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54
Nonmonotone spectral projected gradient methods on convex sets
 SIAM Journal on Optimization
, 2000
"... Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone lin ..."
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Cited by 147 (28 self)
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Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the onedimensional search process. Convergence properties and extensive numerical results are presented.
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 55 (6 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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GradientBased Optimization of Custom Circuits Using a StaticTiming Formulation
, 1999
"... This paper describes a method of optimally sizing digital circuits on a statictiming basis. All paths through the logic are considered simultaneously and no input patterns need be specified by the user. The method is unique in that it is based on gradientbased, nonlinear optimization and can accom ..."
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Cited by 27 (4 self)
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This paper describes a method of optimally sizing digital circuits on a statictiming basis. All paths through the logic are considered simultaneously and no input patterns need be specified by the user. The method is unique in that it is based on gradientbased, nonlinear optimization and can accommodate transistorlevel schematics without the need for precharacterization. It employs efficient timedomain simulation and gradient computation for each channelconnected component. A largescale, generalpurpose, nonlinear optimization package is used to solve the tuning problem. A prototype tuner has been developed that accommodates combinational circuits consisting of parameterized library cells. Numerical results are presented.
A new active set algorithm for box constrained optimization
 SIAM Journal on Optimization
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A globally convergent linearly constrained Lagrangian method for nonlinear optimization
 SIAM J. Optim
, 2002
"... Abstract. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form “minimize an augmented Lagrangian function subject to linearized constraints. ” Such methods converge rapidly near a solution but may not be relia ..."
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Cited by 24 (5 self)
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Abstract. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form “minimize an augmented Lagrangian function subject to linearized constraints. ” Such methods converge rapidly near a solution but may not be reliable from arbitrary starting points. Nevertheless, the wellknown software package MINOS has proved effective on many large problems. Its success motivates us to derive a related LCL algorithm that possesses three important properties: it is globally convergent, the subproblem constraints are always feasible, and the subproblems may be solved inexactly. The new algorithm has been implemented in Matlab, with an option to use either MINOS or SNOPT (Fortran codes) to solve the linearly constrained subproblems. Only first derivatives are required. We present numerical results on a subset of the COPS, HS, and CUTE test problems, which include many large examples. The results demonstrate the robustness and efficiency of the stabilized LCL procedure.
Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters
, 1997
"... A model algorithm based on the successive quadratic programming method for solving the general nonlinear programming problem is presented. The objective function and the constraints of the problem are only required to be differentiable and their gradients to satisfy a Lipschitz condition. The strate ..."
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Cited by 20 (8 self)
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A model algorithm based on the successive quadratic programming method for solving the general nonlinear programming problem is presented. The objective function and the constraints of the problem are only required to be differentiable and their gradients to satisfy a Lipschitz condition. The strategy for obtaining global convergence is based on the trust region approach. The merit function is a type of augmented Lagrangian. A new updating scheme is introduced for the penalty parameter, by means of which monotone increase is not necessary. Global convergence results are proved and numerical experiments are presented.
An Adaptive Algorithm for Bound Constrained Quadratic Minimization
, 1997
"... A general algorithm for minimizing a quadratic function with bounds on the variables is presented. The new algorithm can use different unconstrained minimization techniques on different faces. At every face, the minimization technique can be chosen according to he structure of the Hessian and the di ..."
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Cited by 20 (9 self)
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A general algorithm for minimizing a quadratic function with bounds on the variables is presented. The new algorithm can use different unconstrained minimization techniques on different faces. At every face, the minimization technique can be chosen according to he structure of the Hessian and the dimension of the face. The strategy for leaving the face is based on a simple scheme that exploits the properties of the "chopped gradient" introduced by Friedlander and Mart'inez in 1989. This strategy guarantees global convergence even in the presence of dual degeneracy, and finite identification in the nondegenerate case. A slight modification of the algorithm satisfies, in addition, an identification property in the case of dual degeneracy. Numerical experiments combining this new strategy with conjugate gradients, gradient with retards and direct solvers are presented. Key words. Quadratic programming, conjugate gradients, gradient with retards, active set methods, sparse Cholesky factor...
Convergence Properties of an Augmented Lagrangian Algorithm for Optimization with a Combination of General Equality and Linear Constraints
 SIAM Journal on Optimization
, 1996
"... We consider the global and local convergence properties of a class of augmented Lagrangian methods for solving nonlinear programming problems. In these methods, linear and more general constraints are handled in different ways. The general constraints are combined with the objective function in an a ..."
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Cited by 19 (0 self)
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We consider the global and local convergence properties of a class of augmented Lagrangian methods for solving nonlinear programming problems. In these methods, linear and more general constraints are handled in different ways. The general constraints are combined with the objective function in an augmented Lagrangian. The iteration consists of solving a sequence of subproblems; in each subproblem the augmented Lagrangian is approximately minimized in the region defined by the linear constraints. A subproblem is terminated as soon as a stopping condition is satisfied. The stopping rules that we consider here encompass practical tests used in several existing packages for linearly constrained optimization. Our algorithm also allows different penalty parameters to be associated with disjoint subsets of the general constraints. In this paper, we analyze the convergence of the sequence of iterates generated by such an algorithm and prove global and fast linear convergence as well as showin...
Partially augmented Lagrangian method for matrix inequalities
 SIAM J. on Optimization
"... Pierre Apkarian k Abstract We discuss a partially augmented Lagrangian method for optimization programs with matrix inequality constraints. A global convergence result is obtained. Applications to hard problems in feedback control are presented to validate the method numerically. ..."
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Cited by 16 (8 self)
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Pierre Apkarian k Abstract We discuss a partially augmented Lagrangian method for optimization programs with matrix inequality constraints. A global convergence result is obtained. Applications to hard problems in feedback control are presented to validate the method numerically.