Leonardo D. Secchin

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

    On the convergence of augmented Lagrangian strategies for nonlinear programming

  • Roberto Andreani
  • A. R. V. Cárdenas
  • Alberto Ramos
  • Ademir A. Ribeiro
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , , , ,

    Augmented Lagrangian algorithms are very popular and successful methods for solving constrained optimization problems. Recently, the global convergence analysis of these methods have been dramatically improved by using the notion of the sequential optimality conditions. Such conditions are optimality conditions independently of the fulfilment of any constraint qualifications and provide theoretical tools to justify stopping … 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

    New sequential optimality conditions for mathematical problems with complementarity constraints and algorithmic consequences

  • Roberto Andreani
  • Gabriel Haeser
  • Paulo J. S. Silva
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization

    In recent years, the theoretical convergence of iterative methods for solving nonlinear constrained optimization problems has been addressed using sequential optimality conditions, which are satisfied by minimizers independently of constraint qualifications (CQs). Even though there is a considerable literature devoted to sequential conditions for standard nonlinear optimization, the same is not true for Mathematical Problems … Read more

    A sequential optimality condition related to the quasinormality constraint qualification and its algorithmic consequences

  • Roberto Andreani
  • N. S. Fazzio
  • María Laura Schuverdt
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization Tags , , ,

    In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasinormality constraint qualification, which is associated with the external penalty theory. For this purpose, a new sequential optimality condition for smooth constrained optimization, called PAKKT, is defined. The new condition takes into account the sign of the dual … Read more

    Bilevel optimization with a multiobjective problem in the lower level

  • Roberto Andreani
  • Sandra Augusta Santos
  • Viviana A. Ramirez
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    Bilevel problems model instances with a hierarchical structure. Aiming at an efficient solution of a constrained multiobjective problem according with some pre-defined criterion, we reformulate this optimization but non standard problem as a classic bilevel one. This reformulation intents to encompass all the objectives, so that the properly efficient solution set is recovered by means … Read more

    Convergence properties of a second order augmented Lagrangian method for mathematical programs with complementarity constraints

  • Roberto Andreani
  • Paulo J. S. Silva
  • Leonardo D. Secchin
    • Categories Complementarity and Variational Inequalities Tags , ,

    Mathematical Programs with Complementarity Constraints (MPCCs) are difficult optimization problems that do not satisfy the majority of the usual constraint qualifications (CQs) for standard nonlinear optimization. Despite this fact, classical methods behaves well when applied to MPCCs. Recently, Izmailov, Solodov and Uskov proved that first order augmented Lagrangian methods, under a natural adaption of the … Read more