Interior point method
()
SIAM J. OPTIMIZATION
Downloaded 02/21/14 to 128.42.163.118. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Vol. 4, No. 1, pp. 208-227, February 1994
1994 Society for Industrial and Applied Mathematics
012
ON THE CONVERGENCE OF A CLASS OF INFEASIBLE
INTERIOR-POINT METHODS FOR THE HORIZONTAL LINEAR
COMPLEMENTARITY PROBLEM*
YIN
ZHANGt
Abstract. Interior-point methods require strictly feasible points as starting points. In theory, this requirement does not seem to be particularly restrictive, but it can be costly in computation.
To overcome this deficiency, most existing practical algorithms allow positive but infeasible starting points and seek feasibility and optimality simultaneously. Algorithms of this type shall be called in]easible interior-point algorithms. Despite their superior performance, existing infeasible interiorpoint algorithms still lack a satisfactory demonstration of theoretical convergence and polynomial complexity. This paper studies a popular infeasible interior-point algorithmic framework that was implemented for linear programming in the highly successful interior-point code OB1 [I. J. Lustig,
R. E. Marsten, and D. F. Shanno, Linear Algebra Appl., 152 (1991), pp. 191-222]. For generality, the analysis is carried out on a horizontal linear complementarity problem that includes linear and quadratic programming, as well as the standard linear complementarity problem. Under minimal assumptions, it is demonstrated that with properly controlled steps the algorithm converges at a global Q-linear rate. Moreover, with properly chosen starting points it is established the algorithm can obtain e-feasibility and -complementarity in at most O(n 2 ln(1/)) iterations.
Key
words,
infeasible interior-point methods, horizontal linear complementarity problem,
global convergence, polynomiality
AMS subject classification. 90C05
1. Introduction. Interior-point
SIAM J. OPTIMIZATION
Downloaded 02/21/14 to 128.42.163.118. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Vol. 4, No. 1, pp. 208-227, February 1994
1994 Society for Industrial and Applied Mathematics
012
ON THE CONVERGENCE OF A CLASS OF INFEASIBLE
INTERIOR-POINT METHODS FOR THE HORIZONTAL LINEAR
COMPLEMENTARITY PROBLEM*
YIN
ZHANGt
Abstract. Interior-point methods require strictly feasible points as starting points. In theory, this requirement does not seem to be particularly restrictive, but it can be costly in computation.
To overcome this deficiency, most existing practical algorithms allow positive but infeasible starting points and seek feasibility and optimality simultaneously. Algorithms of this type shall be called in]easible interior-point algorithms. Despite their superior performance, existing infeasible interiorpoint algorithms still lack a satisfactory demonstration of theoretical convergence and polynomial complexity. This paper studies a popular infeasible interior-point algorithmic framework that was implemented for linear programming in the highly successful interior-point code OB1 [I. J. Lustig,
R. E. Marsten, and D. F. Shanno, Linear Algebra Appl., 152 (1991), pp. 191-222]. For generality, the analysis is carried out on a horizontal linear complementarity problem that includes linear and quadratic programming, as well as the standard linear complementarity problem. Under minimal assumptions, it is demonstrated that with properly controlled steps the algorithm converges at a global Q-linear rate. Moreover, with properly chosen starting points it is established the algorithm can obtain e-feasibility and -complementarity in at most O(n 2 ln(1/)) iterations.
Key
words,
infeasible interior-point methods, horizontal linear complementarity problem,
global convergence, polynomiality
AMS subject classification. 90C05
1. Introduction. Interior-point
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