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We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufﬁcient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x∗ to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisﬁed at x∗.

@article{wang_properties_2009, title = {Properties of equation reformulation of the {Karush}–{Kuhn}–{Tucker} condition for nonlinear second order cone optimization problems}, volume = {70}, issn = {1432-2994, 1432-5217}, url = {http://link.springer.com/10.1007/s00186-008-0241-x}, doi = {10.1007/s00186-008-0241-x}, abstract = {We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufﬁcient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x∗ to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisﬁed at x∗.}, language = {en}, number = {2}, urldate = {2022-01-19}, journal = {Mathematical Methods of Operations Research}, author = {Wang, Yun and Zhang, Liwei}, month = oct, year = {2009}, keywords = {/unread}, pages = {195--218}, }

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