A simplified homogeneous and self-dual linear programming algorithm and its
- Xu, Hung, et al.
- 1996
(Show Context)
Citation Context ...has nonempty interior), whereby C ∗ is also a regular cone. We say that P (D) is strictly feasible if there exists ¯x ∈ intC (¯y and ¯z ∈ intC ∗ ) that is feasible for P (D). Following [11] (also see =-=[10]-=-) we consider the following homogeneous selfdual (HSD) embedding of P and D. Given initial values (x 0 ,y 0 ,z 0 ) satisfying x 0 ∈ intC, z 0 ∈ intC ∗ , as well as initial constants τ 0 > 0,κ 0 > 0,θ ... |

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
- 1995
(Show Context)
Citation Context ...e τC denotes the “min-width” constant of C: τC := max{dist(x, ∂C) :x ∈ C, �x� ≤1} ; x (5) (6)sBEHAVIOR OF HOMOGENEOUS SELF-DUAL MODEL 3 this follows directly from Theorem 1.1 of [7] and Theorem 17 of =-=[4]-=-. It is shown in [3] that r P ε and R D ε obey the following inequalities and so are nearly inversely proportional for fixed ε>0: τC · ε ≤ r P ε · R D ε ≤ 2 · ε, provided that r P ε and R D ε are both... |

37 |
A centered projective algorithm for linear programming
- Todd, Ye
- 1990
(Show Context)
Citation Context ...onal consistency. Because P (D) can be recast equivalently as the problem of minimizing a linear function of a (regular) cone variable over the intersection of the regular cone and an affine set (see =-=[9]-=-, [5]), we will focus on the behavior of the regular cone variables x and z and will effectively ignore the unrestricted variables y. One natural measure of of the behavior of P /D is the size of the ... |

21 |
Implementation of interior point methods for mixed semidefinite and second order cone optimization problems
- Sturm
(Show Context)
Citation Context ...t H is inherently well-behaved in these measures in this norm. Note also that RH ε and rH ε are linear in ε. We also develop a stopping-rule theory for HSD-based interior-point methods such as SeDuMi =-=[8]-=-. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the norms of the ε-optimal primal and dual solutions (where these norms are also defined by the starting ... |

9 | On the primal-dual geometry of level sets in linear and conic optimization
- Freund
- 2003
(Show Context)
Citation Context ...n-width” constant of C: τC := max{dist(x, ∂C) :x ∈ C, �x� ≤1} ; x (5) (6)sBEHAVIOR OF HOMOGENEOUS SELF-DUAL MODEL 3 this follows directly from Theorem 1.1 of [7] and Theorem 17 of [4]. It is shown in =-=[3]-=- that r P ε and R D ε obey the following inequalities and so are nearly inversely proportional for fixed ε>0: τC · ε ≤ r P ε · R D ε ≤ 2 · ε, provided that r P ε and R D ε are both finite and positive... |

4 |
On two measures of problem complexity and their explanatory value for the performance of SeDuMi on second-order cone problems
- Cai, Freund
- 2004
(Show Context)
Citation Context ...able 1. We also computed β for a set of 144 second-order cone problem instances that were generated specifically to have have a wide range of condition measure values C(d), intherange10 2 -10 9 , see =-=[2]-=- for details how these problems were generated. Here we observed β in the range 0.06-0.55, see Table 2, with larger values roughly corresponding to problems with larger values of R P ε + R D ε and wit... |