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39
LOCAL CONVERGENCE OF EXACT AND INEXACT AUGMENTED LAGRANGIAN METHODS UNDER THE SECONDORDER SUFFICIENT OPTIMALITY CONDITION
, 2012
"... We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the secondorder sufficient optimality condition. In particular, no constraint qualifications of any kind ..."
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Cited by 15 (5 self)
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We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the secondorder sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the secondorder sufficient condition. That said, the classical results allow the initial multiplier estimate to be far from the optimal one, at the expense of proportionally increasing the threshold value for the penalty parameters. Although our primary goal is to avoid constraint qualifications, if the stronger assumptions are introduced, then starting points far from the optimal multiplier are allowed within our analysis as well. Using only the secondorder sufficient optimality condition, for penalty parameters large enough we prove primaldual Qlinear convergence rate, which becomes superlinear if the parameters are allowed to go to infinity. Both exact and inexact solutions of subproblems are considered. In the exact case, we further show that the primal convergence rate is of the same Qorder as the primaldual rate. Previous assertions for the primal sequence all had to do with the weaker Rrate of convergence and required the stronger assumptions cited above. Finally, we show that under our assumptions one of the popular rules of controlling the penalty parameters ensures their boundedness.
Improving ultimate convergence of an Augmented Lagrangian method
, 2007
"... Optimization methods that employ the classical PowellHestenesRockafellar Augmented Lagrangian are useful tools for solving Nonlinear Programming problems. Their reputation decreased in the last ten years due to the comparative success of InteriorPoint Newtonian algorithms, which are asymptoticall ..."
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Cited by 14 (0 self)
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Optimization methods that employ the classical PowellHestenesRockafellar Augmented Lagrangian are useful tools for solving Nonlinear Programming problems. Their reputation decreased in the last ten years due to the comparative success of InteriorPoint Newtonian algorithms, which are asymptotically faster. In the present research a combination of both approaches is evaluated. The idea is to produce a competitive method, being more robust and efficient than its “pure” counterparts for critical problems. Moreover, an additional hybrid algorithm is defined, in which the Interior Point method is replaced by the Newtonian resolution of a KKT system identified by the Augmented Lagrangian algorithm. The software used in this work is freely available through the Tango Project web page:
Low OrderValue Optimization and Applications
, 2005
"... Given r real functions F1(x),..., Fr(x) and an integer p between 1 and r, the Low OrderValue Optimization problem (LOVO) consists of minimizing the sum of the functions that take the p smaller values. If (y1,..., yr) is a vector of data and T (x, ti) is the predicted value of the observation i with ..."
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Cited by 9 (5 self)
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Given r real functions F1(x),..., Fr(x) and an integer p between 1 and r, the Low OrderValue Optimization problem (LOVO) consists of minimizing the sum of the functions that take the p smaller values. If (y1,..., yr) is a vector of data and T (x, ti) is the predicted value of the observation i with the parameters x ∈ IR n, it is natural to define Fi(x) = (T (x, ti) − yi) 2 (the quadratic error at observation i under the parameters x). When p = r this LOVO problem coincides with the classical nonlinear leastsquares problem. However, the interesting situation is when p is smaller than r. In that case, the solution of LOVO allows one to discard the influence of an estimated number of outliers. Thus, the LOVO problem is an interesting tool for robust estimation of parameters of nonlinear models. When p ≪ r the LOVO problem may be used to find hidden structures in data sets. One of the best succeeded applications include the Protein Alignment problem. Fully documented algorithms for this application are available at www.ime.unicamp.br/∼martinez/lovoalign. In this paper optimality conditions are discussed, algorithms for solving the LOVO problem are introduced and convergence theorems are proved. Finally, numerical experiments are presented.
A new sequential optimality condition for constrained optimization and algorithmic consequences
, 2009
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On the Boundedness of Penalty Parameters in an Augmented Lagrangian Method with Constrained Subproblems
, 2011
"... Augmented Lagrangian methods are effective tools for solving largescale nonlinear programming problems. At each outer iteration a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When t ..."
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Cited by 8 (1 self)
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Augmented Lagrangian methods are effective tools for solving largescale nonlinear programming problems. At each outer iteration a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When the penalty parameter becomes very large the subproblem is difficult, therefore the effectiveness of this approach is associated with boundedness of penalty parameters. In this paper it is proved that, under more natural assumptions than the ones up to now employed, penalty parameters are bounded. For proving the new boundedness result, the original algorithm has been slightly modified. Numerical consequences of the modifications are discussed and computational experiments are presented.
Derivativefree methods for nonlinear programming with general lowerlevel constraints
, 2010
"... Augmented Lagrangian methods for derivativefree continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martínez and Schuverdt for the case in which analytic derivatives exist and are available. In particula ..."
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Cited by 7 (0 self)
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Augmented Lagrangian methods for derivativefree continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martínez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. The form of our main algorithm allows us to employ well established derivativefree subalgorithms for solving lowerlevel constrained subproblems. Numerical experiments are presented.
A cover partitioning method for bound constrained global optimization
 Optim. Meth. Softw
"... A stochastic algorithm for global optimization subject to simple bounds is described. The method is applicable to blackbox functions which may be nonsmooth or discontinuous. The algorithm is in the spirit of the deterministic algorithm direct of Jones, Perttunen, and Stuckman. Like direct, it gene ..."
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Cited by 4 (2 self)
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A stochastic algorithm for global optimization subject to simple bounds is described. The method is applicable to blackbox functions which may be nonsmooth or discontinuous. The algorithm is in the spirit of the deterministic algorithm direct of Jones, Perttunen, and Stuckman. Like direct, it generates successively finer covers of the feasible region, where each cover consists of a finite number of boxes, and each box is defined by simple bounds. Its principal difference is that it calculates the objective at a randomly selected point in each unpopulated box, rather than at the centre of the box. A limited storage version of the algorithm is also presented. The sequence of bestknown function values is shown to converge to the essential minimum with probability 1 for both versions of the algorithm. A worst case expected rate theorem is established. Numerical results are presented which show the methods are effective in practice.
An Inexact Modified Subgradient Algorithm for Nonconvex Optimization ∗
, 2008
"... We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence ..."
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Cited by 4 (1 self)
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We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang–bang control problem, under several different inexactness schemes.