Interior-point methods based on kernel functions for symmetric optimization

2012 ◽  
Vol 27 (3) ◽  
pp. 513-537 ◽  
Author(s):  
Manuel V.C. Vieira
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Kernel Functions ◽  
Symmetric Optimization

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.

Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization

2015 ◽  
Vol 08 (04) ◽  
pp. 1550071 ◽  
Author(s):  
Behrouz Kheirfam
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Interior Point Methods ◽  
Point Method ◽  
Central Path ◽  
Complexity Bound ◽  
Symmetric Optimization ◽  
Iteration Bound ◽  
Infeasible Interior Point Method ◽  
Simplified Analysis

We give a simplified analysis and an improved iteration bound of a full Nesterov–Todd (NT) step infeasible interior-point method for solving symmetric optimization. This method shares the features as, it (i) requires strictly feasible iterates on the central path of a perturbed problem, (ii) uses the feasibility steps to find strictly feasible iterates for the next perturbed problem, (iii) uses the centering steps to obtain a strictly feasible iterate close enough to the central path of the new perturbed problem, and (iv) reduces the size of the residual vectors with the same speed as the duality gap. Furthermore, the complexity bound coincides with the currently best-known iteration bound for full NT step infeasible interior-point methods.

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.

scholarly journals An interior point method for linear programming based on a class of Kernel functions

2005 ◽  
Vol 71 (1) ◽  
pp. 139-153 ◽  
Author(s):  
K. Amini ◽  
M. R. Peyghami
Keyword(s):  
Linear Programming ◽  
Interior Point ◽  
Time Complexity ◽  
Interior Point Method ◽  
Interior Point Methods ◽  
Kernel Functions ◽  
Point Method ◽  
New Family ◽  
Linear Programming Problems ◽  
Simplified Analysis

Interior point methods are not only the most effective methods for solving optimisation problems in practice but they also have polynomial time complexity. However, there is still a gap between the practical behavior of the interior point method algorithms and their theoretical complexity results. In this paper, by focusing on linear programming problems, we introduce a new family of kernel functions that have some simple and easy to check properties. We present a simplified analysis to obtain the complexity of generic interior point methods based on the proximity functions induced by these kernel functions. Finally, we prove that this family of kernel functions leads to improved iteration bounds of the large-update interior point methods.

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