Leonardo D. Secchin

On the boundedness of multipliers in augmented Lagrangian methods for mathematical programs with complementarity constraints

  • Roberto Andreani
  • Mariana da Rosa
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , , ,

    In this paper, we present a theoretical analysis of augmented Lagrangian (AL) methods applied to mathematical programs with complementarity constraints (MPCCs). Our focus is on a variant that reformulates the complementarity constraints using slack variables, where these constraints are handled directly in the subproblems rather than being penalized. We introduce specialized constraint qualifications (CQs) of … Read more

    Primal-dual global convergence of an augmented Lagrangian method under the error bound condition

  • Roberto Andreani
  • Gabriel Haeser
  • Renan W. Prado
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    This work investigates global convergence properties of a safeguarded augmented Lagrangian method applied to nonlinear programming problems, with an emphasis on the role of constraint qualifications in ensuring boundedness of the Lagrange multiplier estimates, also known as dual sequences. When functions with locally Lipschitz continuous derivatives define the constraint set, we prove that the Error … Read more

    A new constant-rank-type condition related to MFCQ and local error bounds

  • Roberto Andreani
  • Mariana da Rosa
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization

    Constraint qualifications (CQs) are fundamental for understanding the geometry of feasible sets and for ensuring the validity of optimality conditions in nonlinear programming. A known idea is that constant-rank type CQs allow one to modify the description feasible set, by eliminating redundant constraints, so that the Mangasarian-Fromovitz CQ (MFCQ) holds. Traditionally, such modifications, called reductions … Read more

    Global convergence of a second-order augmented Lagrangian method under an error bound condition

  • Renan W. Prado
  • Leonardo D. Secchin
  • Gabriel Haeser
  • Roberto Andreani
  • María Laura Schuverdt
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    This work deals with convergence to points satisfying the weak second-order necessary optimality conditions of a second-order safeguarded augmented Lagrangian method from the literature. To this end, we propose a new second-order sequential optimality condition that is, in a certain way, based on the iterates generated by the algorithm itself. This also allows us to … Read more

    Strong global convergence properties of algorithms for nonlinear symmetric cone programming

  • Roberto Andreani
  • Gabriel Haeser
  • Alberto Ramos
  • Daiana O. Santos
  • Leonardo D. Secchin
  • Ariel Serranoni
    • Categories Cone Programming, Second-Order Cone Programming, Semi-definite Programming Tags , , ,

    Sequential optimality conditions have played a major role in proving strong global convergence properties of numerical algorithms for many classes of optimization problems. In particular, the way complementarity is dealt is fundamental to achieve a strong condition. Typically, one uses the inner product structure to measure complementarity, which gives a very general approach to a … Read more

    A relaxed quasinormality condition and the boundedness of dual augmented Lagrangian sequences

  • Roberto Andreani
  • Gabriel Haeser
  • María Laura Schuverdt
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , , ,

    Global convergence of augmented Lagrangian methods to a first-order stationary point is well-known to hold under considerably weak constraint qualifications. In particular, several constant rank-type conditions have been introduced for this purpose which turned out to be relevant also beyond this scope. In this paper we show that in fact under these conditions subsequences of … Read more

    On enhanced KKT optimality conditions for smooth nonlinear optimization

  • Roberto Andreani
  • María Laura Schuverdt
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , , , ,

    The Fritz-John (FJ) and KKT conditions are fundamental tools for characterizing minimizers and form the basis of almost all methods for constrained optimization. Since the seminal works of Fritz John, Karush, Kuhn and Tucker, FJ/KKT conditions have been enhanced by adding extra necessary conditions. Such an extension was initially proposed by Hestenes in the 1970s … Read more

    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

    Improving the global convergence of Inexact Restoration methods for constrained optimization problems

  • Roberto Andreani
  • Alberto Ramos
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization Tags , , ,

    Inexact restoration (IR) methods are an important family of numerical methods for solving constrained optimization problems with applications to electronic structures and bilevel programming among others areas. In these methods, the minimization is divided in two phases: decreasing infeasibility (feasibility phase) and improving optimality (optimality phase). The feasibility phase does not require the generated points … Read more

    An extended delayed weighted gradient algorithm for solving strongly convex optimization problems

  • Roberto Andreani
  • Harry Oviedo
  • Marcos Raydan
  • Leonardo D. Secchin
    • Categories Unconstrained Optimization Tags , , ,

    The recently developed delayed weighted gradient method (DWGM) is competitive with the well-known conjugate gradient (CG) method for the minimization of strictly convex quadratic functions. As well as the CG method, DWGM has some key optimality and orthogonality properties that justify its practical performance. The main difference with the CG method is that, instead of … Read more