augmented lagrangian method

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

    A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Filter Line Search Algorithm

  • Christof Büskens
  • Renke Kuhlmann
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    Interior-point methods have been shown to be very efficient for large-scale nonlinear programming. The combination with penalty methods increases their robustness due to the regularization of the constraints caused by the penalty term. In this paper a primal-dual penalty-interior-point algorithm is proposed, that is based on an augmented Lagrangian approach with an l2-exact penalty function. … 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

    On Relaxation of Some Customized Proximal Point Algorithms for Convex Minimization: From Variational Inequality Perspective

  • Feng Ma
    • Categories Convex Optimization Tags , , ,

    The proximal point algorithm (PPA) is a fundamental method for convex programming. When PPA applied to solve linearly constrained convex problems, we may prefer to choose an appropriate metric matrix to define the proximal regularization, so that the computational burden of the resulted PPA can be reduced, and in most cases, even admit closed form … Read more

    Optimality conditions for problems over symmetric cones and a simple augmented Lagrangian method

  • Ellen H. Fukuda
  • Bruno F. Lourenço
  • Masao Fukushima
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , ,

    In this work we are interested in nonlinear symmetric cone problems (NSCPs), which contain as special cases nonlinear semidefinite programming, nonlinear second order cone programming and the classical nonlinear programming problems. We explore the possibility of reformulating NSCPs as common nonlinear programs (NLPs), with the aid of squared slack variables. Through this connection, we show … Read more

    Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points

  • Ernesto G. Birgin
  • Gabriel Haeser
  • Alberto Ramos
    • Categories Constrained Nonlinear Optimization Tags , ,

    Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the … Read more

    Global Convergence of ADMM in Nonconvex Nonsmooth Optimization

  • Yu Wang
  • Wotao Yin
  • Jinshan Zeng
    • Categories Nonsmooth Optimization Tags , , , ,

    In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, $\phi(x_0,\ldots,x_p,y)$, subject to coupled linear equality constraints. Our ADMM updates each of the primal variables $x_0,\ldots,x_p,y$, followed by updating the dual variable. We separate the variable $y$ from $x_i$’s as it … Read more

    A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

  • Natashia Boland
  • Brian Dandurand
  • Andrew C. Eberhard
  • Fabricio Oliveira
  • Jeffrey Christiansen
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Parallel Algorithms Tags , , , ,

    We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. The dual solution approach needs … Read more

    Positive-Indefinite Proximal Augmented Lagrangian Method and its Application to Full Jacobian Splitting for Multi-block Separable Convex Minimization Problems

  • Bingsheng He
  • Feng Ma
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , , ,

    The augmented Lagrangian method (ALM) is fundamental for solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper, we show that it is not necessary … Read more

    Improving an ADMM-like Splitting Method via Positive-Indefinite Proximal Regularization for Three-Block Separable Convex Minimization

  • Bingsheng He
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , ,

    The augmented Lagrangian method (ALM) is fundamental for solving convex minimization models with linear constraints. When the objective function is separable such that it can be represented as the sum of more than one function without coupled variables, various splitting versions of the ALM have been well studied in the literature such as the alternating … Read more