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22
On Augmented Lagrangian methods with general lowerlevel constraints
, 2005
"... Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. In ..."
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Cited by 59 (7 self)
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Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. Inexact resolution of the lowerlevel constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot. All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lowerlevel set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.
Augmented Lagrangian methods under the Constant Positive Linear Dependence constraint qualification
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Global minimization using an Augmented Lagrangian method with variable lowerlevel constraints
, 2007
"... A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global c ..."
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Cited by 21 (1 self)
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A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an εglobal minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.
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 6 (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.
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 6 (4 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.
Secondorder negativecurvature methods for boxconstrained and general constrained optimization
, 2009
"... A Nonlinear Programming algorithm that converges to secondorder stationary points is introduced in this paper. The main tool is a secondorder negativecurvature method for boxconstrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is ..."
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Cited by 6 (0 self)
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A Nonlinear Programming algorithm that converges to secondorder stationary points is introduced in this paper. The main tool is a secondorder negativecurvature method for boxconstrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (PowellHestenesRockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to secondorder stationary points in situations in which firstorder methods fail are exhibited.
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 4 (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:
On secondorder optimality conditions for nonlinear programming
 Optimization
"... A new SecondOrder condition is given, which depends on a weak constant rank constraint requirement. We show that practical and publicly available algorithms (www.ime.usp.br/∼egbirgin/tango) of Augmented Lagrangian type converge, after slight modifications, to stationary points defined by the new co ..."
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Cited by 4 (0 self)
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A new SecondOrder condition is given, which depends on a weak constant rank constraint requirement. We show that practical and publicly available algorithms (www.ime.usp.br/∼egbirgin/tango) of Augmented Lagrangian type converge, after slight modifications, to stationary points defined by the new condition.
On sequential optimality conditions for smooth constrained optimization
, 2009
"... Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between differen ..."
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Cited by 4 (1 self)
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Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counterexamples will be shown. Algorithmic consequences will be discussed.
Two new weak constraint qualifications and applications
"... We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Dependence (RCPLD) constraint qualification. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependen ..."
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Cited by 3 (0 self)
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We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Dependence (RCPLD) constraint qualification. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new constraint qualification, that we call Constant Rank of the Subspace Component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, like local stability and the validity of an error bound. We also introduce an even weaker CQ, called Constant Positive Generator (CPG), that can replace RCPLD in the analysis of the global convergence of algorithms. We close this work extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: SQP, augmented Lagrangians, interior point algorithms, and inexact restoration. ∗ This work was supported by PRONEXOptimization (PRONEXCNPq/FAPERJ E