scholarly journals A note on strict complementarity for the doubly non-negative cone

Optimization ◽  
2018 ◽  
Vol 68 (2-3) ◽  
pp. 457-464
Author(s):  
Bolor Jargalsaikhan ◽  
Jan-J. Rückmann
Keyword(s):  
Strict Complementarity

scholarly journals Convergence of a short-step primal-dual algorithm based on the Gauss-Newton direction

2003 ◽  
Vol 2003 (10) ◽  
pp. 517-534 ◽  
Author(s):  
Serge Kruk ◽  
Henry Wolkowicz
Keyword(s):  
Optimal Solution ◽  
Path Following ◽  
Singular Value ◽  
Dual Algorithm ◽  
Strict Complementarity ◽  
Semidefinite Programs ◽  
Primal Dual ◽  
Newton Direction ◽  
Proof Of Convergence ◽  
Smallest Singular Value

We prove the theoretical convergence of a short-step, approximate path-following, interior-point primal-dual algorithm for semidefinite programs based on the Gauss-Newton direction obtained from minimizing the norm of the perturbed optimality conditions. This is the first proof of convergence for the Gauss-Newton direction in this context. It assumes strict complementarity and uniqueness of the optimal solution as well as an estimate of the smallest singular value of the Jacobian.

NON-INTERIOR CONTINUATION METHOD FOR COMPLEMENTARITY PROBLEMS IN ABSENCE OF STRICT COMPLEMENTARITY

2006 ◽  
Vol 23 (01) ◽  
pp. 107-122 ◽  
Author(s):  
MIN SUN ◽  
ZHEN-JUN SHI
Keyword(s):  
Line Search ◽  
Convergence Rates ◽  
Complementarity Problems ◽  
Continuation Method ◽  
New Method ◽  
Smoothing Function ◽  
Strict Complementarity ◽  
Linear System Of Equations ◽  
Lipschitz Continuous ◽  
Quadratical Convergence

In this paper, by using a modified smoothing function, we propose a new continuation method for complementarity problems with R0-function and P0-function in the absence of strict complementarity. At each iteration, the continuation method solves one linear system of equations and performs one line search. When the underlying mapping is both a P0-function and a R0-function and its Hessian is Lipschitz continuous, we prove the global convergence of the new method. The new method also has global Q-linear and local Q-quadratical convergence rates under the same conditions.

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