## A Unifying Investigation of Interior-Point Methods for Convex Programming (1992)

Venue: | Faculty of Mathematics and Informatics, TU Delft, NL--2628 BL |

Citations: | 5 - 4 self |

### BibTeX

@TECHREPORT{Hertog92aunifying,

author = {D. Den Hertog and C. Roos and C. Roos and T. Terlaky and T. Terlaky and F. Jarre and F. Jarre and F. Jarre},

title = {A Unifying Investigation of Interior-Point Methods for Convex Programming},

institution = {Faculty of Mathematics and Informatics, TU Delft, NL--2628 BL},

year = {1992}

}

### OpenURL

### Abstract

In the recent past a number of papers were written that present low complexity interior-point methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is self-concordant, and that the analyses of interiorpoint methods for these programs can thus be reduced to the analysis of interior-point methods with self-concordant barrier functions. Key words: interior-point method, barrier function, dual geometric programming, (extended) entropy programming, primal and dual l p -programming, relative Lipschitz condition, scaled Lipschitz condition, self-concordance. 1 Introduction The efficiency of a barrier method for solving convex programs strongly depends on the properties of the barrier function used. A key property that is sufficient to prove fast convergence for barrier methods is the property of self-concordance introduced in [17]. This condition not only allows a proof of polynomial convergen...

### Citations

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Citation Context ... This observation can be used to prove that for the classes of problems considered in this paper not only the logarithmic barrier function but also the center function of Huard [8] (also used in e.g. =-=[21, 9, 10, 6]-=-) is self-concordant. 3 The dual geometric programming problem Let fI k g k=1;\Delta\Delta\Delta;r be a partition of f1; \Delta \Delta \Delta ; ng (i.e. [ r k=1 I k = f1; \Delta \Delta \Delta ; ng and... |

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Citation Context ...is bounded by 2. The associated norm to measure the relative change is given by r 2 '(x), i.e. for h 2 IR n the norm associated with the point x is khk r 2 '(x) := (h T r 2 '(x)h) 1=2 . (See [10] and =-=[6]-=- for example, where also a brief analysis is given, showing that the property of self-concordance of the barrier function of a convex program is sufficient to prove polynomial convergence. A more deta... |

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Citation Context ...\Gamma P n i=1 ln x i : This observation can be used to prove that for the classes of problems considered in this paper not only the logarithmic barrier function but also the center function of Huard =-=[8]-=- (also used in e.g. [21, 9, 10, 6]) is self-concordant. 3 The dual geometric programming problem Let fI k g k=1;\Delta\Delta\Delta;r be a partition of f1; \Delta \Delta \Delta ; ng (i.e. [ r k=1 I k =... |

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Citation Context ...rams strongly depends on the properties of the barrier function used. A key property that is sufficient to prove fast convergence for barrier methods is the property of self-concordance introduced in =-=[17]-=-. This condition not only allows a proof of polynomial convergence, but numerical experiments in [1, 11, 14] and others further indicate that numerical algorithms based on self-concordant barrier func... |

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Citation Context ...of r 2 ' is bounded by 2. The associated norm to measure the relative change is given by r 2 '(x), i.e. for h 2 IR n the norm associated with the point x is khk r 2 '(x) := (h T r 2 '(x)h) 1=2 . (See =-=[10]-=- and [6] for example, where also a brief analysis is given, showing that the property of self-concordance of the barrier function of a convex program is sufficient to prove polynomial convergence. A m... |

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Citation Context ...fI k g k=1;\Delta\Delta\Delta;r be a partition of f1; \Delta \Delta \Delta ; ng (i.e. [ r k=1 I k = f1; \Delta \Delta \Delta ; ng and I k " I l =6O for k 6= l). The dual geometric programming pro=-=blem [2]-=- is then given by (DGP) 8 ? ? ? ? ! ? ? ? ? : min c T x + P r k=1 h P i2I k x i ln x i \Gamma i P i2I k x i j ln i P i2I k x i ji Ax = b xs0; where A is an m \Theta n matrix and c and b are n- and m-d... |

