scholarly journals Complexity Analysis of Primal-Dual Interior-Point Methods for Linear Optimization Based on a New Parametric Kernel Function with a Trigonometric Barrier Term

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
X. Z. Cai ◽  
G. Q. Wang ◽  
M. El Ghami ◽  
Y. J. Yue
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Interior Point Methods ◽  
Complexity Analysis ◽  
Linear Optimization ◽  
Kernel Functions ◽  
Primal Dual

We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods,O(n2/3log⁡(n/ε)), and small-update methods,O(nlog⁡(n/ε)). These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions.

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 A new parameterized logarithmic kernel function for linear optimization with a double barrier term yielding the best known iteration bound

2020 ◽  
Vol 28 (1) ◽  
pp. 27-41
Author(s):  
Benhadid Ayache ◽  
Saoudi Khaled
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Linear Optimization ◽  
Kernel Functions ◽  
Special Choice ◽  
Interior Point Algorithm ◽  
Double Barrier ◽  
New Class ◽  
Iteration Bound ◽  
Primal Dual

AbstractIn this paper, we propose a large-update primal-dual interior point algorithm for linear optimization. The method is based on a new class of kernel functions which differs from the existing kernel functions in which it has a double barrier term. The investigation according to it yields the best known iteration bound O\sqrt n \log (n)\log \left( {{n \over \in }} \right) for large-update algorithm with the special choice of its parameter m and thus improves the iteration bound obtained in Bai et al. [2] for large-update algorithm.

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.

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