scholarly journals A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods

2007 ◽  
Vol 49 (2) ◽  
pp. 259-270 ◽  
Author(s):  
Keyvan Aminis ◽  
Arash Haseli
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Point Of View ◽  
Theory And Practice ◽  
Theoretical Point ◽  
Barrier Functions ◽  
Iteration Bound ◽  
Linear Optimization Problems

AbstractInterior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.

An improved and modified infeasible interior-point method for symmetric optimization

2016 ◽  
Vol 09 (03) ◽  
pp. 1650059 ◽  
Author(s):  
Behrouz Kheirfam
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Interior Point Methods ◽  
Linear Optimization ◽  
Point Method ◽  
Semidefinite Optimization ◽  
Symmetric Optimization ◽  
Second Order Cone ◽  
Iteration Bound ◽  
Infeasible Interior Point Method

In this paper an improved and modified version of full Nesterov–Todd step infeasible interior-point methods for symmetric optimization published in [A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res. 211(1) (2013) 209–224; G. Gu, M. Zangiabadi and C. Roos, Full Nesterov–Todd step infeasible interior-point method for symmetric optimization, European J. Oper. Res. 214(3) (2011) 473–484; Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math. 8(4) (2015) 1550071, 14 pp.] is considered. Each main iteration of our algorithm consisted of only a feasibility step, whereas in the earlier versions each iteration is composed of one feasibility step and several — at most three — centering steps. The algorithm finds an [Formula: see text]-solution of the underlying problem in polynomial-time and its iteration bound improves the earlier bounds factor from [Formula: see text] and [Formula: see text] to [Formula: see text]. Moreover, our method unifies the analysis for linear optimization, second-order cone optimization and semidefinite optimization.

AN INTERIOR POINT APPROACH FOR SEMIDEFINITE OPTIMIZATION USING NEW PROXIMITY FUNCTIONS

2009 ◽  
Vol 26 (03) ◽  
pp. 365-382 ◽  
Author(s):  
M. REZA PEYGHAMI
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Linear Optimization ◽  
Kernel Functions ◽  
Semidefinite Optimization ◽  
Special Choice ◽  
Simple Analysis ◽  
New Class ◽  
Iteration Bound ◽  
Primal Dual

Kernel functions play an important role in interior point methods (IPMs) for solving linear optimization (LO) problems to define a new search direction. In this paper, we consider primal-dual algorithms for solving Semidefinite Optimization (SDO) problems based on a new class of kernel functions defined on the positive definite cone [Formula: see text]. Using some appealing and mild conditions of the new class, we prove with simple analysis that the new class-based large-update primal-dual IPMs enjoy an [Formula: see text] iteration bound to solve SDO problems with special choice of the parameters of the new class.

scholarly journals An interior point algorithm for solving linear optimization problems using a new trigonometric kernel function

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1471-1486
Author(s):  
S. Fathi-Hafshejani ◽  
Reza Peyghami
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Interior Point Algorithm ◽  
Worst Case ◽  
Complexity Bounds ◽  
Linear Optimization Problems ◽  
Primal Dual ◽  
The Given

In this paper, a primal-dual interior point algorithm for solving linear optimization problems based on a new kernel function with a trigonometric barrier term which is not only used for determining the search directions but also for measuring the distance between the given iterate and the ?-center for the algorithm is proposed. Using some simple analysis tools and prove that our algorithm based on the new proposed trigonometric kernel function meets O (?n log n log n/?) and O (?n log n/?) as the worst case complexity bounds for large and small-update methods. Finally, some numerical results of performing our algorithm are presented.

scholarly journals A Full Nesterov-Todd Step Infeasible Interior-point Method for Symmetric Optimization in the Wider Neighborhood of the Central Path

2021 ◽  
Vol 9 (2) ◽  
pp. 250-267
Author(s):  
Lesaja Goran ◽  
G.Q. Wang ◽  
A. Oganian
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Point Method ◽  
Central Path ◽  
Symmetric Optimization ◽  
Statistical Disclosure Limitation ◽  
Starting Point ◽  
Linear Optimization Problems

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to ε-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.

scholarly journals A PRIMAL-DUAL INTERIOR-POINT ALGORITHM BASED ON A NEW KERNEL FUNCTION

2010 ◽  
Vol 51 (4) ◽  
pp. 476-491 ◽  
Author(s):  
G. M. CHO ◽  
Y. Y. CHO ◽  
Y. H. LEE
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Interior Point Algorithm ◽  
Linear Optimization Problems ◽  
Primal Dual ◽  
Search Directions

AbstractWe propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has $\mathcal {O}(\sqrt {n} \log n \log ({n}/{\epsilon }))$ and $\mathcal {O}(\sqrt {n}\log ({n}/{\epsilon }))$ iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.

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