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Citation Context ...ems for which polynomiality can be proved. (In [12] only a convergence analysis is given.) Moreover, we show that some other smoothness conditions used in the literature (relative Lipschitz condition =-=[9, 7]-=-, scaled Lipschitz condition [25, 13], Monteiro and Adler's condition [16]) are also covered by this self-concordance condition. These observations allow a unification of the analyses of interior-poin... |

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Citation Context ... of f1; \Delta \Delta \Delta ; mg (i.e. [ r k=1 I k = f1; \Delta \Delta \Delta ; mg and I k " I l =6O for k 6= l). Let p is1, i = 1; \Delta \Delta \Delta ; m. Then the primal l p -programming pro=-=blem [18, 22]-=- can be formulated as (PL p ) 8 ? ! ? : max j T x P i2I k 1 p i ja T i x \Gamma c i j p i + b T k x \Gamma d ks0; k = 1; \Delta \Delta \Delta ; r; where (for all i and k) a i , b k , and j are n-dimen... |

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Citation Context ...et the rows of a matrix A be a i , i = 1; \Delta \Delta \Delta ; m, and the rows of a matrix B be b k , k = 1; \Delta \Delta \Delta ; r. Then, the dual of the l p -programming problem (PL p ) is (see =-=[18]-=---[22]) (DL p ) 8 ? ? ? ? ! ? ? ? ? : min c T y + d T z + P r k=1 z k P i2I k 1 q i fi fi fi y i z k fi fi fi q i A T y +B T z = j zs0: (If y i 6= 0 and z k = 0, then z k jy i =z k j q i is defined as... |

9 |
A Path-Following Algorithm for a Class of Convex Programing Problems, Zeitschrift fur
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Citation Context ...proved. (In [12] only a convergence analysis is given.) Moreover, we show that some other smoothness conditions used in the literature (relative Lipschitz condition [9, 7], scaled Lipschitz condition =-=[25, 13]-=-, Monteiro and Adler's condition [16]) are also covered by this self-concordance condition. These observations allow a unification of the analyses of interior-point methods for a number of convex prob... |

8 |
The logarithmic potential method for solving linear programming problems
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Citation Context ...practical interest and effectively exploit the structure of the underlying problem. A well-known barrier function for solving convex programs is the logarithmic barrier function, introduced by Frisch =-=[4]-=- and Fiacco and McCormick [3]. To describe the logarithmic barrier function more precisely, we will first give a general form for the classes of problems considered in this paper: (CP) 8 ? ? ? ? ! ? ?... |

7 |
On the classical logarithmic barrier method for a class of smooth convex programming problems
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- 1992
(Show Context)
Citation Context ...ems for which polynomiality can be proved. (In [12] only a convergence analysis is given.) Moreover, we show that some other smoothness conditions used in the literature (relative Lipschitz condition =-=[9, 7]-=-, scaled Lipschitz condition [25, 13], Monteiro and Adler's condition [16]) are also covered by this self-concordance condition. These observations allow a unification of the analyses of interior-poin... |

6 |
On Interior–Point Algorithms for Some Entropy Optimization Problems
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Citation Context ...In Section 2 we give the definition of self-concordance and state some basic lemmas about self-concordant functions. In Sections 3 - 6 we prove self-concordance for the classes of problems treated in =-=[5, 12, 23]-=-, and in Section 7 we show that the smoothness conditions used in [7, 9, 13, 16, 25] imply self-concordance of the barrier function. 2 Some general composition rules Let us first give the precise defi... |

6 |
A second order affine scaling algorithm for the geometric programming dual with logarithmic barrier
- Kortanek, NO
(Show Context)
Citation Context ...ometric programming and dual l p -programming no complexity results are known in the literature, these self-concordance proofs enlarge the class of problems for which polynomiality can be proved. (In =-=[12]-=- only a convergence analysis is given.) Moreover, we show that some other smoothness conditions used in the literature (relative Lipschitz condition [9, 7], scaled Lipschitz condition [25, 13], Montei... |

