Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming

Roberto Andreani (), Gabriel Haeser (), Leonardo M. Mito (), C. Héctor Ramírez () and Thiago P. Silveira ()
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Roberto Andreani: University of Campinas
Gabriel Haeser: University of São Paulo
Leonardo M. Mito: University of São Paulo
C. Héctor Ramírez: Universidad de Chile
Thiago P. Silveira: University of São Paulo

Journal of Optimization Theory and Applications, 2022, vol. 195, issue 1, No 2, 42-78

Abstract: Abstract In Andreani et al. (Weak notions of nondegeneracy in nonlinear semidefinite programming, 2020), the classical notion of nondegeneracy (or transversality) and Robinson’s constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. In this paper, we extend these ideas to the context of nonlinear second-order cone programming. For instance, for an m-dimensional second-order cone, instead of stating nondegeneracy at the vertex as the linear independence of m derivative vectors, we do it in terms of several statements of linear independence of 2 derivative vectors. This allows embedding the structure of the second-order cone into the formulation of nondegeneracy and, by extension, Robinson’s constraint qualification as well. This point of view is shown to be crucial in defining significantly weaker constraint qualifications such as the constant rank constraint qualification and the constant positive linear dependence condition. Also, these conditions are shown to be sufficient for guaranteeing global convergence of several algorithms, while still implying metric subregularity and without requiring boundedness of the set of Lagrange multipliers.

Keywords: Second-order cone programming; Constraint qualifications; Algorithms; Global convergence; Constant rank.; 90C46; 90C30 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-022-02056-5

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