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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Vydané v:	Mathematical programming Ročník 205; číslo 1-2; s. 1 - 32
Hlavní autori:	Andreani, Roberto, Haeser, Gabriel, Mito, Leonardo M., Ramírez, Héctor
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2024 Springer
Predmet:	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
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Abstract	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.
AbstractList	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, [Formula omitted]-dimensional kernel of the constraint matrix, by the linear independence of a set of [Formula omitted] derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of [Formula omitted] 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. 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.
Audience	Academic
Author	Ramírez, Héctor Mito, Leonardo M. Andreani, Roberto Haeser, Gabriel
Author_xml	– sequence: 1 givenname: Roberto surname: Andreani fullname: Andreani, Roberto organization: Department of Applied Mathematics, State University of Campinas – sequence: 2 givenname: Gabriel surname: Haeser fullname: Haeser, Gabriel organization: Department of Applied Mathematics, University of São Paulo – sequence: 3 givenname: Leonardo M. orcidid: 0000-0003-2851-8285 surname: Mito fullname: Mito, Leonardo M. email: leonardommito@gmail.com organization: Department of Applied Mathematics, University of São Paulo – sequence: 4 givenname: Héctor surname: Ramírez fullname: Ramírez, Héctor organization: Department of Mathematical Engineering and Center for Mathematical Modeling (CNRS IRL 2807), University of Chile
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Cites_doi	10.1007/s11075-021-01212-8 10.1137/120903221 10.1007/s11590-021-01737-w 10.1287/moor.2014.0669 10.1007/s10107-015-0915-0 10.1007/PL00011370 10.1137/120887035 10.1007/s00186-013-0429-6 10.1016/j.orl.2012.11.009 10.1137/0130053 10.1007/978-1-4614-7621-4_28 10.1080/02331930903578700 10.1137/S1052623495279785 10.1137/110843939 10.1007/s10107-018-1354-5 10.1137/0805028 10.1007/978-1-4612-1394-9 10.1007/s10107-011-0456-0 10.1080/02331930902971377 10.1137/S1052623497326629 10.1287/moor.1060.0195 10.1007/BF02614439 10.1007/s11228-021-00573-5. 10.1007/978-1-4615-4381-7 10.1017/CBO9780511810817 10.1007/s11228-015-0328-5 10.1007/978-1-4614-0769-0 10.1007/s00186-005-0054-0 10.1007/BFb0121215
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Snippet	The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be...
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SubjectTerms	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
Title	Weak notions of nondegeneracy in nonlinear semidefinite programming
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Volume	205
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