5 |
On implementing Mehrotra's predictor-corrector interior-point method for linear programming
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- 1992
(Show Context)
Citation Context ...nt to prove fast convergence for barrier methods is the property of self-concordance introduced in [17]. This condition not only allows a proof of polynomial convergence, but numerical experiments in =-=[1, 11, 14]-=- and others further indicate that numerical algorithms based on self-concordant barrier functions are of practical interest and effectively exploit the structure of the underlying problem. A well-know... |

4 |
Optimization over the positive definite cone: interior point methods and combinatorial applications
- Alizadeh
- 1992
(Show Context)
Citation Context ...nt to prove fast convergence for barrier methods is the property of self-concordance introduced in [17]. This condition not only allows a proof of polynomial convergence, but numerical experiments in =-=[1, 11, 14]-=- and others further indicate that numerical algorithms based on self-concordant barrier functions are of practical interest and effectively exploit the structure of the underlying problem. A well-know... |

4 |
A polynomial barrier algorithm for linearly constrained convex programming problems
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- 1993
(Show Context)
Citation Context ...proved. (In [12] only a convergence analysis is given.) Moreover, we show that some other smoothness conditions used in the literature (relative Lipschitz condition [9, 7], scaled Lipschitz condition =-=[25, 13]-=-, Monteiro and Adler's condition [16]) are also covered by this self-concordance condition. These observations allow a unification of the analyses of interior-point methods for a number of convex prob... |

4 |
An Extension of Karmarkar Type Algorithms to a Class of Convex Separable Programming Problems with Global Rate of Convergence
- Monteiro, Adler
- 1990
(Show Context)
Citation Context ...sis is given.) Moreover, we show that some other smoothness conditions used in the literature (relative Lipschitz condition [9, 7], scaled Lipschitz condition [25, 13], Monteiro and Adler's condition =-=[16]-=-) are also covered by this self-concordance condition. These observations allow a unification of the analyses of interior-point methods for a number of convex problems. The article is divided in three... |

4 |
Convergence property of the Iri-Imai algorithm for some smooth convex programming problems
- Zhang
- 1994
(Show Context)
Citation Context ...tax T r 2 f(x)\Deltax; which is exactly relation (1). Q:E:D: Before we conclude this work we would like to briefly point out a class of problems considered by Mehrotra and Sun [15] (and also by Zhang =-=[24]-=-) which does not have a self-concordant logarithmic barrier function. Mehrotra and Sun introduced a curvature constraint of the following form: There exists a numbers1 such that for all x, y and h in ... |

3 |
Practical Aspects of an Interior-Point Method for Convex Programming
- Jarre, Saunders
- 1991
(Show Context)
Citation Context ...nt to prove fast convergence for barrier methods is the property of self-concordance introduced in [17]. This condition not only allows a proof of polynomial convergence, but numerical experiments in =-=[1, 11, 14]-=- and others further indicate that numerical algorithms based on self-concordant barrier functions are of practical interest and effectively exploit the structure of the underlying problem. A well-know... |

3 |
An Interior–Point Algorithm for Solving Entropy Optimization Problems with Globally Linear and Locally Quadratic Convergence Rate, Working Paper Series No
- Ye, Potra
- 1990
(Show Context)
Citation Context ...In Section 2 we give the definition of self-concordance and state some basic lemmas about self-concordant functions. In Sections 3 - 6 we prove self-concordance for the classes of problems treated in =-=[5, 12, 23]-=-, and in Section 7 we show that the smoothness conditions used in [7, 9, 13, 16, 25] imply self-concordance of the barrier function. 2 Some general composition rules Let us first give the precise defi... |

1 |
An Interior Point Algorithm for Solving
- Mehrotra, Sun
- 1990
(Show Context)
Citation Context ...MkX \Gamma1 \Deltaxk\Deltax T r 2 f(x)\Deltax; which is exactly relation (1). Q:E:D: Before we conclude this work we would like to briefly point out a class of problems considered by Mehrotra and Sun =-=[15]-=- (and also by Zhang [24]) which does not have a self-concordant logarithmic barrier function. Mehrotra and Sun introduced a curvature constraint of the following form: There exists a numbers1 such tha... |