On Optimality Conditions for Nonlinear Conic Programming

Roberto Andreani (), Walter Gómez (), Gabriel Haeser (), Leonardo M. Mito () and Alberto Ramos ()
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Roberto Andreani: Department of Applied Mathematics, University of Campinas, Campinas 13083-970, Brazil
Walter Gómez: Department of Mathematical Engineering, Universidad de la Frontera, Temuco 4811230, Chile
Gabriel Haeser: Department of Applied Mathematics, University of São Paulo, São Paulo 05508-090, Brazil
Leonardo M. Mito: Department of Applied Mathematics, University of São Paulo, São Paulo 05508-090, Brazil
Alberto Ramos: Department of Mathematics, Federal University of Paraná, Curitiba 81530-015, Brazil

Mathematics of Operations Research, 2022, vol. 47, issue 3, 2160-2185

Abstract: Sequential optimality conditions play a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions are described in conic contexts, in which many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the augmented Lagrangian method satisfy the so-called approximate gradient projection optimality condition and, under an additional smoothness assumption, the so-called complementary approximate Karush–Kuhn–Tucker condition. The first result was unknown even for nonlinear programming, and the second one was unknown, for instance, for semidefinite programming.

Keywords: Primary: 90C46; secondary: 90C48; 90C30; 90C22; nonlinear conic optimization; optimality conditions; numerical methods; global convergence; constraint qualifications (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:3:p:2160-2185

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