constraint qualifications

A practical second-order optimality condition for cardinality-constrained problems with application to an augmented Lagrangian method

  • Jean C. A. Medeiros
  • Ademir A. Ribeiro
  • Mael Sachine
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , ,

    This paper addresses the mathematical programs with cardinality constraints (MPCaC). We first define two new tailored (strong and weak) second-order necessary conditions, MPCaC-SSONC and MPCaC-WSONC. We then propose a constraint qualification (CQ), namely, MPCaC-relaxed constant rank constraint qualification (MPCaC-RCRCQ), and establish the validity of MPCaC-SSONC at minimizers under this new CQ. All the concepts proposed … Read more

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

  • Roberto Andreani
  • Gabriel Haeser
  • Leonardo M. Mito
  • Héctor Ramírez
  • Thiago P. Silveira
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming Tags , , , ,

    In [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 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 … Read more

    A novel approach for bilevel programs based on Wolfe duality

  • Yuwei Li
  • Guihua Lin
  • Jin Zhang
  • Xide Zhu
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    This paper considers a bilevel program, which has many applications in practice. To develop effective numerical algorithms, it is generally necessary to transform the bilevel program into a single-level optimization problem. The most popular approach is to replace the lower-level program by its KKT conditions and then the bilevel program can be transformed into a … Read more

    Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

  • Roberto Andreani
  • Gabriel Haeser
  • Héctor Ramírez
  • Leonardo M. Mito
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming, Semi-definite Programming Tags , , ,

    We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange … Read more

    Weak notions of nondegeneracy in nonlinear semidefinite programming

  • Roberto Andreani
  • Gabriel Haeser
  • Leonardo M. Mito
  • Héctor Ramírez
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , ,

    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, $\ell$-dimensional kernel of the constraint matrix, by the linear independence of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this … Read more

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

  • Roberto Andreani
  • Ellen H. Fukuda
  • Gabriel Haeser
  • Daiana O. Santos
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization, Semi-definite Programming Tags , , , ,

    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 … Read more

    Constraint Qualifications for Karush-Kuhn-Tucker Conditions in Constrained Multiobjective Optimization

  • Gabriel Haeser
  • Alberto Ramos
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is … Read more

    On optimality conditions for nonlinear conic programming

  • Roberto Andreani
  • Walter Gómez
  • Gabriel Haeser
  • Alberto Ramos
  • Leonardo M. Mito
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    Sequential optimality conditions have played a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions have been described in conic contexts, where 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 … Read more

    On Constraint Qualifications for Second-Order Optimality Conditions Depending on a Single Lagrange Multiplier.

  • Gabriel Haeser
  • Alberto Ramos
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    Second-order optimality conditions play an important role in continuous optimization. In this paper, we present and discuss new constraint qualifications to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single … Read more

    Optimality conditions for nonlinear second-order cone programming and symmetric cone programming

  • Roberto Andreani
  • Ellen H. Fukuda
  • Gabriel Haeser
  • Daiana O. Santos
  • Leonardo D. Secchin
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization, Second-Order Cone Programming Tags , , , ,

    Nonlinear symmetric cone programming (NSCP) generalizes important optimization problems such as nonlinear programming, nonlinear semidefinite programming and nonlinear second-order cone programming (NSOCP). In this work, we present two new optimality conditions for NSCP without constraint qualifications, which implies the Karush-Kuhn-Tucker conditions under a condition weaker than Robinson’s constraint qualification. In addition, we show the relationship … Read more