## "Cone-free" primal-dual path-following and potential reduction polynomial time interior-point methods (2005)

Venue: | MATH. PROG |

Citations: | 1 - 1 self |

### BibTeX

@ARTICLE{Nemirovski05"cone-free"primal-dual,

author = {Arkadi Nemirovski and Levent Tunçel},

title = {"Cone-free" primal-dual path-following and potential reduction polynomial time interior-point methods},

journal = {MATH. PROG},

year = {2005},

volume = {102},

pages = {261--294}

}

### OpenURL

### Abstract

We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primal-dual interior-point techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.

### Citations

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Citation Context ...etter than the “standard” O( √ m + n)-complexity bound. 7.2. Long steps. We consider three related viewpoints: (a) α-regularity of a s.c.b. [17]; (b) convexity of the “gradient product” 〈−H ′ (x), y〉 =-=[18, 19]-=-; (c) β-normality of a s.c.b. [13]. All of these properties are strengthenings of the fundamental property of the self-concordant barriers which states that the Hessian of a s.c.b. behaves very well i... |

179 |
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Citation Context ...) � H ′′ (x − th) � 1 [1 − tσx(h)] for every x ∈ int K, h ∈ E and t ∈ [0, 1/σx(h)) . This property was proven via establishing the convexity of the function 〈−H ′ (x), y〉 : int K → R, for every y ∈ K =-=[18]-=-. Later, this property was extended to all hyperbolic barriers [8]. (c) f is β-normal if for every x, z ∈ Q, r ≡ πQ,x(z − x) < 1 implies (1 − r) β � � � � � � 2 d � 2 � d � f(x + th) ≤ � 1 f(z + th) ≤... |

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Citation Context ...reserves 1., 2. and 3. above? In this last section, we comment on the above issues. 7.1. Potential applications. Geometric Programming provides an interesting class of applications (for a survey, see =-=[5]-=-; for a set of test problems see [4]; interesting recent applications in Engineering are presented in [2]). We have seen in the Introduction that this problem class fits our primal-dual framework. At ... |

39 | Hyperbolic polynomials and interior point methods for convex programming
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Citation Context ...is 2-regular by the results of [17], it follows that − ln Det(x − uu u x T ) is also 2-regular for its domain. Actually, it is now known that all hyperbolic barriers are 2-regular (see Theorem 4.2 of =-=[8]-=-). The above fact can also be easily obtained using an affine restriction of this theorem. As a final remark on α-regularity, we note that this property behaves very nicely under the symmetries of the... |

37 |
Y.Ye, "An infeasible interior-point algorithm for solving primal and dual geometric programs
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Citation Context ...framework. At the same time, it is not directly covered by the existing primal-dual polynomial time algorithms. Moreover, the only previous primal-dual interior-point method for Geometric Programming =-=[11]-=-, although globally convergent, is not known to be a polynomial time one. Note that, essentially, the only feature of Geometric Programming which is responsible for the possibility to process this cla... |

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Citation Context ...chniques as given in [16] can be applied. We could also apply the surface-following idea developed in [17]. However, a particularly attractive choice would be an effective analogue of the approach of =-=[21]-=-. Such analogues seem possible and the development of such techniques is left for future work. Appendix A. 3) The idea to solve the problem by tracing the primal path is, of course, a common place. Th... |

31 | On the Riemannian Geometry Defined by self-Concordant
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(Show Context)
Citation Context ...tandard conic ones to justify “enforcement” of the standard techniques. It should be added that, at the time of this writing, there is neither clear theoretical reasons (perhaps with the exception of =-=[20]-=-) nor computational experience in favour of the standard primal-dual interiorpoint techniques beyond the scope of problems on self-scaled cones, i.e., beyond the scope of linear, conic, quadratic, and... |

25 |
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(Show Context)
Citation Context ...ze whether a given primal-dual pair (ξ, y) is close to a given “target pair” (ξ∗(t), y∗(t)) on the primal-dual central path. This allows for theoretically valid long-step path-following policies (see =-=[15]-=-). In contrast to this, in the “purely primal” framework it seems to be impossible to realize, at a low computational cost, whether a given primal solution x is close to a given target point x∗(t) on ... |

