Weak notions of nondegeneracy in nonlinear semidefinite programming

The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, ℓ -dimensional kernel of the constraint matrix, by the linear independenc...

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Bibliographic Details
Published in:Mathematical programming Vol. 205; no. 1-2; pp. 1 - 32
Main Authors: Andreani, Roberto, Haeser, Gabriel, Mito, Leonardo M., Ramírez, Héctor
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2024
Springer
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Semidefinite programming
90C30
90C46
Constraint nondegeneracy
Constraint qualifications
90C22
90C26
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, ℓ -dimensional kernel of the constraint matrix, by the linear independence of a set of ℓ ( ℓ + 1 ) / 2 derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of ℓ derivative vectors, by considering all orthonormal bases of the kernel instead. This allows us to identify that not all bases are relevant for a constraint qualification to be defined, giving rise to a strictly weaker variant of nondegeneracy related to the global convergence of an external penalty method. We use some of these ideas to revisit an approach of Forsgren (Math Program 88, 105–128, 2000) for exploiting the sparsity structure of a transformation of the constraints to define a constraint qualification, which led us to develop another relaxed notion of nondegeneracy using a simpler transformation. If the zeros of the derivatives of the constraint function at a given point are considered, instead of the zeros of the function itself in a neighborhood of that point, we obtain an even weaker constraint qualification that connects Forsgren’s condition and ours.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01970-4