On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming

Autor(es)

Andreani, R. ^[1] ; Fukuda, E. H. ^[2] ; Haeser, G. ^[3] ; Santos, D. O. ^[4] ; Secchin, L. D. ^[5]

Afiliação do(s) autor(es)

^[1] Univ Estadual Campinas, Dept Appl Math, Campinas, SP - Brazil

^[2] Kyoto Univ, Grad Sch Informat, Kyoto - Japan

^[3] Univ Sao Paulo, Dept Appl Math, Sao Paulo, SP - Brazil

^[4] Univ Fed Sao Paulo, Inst Sci & Technol, Sao Jose Dos Campos, SP - Brazil

^[5] Univ Fed Espirito Santo, Dept Appl Math, Sao Mateus, ES - England

• Tipo de documento: Artigo Científico
• Fonte: COMPUTATIONAL OPTIMIZATION AND APPLICATIONS; v. 79, n. 3, p. 633-648, JUL 2021.
• Citações Web of Science: 0

Resumo

Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed; in particular, a global convergence result is available even when Lagrange multipliers are unbounded. (AU)

• Processo FAPESP: 13/07375-0 - CeMEAI - Centro de Ciências Matemáticas Aplicadas à Indústria
• Beneficiário: Francisco Louzada Neto
• Modalidade de apoio: Auxílio à Pesquisa - Centros de Pesquisa, Inovação e Difusão - CEPIDs
• Processo FAPESP: 18/24293-0 - Métodos computacionais de otimização
• Beneficiário: Sandra Augusta Santos
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 17/18308-2 - Condições de otimalidade e algoritmos de segunda-ordem
• Beneficiário: Gabriel Haeser
• Modalidade de apoio: Auxílio à Pesquisa - Regular