21 |
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Citation Context ...veloped to get new families of polynomial time interior-point methods for Entropy Optimization, an important problem class which, in particular, has very interesting applications in graph theory (see =-=[3, 10]-=-). (At the moment, there exists just one dedicated polynomial time algorithm for Entropy Minimization [22]). Another application worth mentioning is minimization of conic combinations of p-norms (this... |

21 |
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Citation Context ...ed barriers is ideal for the strongest use of primal-dual symmetry in interior-point algorithms. However, taking all of these nice properties beyond symmetric cones is not possible (see, for instance =-=[23]-=-). In most applications, the importance of generating good bounds (via good dual feasible solutions) on the optimal objective value of the problem at hand cannot be denied. In the self-scaled case, th... |

17 |
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(Show Context)
Citation Context ..., q} with properly chosen A, b and p = dim h + m, q = mL. Now, the set cl D admits the following (p + 2q)self-concordant barrier (1.11) Φ(t, y, s) = − p� ln ti − i=1 q� [ln(ln(si) − yi) + ln si] (see =-=[16]-=-, Section 5.3.2). One can easily compute the Legendre transformation of Φ: (1.12) Φ∗(τ, η, σ) = −(p + 2q) − p� ln(−τi) − i=1 i=1 q� � � � � −σi (ηi + 1) ln + ln ηi + ηi ηi + 1 (from now on, unless sta... |

17 | Generalization of primal-dual interior-point methods to convex optimization problems in conic form
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(Show Context)
Citation Context ...ns for both primal and the dual problems. Some properties of primal-dual joint scaling interior-point methods have been generalized and extended to all convex optimization problems in conic form (see =-=[24]-=-). We can use analogous search directions in our set-up as well. An important advantage of the current set-up is that when we are in the “Complete Formulation Case”, the primal and the dual paths are ... |

16 |
Entropy splitting for antiblocking corners and perfect graphs
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(Show Context)
Citation Context ...veloped to get new families of polynomial time interior-point methods for Entropy Optimization, an important problem class which, in particular, has very interesting applications in graph theory (see =-=[3, 10]-=-). (At the moment, there exists just one dedicated polynomial time algorithm for Entropy Minimization [22]). Another application worth mentioning is minimization of conic combinations of p-norms (this... |

13 | An efficient algorithm for minimizing a sum of p-norms
- Xue, Ye
- 1998
(Show Context)
Citation Context ...y Minimization [22]). Another application worth mentioning is minimization of conic combinations of p-norms (this problem has many applications, including “p-norm multi-facility location problem” see =-=[25]-=-). In [25], Xue and Ye present an interior-point method approach to this problem. Their development however, follows the general approach of converting the given problem to conic form and homogenizing... |

12 |
A set of Geometric Programming Test Problems and their
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(Show Context)
Citation Context ...s last section, we comment on the above issues. 7.1. Potential applications. Geometric Programming provides an interesting class of applications (for a survey, see [5]; for a set of test problems see =-=[4]-=-; interesting recent applications in Engineering are presented in [2]). We have seen in the Introduction that this problem class fits our primal-dual framework. At the same time, it is not directly co... |

12 |
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(Show Context)
Citation Context ... In the above, (a) is given by (2.3.2) of [16]. (b) was first proven in [16] with a larger constant (3ϑ+1), see Proposition 2.3.2 in [16]. The better bound (ϑ + 2 √ ϑ) follows from Lemma 2.8 of Jarre =-=[9]-=-, see Lemma 3.2.1 in [12]. 2) f is bounded below on Q if and only if Q is bounded, and in this case (2.8) {y : �y − xf � f ′′ (xf ) < 1} ⊂ Q ⊂ {y : �y − xf � f ′′ (xf ) < ϑ + 2 √ ϑ}. This fact was als... |

10 |
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(Show Context)
Citation Context ...eneous barrier (for the conic hull of the origin in R × E and D) Φ + (x, t) = κ [Φ(x/t) − ϑ ln(t)] (κ is an appropriate absolute constant — for instance, 25 [Φ(x/t) − 7ϑ ln(t)] works for every Φ, see =-=[6]-=-). Note that the original barrier Φ, up to absolute constant factor, can be obtained from Φ + by an affine substitution of the argument. Further, if Φ∗(·) is available, then it is not that difficult t... |

