Roberto Andreani

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

    Global convergence of an augmented Lagrangian method for nonlinear programming via Riemannian optimization

  • Roberto Andreani
  • Kelvin Rodrigues Couto
  • Orizon Pereira Ferreira
  • Gabriel Haeser
  • Leandro Fonseca Prudente
    • Categories Nonlinear Optimization Tags , , ,

    Considering a standard nonlinear programming problem, one may view a subset of the equality constraints as an embedded Riemannian manifold. In this paper we investigate the differences between the Euclidean and the Riemannian approach for this problem. It is well known that the linear independence constraint qualification for both approaches are equivalent. However, when considering … 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

    Constraint qualifications and strong global convergence properties of an augmented Lagrangian method on Riemannian manifolds

  • Roberto Andreani
  • Kelvin Rodrigues Couto
  • Orizon Pereira Ferreira
  • Gabriel Haeser
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In the past years, augmented Lagrangian methods have been successfully applied to several classes of non-convex optimization problems, inspiring new developments in both theory and practice. In this paper we bring most of these recent developments from nonlinear programming to the context of optimization on Riemannian manifolds, including equality and inequality constraints. Many research have … Read more

    A minimal face constant rank constraint qualification for reducible conic programming

  • Roberto Andreani
  • Gabriel Haeser
  • leomakoto
  • Héctor Ramírez
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , ,

    In a previous paper [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez, T. P. Silveira. First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition. Mathematical Programming, 2023. DOI: 10.1007/s10107-023-01942-8] we introduced a constant rank constraint qualification for nonlinear semidefinite and second-order cone programming by considering all faces … 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