scholarly journals On methods for solving nonlinear semidefinite optimization problems

2011 ◽  
Vol 1 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jie Sun ◽  
Keyword(s):  
Optimization Problems ◽  
Semidefinite Optimization

A corrector–predictor path-following algorithm for semidefinite optimization

2014 ◽  
Vol 07 (02) ◽  
pp. 1450028 ◽  
Author(s):  
Behrouz Kheirfam
Keyword(s):  
Line Search ◽  
Optimization Problems ◽  
Path Following ◽  
Central Path ◽  
Semidefinite Optimization ◽  
Iteration Complexity ◽  
Proximity Measure ◽  
Corrector Step ◽  
Predictor Step ◽  
Path Following Algorithm

A corrector–predictor algorithm is proposed for solving semidefinite optimization problems. In each two steps, the algorithm uses the Nesterov–Todd directions. The algorithm produces a sequence of iterates in a neighborhood of the central path based on a new proximity measure. The predictor step uses line search schemes requiring the reduction of the duality gap, while the corrector step is used to restore the iterates to the neighborhood of the central path. Finally, the algorithm has [Formula: see text] iteration complexity.

Semidefinite optimization

Acta Numerica ◽  
2001 ◽  
Vol 10 ◽  
pp. 515-560 ◽  
Author(s):  
M. J. Todd
Keyword(s):  
Combinatorial Optimization ◽  
Differential Inclusion ◽  
Computational Methods ◽  
Duality Theory ◽  
Optimization Problems ◽  
Symmetric Matrix ◽  
Positive Semidefinite ◽  
Semidefinite Optimization ◽  
Combinatorial Optimization Problems ◽  
The Stability

Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps ‘or’ would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.

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