9 | Optimal allocation of local feedback in multistage amplifiers via geometric programming
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(Show Context)
Citation Context ...ations. Geometric Programming provides an interesting class of applications (for a survey, see [5]; for a set of test problems see [4]; interesting recent applications in Engineering are presented in =-=[2]-=-). We have seen in the Introduction that this problem class fits our primal-dual framework. At the same time, it is not directly covered by the existing primal-dual polynomial time algorithms. Moreove... |

9 | Multiparameter surfaces of analytic centers and long-step path-following interior point methods
- Nesterov, Nemirovski
- 1998
(Show Context)
Citation Context ...ow reproduces in our “complete formulation case” setting the construction developed in [15] for the Standard case (and in fact it was investigated, even in a more general “surface-following” form, in =-=[17]-=-). Let us say that a triple (x ∈ D, y ∈ D + ∗ , t > 0) is close to the primal-dual path if (5.1) A ∗ y = −tc & max[λ(Ft, x), λ∗(y)] ≤ 0.1. Assume that we are given a starting triple (x0, t0, y0) 1) , ... |

8 |
A.: Interior Point Polynomial Time Methods
- Nesterov, Nemirovski
- 1994
(Show Context)
Citation Context ...x), y − x〉 + ρ(−r) (b.1) y ∈ Q, r ≡ �y − x� f ′′ (x) ⇒ f(y) ≥ f(x) + 〈f ′ (x), y − x〉 + ρ(r). (b.2) In the above, (a) is given in Theorem 2.1.1 of [16], and (b.1-2) is relation (2.4) in Lecture Notes =-=[12]-=- (a simplified version of [16] with all necessary proofs). 2) For x ∈ Q, we define the damped Newton iterate of x as For every x ∈ Q we have (2.5) x+ = x − 1 1 + λ(f, x) [f ′′ (x)] −1 f ′ (x). x+ ∈ Q ... |

7 | On the self-concordance of the universal barrier function
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(Show Context)
Citation Context ...his, α-regularity is preserved under the summation of barriers and an affine substitution of argument, see [17]. The fact that the universal barrier for a convex set is O � ϑ 2� -regular was shown in =-=[7]-=-. We note that the barrier � − ln Det � x − uu T �� with the domain � (x, u) ∈ S m × R m×n : � x − uu T � � 0 � . (see above) is also 2-regular. Indeed, we have � I 0 0 x − uuT � � I = −u � � 0 I I u ... |

7 |
A quadratically convergent polynomial algorithm for solving entropy optimization problems
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(Show Context)
Citation Context ...problem class which, in particular, has very interesting applications in graph theory (see [3, 10]). (At the moment, there exists just one dedicated polynomial time algorithm for Entropy Minimization =-=[22]-=-). Another application worth mentioning is minimization of conic combinations of p-norms (this problem has many applications, including “p-norm multi-facility location problem” see [25]). In [25], Xue... |

1 |
On normal self-concordant barriers and long-step interior-point methods
- Nemirovski
- 1997
(Show Context)
Citation Context ...complexity bound. 7.2. Long steps. We consider three related viewpoints: (a) α-regularity of a s.c.b. [17]; (b) convexity of the “gradient product” 〈−H ′ (x), y〉 [18, 19]; (c) β-normality of a s.c.b. =-=[13]-=-. All of these properties are strengthenings of the fundamental property of the self-concordant barriers which states that the Hessian of a s.c.b. behaves very well inside the Dikin ellipsoid (see SC.... |

1 |
The method for Linear Programming which requires O(n L) operations", Ekonomika i Matem. Metody v
- Nesterov
- 1988
(Show Context)
Citation Context ...Appendix A. 3) The idea to solve the problem by tracing the primal path is, of course, a common place. The idea to trace what we call here the dual path is not new either (it originates from Nesterov =-=[14]-=-; for a more general treatment, see [16], Section 3.4). What is seemingly new (beyond the scope of the Standard case, of course), is the idea to work with both of these paths simultaneously.sComputing... |

1 | On the Riemannian geometry de by self-concordant barriers and interior-point methods - Todd, J |

1 | An ecient algorithm for minimizing a sum of p norms - Xue, Ye |