A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

2018 ◽  
Vol 79 (1) ◽  
pp. 94-105 ◽  
Author(s):  
Behrouz Kheirfam ◽  
Masoumeh Haghighi
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Linear Optimization ◽  
Interior Point Algorithm ◽  
Wide Neighborhood

A FULL-NEWTON STEP INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR OPTIMIZATION BASED ON AN EXPONENTIAL KERNEL FUNCTION

2014 ◽  
Vol 07 (01) ◽  
pp. 1450018
Author(s):  
Behrouz Kheirfam ◽  
Fariba Hasani
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Interior Point Method ◽  
Polynomial Complexity ◽  
Linear Optimization ◽  
Interior Point Algorithm ◽  
Math Model ◽  
Newton Step ◽  
Infeasible Interior Point Method ◽  
Full Newton Step

This paper deals with an infeasible interior-point algorithm with full-Newton step for linear optimization based on a kernel function, which is an extension of the work of the first author and coworkers (J. Math. Model Algorithms (2013); DOI 10.1007/s10852-013-9227-7). The main iteration of the algorithm consists of a feasibility step and several centrality steps. The centrality step is based on Darvay's direction, while we used a kernel function in the algorithm to induce the feasibility step. For the kernel function, the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods.

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 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 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.

scholarly journals A polynomial-time algorithm for linear optimization based on a new kernel function with trigonometric barrier term

2015 ◽  
Vol 25 (2) ◽  
pp. 233-250 ◽  
Author(s):  
B. Kheirfam ◽  
M. Moslemi
Keyword(s):  
Kernel Function ◽  
Polynomial Time ◽  
Interior Point ◽  
Linear Optimization ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Interior Point Algorithm ◽  
Iteration Complexity ◽  
Proximity Measure ◽  
Starting Point

In this paper, we propose a large-update interior-point algorithm for linear optimization based on a new kernel function. New search directions and proximity measure are defined based on this kernel function. We show that if a strictly feasible starting point is available, then the new algorithm has O(3/4log n/?) iteration complexity.

A New Predictor-corrector Infeasible Interior-point Algorithm for Linear Optimization in aWide Neighborhood

2020 ◽  
Vol 177 (2) ◽  
pp. 141-156
Author(s):  
Behrouz Kheirfam
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Linear Optimization ◽  
Point Method ◽  
Central Path ◽  
Interior Point Algorithm ◽  
Positive Direction ◽  
Complexity Result ◽  
Infeasible Interior Point Method ◽  
Predictor Corrector

In this paper, we propose a Mizuno-Todd-Ye type predictor-corrector infeasible interior-point method for linear optimization based on a wide neighborhood of the central path. According to Ai-Zhang’s original idea, we use two directions of distinct and orthogonal corresponding to the negative and positive parts of the right side vector of the centering equation of the central path. In the predictor stage, the step size along the corresponded infeasible directions to the negative part is chosen. In the corrector stage by modifying the positive directions system a full-Newton step is removed. We show that, in addition to the predictor step, our method reduces the duality gap in the corrector step and this can be a prominent feature of our method. We prove that the iteration complexity of the new algorithm is 𝒪(n log ɛ−1), which coincides with the best known complexity result for infeasible interior-point methods, where ɛ > 0 is the required precision. Due to the positive direction new system, we improve the theoretical complexity bound for this kind of infeasible interior-point method [1] by a factor of n . Numerical results are also provided to demonstrate the performance of the proposed algorithm.

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