Interior Point Methods for Sufficient Horizontal LCP in a Wide Neighborhood of the Central Path with Best Known Iteration Complexity

2014 ◽  
Vol 24 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Florian A. Potra
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Central Path ◽  
Iteration Complexity ◽  
Wide Neighborhood

scholarly journals An Arc Search Interior-Point Algorithm for Monotone Linear Complementarity Problems over Symmetric Cones

2018 ◽  
Vol 23 (1) ◽  
pp. 1-16
Author(s):  
Mohammad Pirhaji ◽  
Maryam Zangiabadi ◽  
Hossein Mansouri ◽  
Saman H. Amin
Keyword(s):  
Interior Point ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Symmetric Cone ◽  
Central Path ◽  
Interior Point Algorithm ◽  
Complexity Bound ◽  
Wide Neighborhood ◽  
Mathematical Problems

An arc search interior-point algorithm for monotone symmetric cone linear complementarity problem is presented. The algorithm estimates the central path by an ellipse and follows an ellipsoidal approximation of the central path to reach an "-approximate solution of the problem in a wide neighborhood of the central path. The convergence analysis of the algorithm is derived. Furthermore, we prove that the algorithm has the complexity bound O ( p rL) using Nesterov-Todd search direction and O (rL) by the xs and sx search directions. The obtained iteration complexities coincide with the best-known ones obtained by any proposed interior- point algorithm for this class of mathematical problems.

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 A FULL NT-STEP INFEASIBLE INTERIOR-POINT ALGORITHM FOR SEMIDEFINITE OPTIMIZATION BASED ON A SELF-REGULAR PROXIMITY

2011 ◽  
Vol 53 (1) ◽  
pp. 48-67 ◽  
Author(s):  
B. KHEIRFAM
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Polynomial Complexity ◽  
Regular Function ◽  
Central Path ◽  
Semidefinite Optimization ◽  
Interior Point Algorithm ◽  
Iteration Bound

AbstractWe introduce a full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular function to provide the feasibility step and to measure proximity to the central path. The result of polynomial complexity coincides with the best known iteration bound for infeasible interior-point methods.

scholarly journals A new second-order corrector interior-point algorithm for P*(k)-LCP

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6379-6391 ◽  
Author(s):  
B. Kheirfam ◽  
M. Chitsaz
Keyword(s):  
Interior Point ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Second Order ◽  
Interior Point Algorithm ◽  
Iteration Complexity ◽  
Improve Performance ◽  
Complexity Bound ◽  
Wide Neighborhood

In this paper, we propose a second-order corrector interior-point algorithm for solving P*(k)- linear complementarity problems. The method generates a sequence of iterates in a wide neighborhood of the central path introduced by Ai and Zhang. In each iteration, the method computes a corrector direction in addition to the Ai-Zhang direction, in an attempt to improve performance. The algorithm does not depend on the handicap k of the problem, so that it can be used for any P*(k)-linear complementarity problems. It is shown that the iteration complexity bound of the algorithm is O ((1+k)3 ? nL). Some numerical results are provided to illustrate the performance of the algorithm.

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