Global minimization using an Augmented Lagrangian method with variable lower-level constraints

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 epsilon(k)-global minimization of the Augmented Lagrangian with simple constraints, where epsilon(k) -> epsilon. Global convergence to an epsilon-global minimizer of the original problem is proved. The subproblems are solved using the alpha BB method. Numerical experiments are presented.

Improving ultimate convergence of an augmented Lagrangian method

Optimization methods that employ the classical Powell-Hestenes-Rockafellar augmented Lagrangian are useful tools for solving nonlinear programming problems. Their reputation decreased in the last 10 years due to the comparative success of interior-point Newtonian algorithms, which are asymptotically faster. In this 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 Karush-Kuhn-Tucker (KKT) system identified by the augmented Lagrangian algorithm. The software used in this work is freely available through the Tango Project web page:http://www.ime.usp.br/similar to egbirgin/tango/.

Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization

At each outer iteration of standard Augmented Lagrangian methods one tries to solve a box-constrained optimization problem with some prescribed tolerance. In the continuous world, using exact arithmetic, this subproblem is always solvable. Therefore, the possibility of finishing the subproblem resolution without satisfying the theoretical stopping conditions is not contemplated in usual convergence theories. However, in practice, one might not be able to solve the subproblem up to the required precision. This may be due to different reasons. One of them is that the presence of an excessively large penalty parameter could impair the performance of the box-constraint optimization solver. In this paper a practical strategy for decreasing the penalty parameter in situations like the one mentioned above is proposed. More generally, the different decisions that may be taken when, in practice, one is not able to solve the Augmented Lagrangian subproblem will be discussed. As a result, an improved Augmented Lagrangian method is presented, which takes into account numerical difficulties in a satisfactory way, preserving suitable convergence theory. Numerical experiments are presented involving all the CUTEr collection test problems.

The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems

Augmented Lagrangian methods are effective tools for solving large-scale 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, solving the subproblem becomes difficult; therefore, the effectiveness of this approach is associated with the boundedness of the penalty parameters. In this paper, it is proved that under more natural assumptions than the ones employed until now, 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.

Reduced gradient method combined with augmented Lagrangian and barrier for the optimal power flow problem

A new approach for solving the optimal power flow (OPF) problem is established by combining the reduced gradient method and the augmented Lagrangian method with barriers and exploring specific characteristics of the relations between the variables of the OPF problem. Computer simulations on IEEE 14-bus and IEEE 30-bus test systems illustrate the method. (c) 2007 Elsevier Inc. All rights reserved.

Primal-dual logarithmic barrier and augmented Lagrangian function to the loss minimization in power systems

This article presents a new approach to minimize the losses in electrical power systems. This approach considers the application of the primal-dual logarithmic barrier method to voltage magnitude and tap-changing transformer variables, and the other inequality constraints are treated by augmented Lagrangian method. The Lagrangian function aggregates all the constraints. The first-order necessary conditions are reached by Newton's method, and by updating the dual variables and penalty factors. Test results are presented to show the good performance of this approach.

Logarithmic barrier-augmented Lagrangian function to the optimal power flow problem

This paper presents a new approach to solve the Optimal Power Flow problem. This approach considers the application of logarithmic barrier method to voltage magnitude and tap-changing transformer variables and the other constraints are treated by augmented Lagrangian method. Numerical test results are presented, showing the effective performance of this algorithm. (C) 2005 Elsevier Ltd. All rights reserved.

Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization

Augmented Lagrangian methods for large-scale optimization usually require efficient algorithms for minimization with box constraints. On the other hand, active-set box-constraint methods employ unconstrained optimization algorithms for minimization inside the faces of the box. Several approaches may be employed for computing internal search directions in the large-scale case. In this paper a minimal-memory quasi-Newton approach with secant preconditioners is proposed, taking into account the structure of Augmented Lagrangians that come from the popular Powell-Hestenes-Rockafellar scheme. A combined algorithm, that uses the quasi-Newton formula or a truncated-Newton procedure, depending on the presence of active constraints in the penalty-Lagrangian function, is also suggested. Numerical experiments using the Cute collection are presented.

A new approach to the solution of the optimal power flow problem based on the modified Newton's method associated to an augmented Lagrangian function

This paper presents a new algorithm for optimal power flow problem. The algorithm is based on Newton's method which it works with an Augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints and is solved using the modified-Newton method. The test results have shown the effectiveness of the approach using the IEEE 30 and 638 bus systems.

An augmented Lagrangian contact algorithm employing a vertex-based finite volume method

Abstract not available

Fast Geodesic Active Fields for Image Registration Based on Splitting and Augmented Lagrangian Approaches.

In this paper, we present an efficient numerical scheme for the recently introduced geodesic active fields (GAF) framework for geometric image registration. This framework considers the registration task as a weighted minimal surface problem. Hence, the data-term and the regularization-term are combined through multiplication in a single, parametrization invariant and geometric cost functional. The multiplicative coupling provides an intrinsic, spatially varying and data-dependent tuning of the regularization strength, and the parametrization invariance allows working with images of nonflat geometry, generally defined on any smoothly parametrizable manifold. The resulting energy-minimizing flow, however, has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach; data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Our approach is more flexible than standard Gaussian regularization, since one can interpolate freely between isotropic Gaussian and anisotropic TV-like smoothing. In this paper, we compare the geodesic active fields method with the popular Demons method and three more recent state-of-the-art algorithms: NL-optical flow, MRF image registration, and landmark-enhanced large displacement optical flow. Thus, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality. Over the range of example applications, it also consistently produces results not far from more dedicated state-of-the-art methods, illustrating the flexibility of the proposed framework.

Outer Trust-Region Method for Constrained Optimization

Given an algorithm A for solving some mathematical problem based on the iterative solution of simpler subproblems, an outer trust-region (OTR) modification of A is the result of adding a trust-region constraint to each subproblem. The trust-region size is adaptively updated according to the behavior of crucial variables. The new subproblems should not be more complex than the original ones, and the convergence properties of the OTR algorithm should be the same as those of Algorithm A. In the present work, the OTR approach is exploited in connection with the ""greediness phenomenon"" of nonlinear programming. Convergence results for an OTR version of an augmented Lagrangian method for nonconvex constrained optimization are proved, and numerical experiments are presented.

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.

Otimização Topológica com Termo-elasticidade Utilizando Elementos Tri3

This work proposes a formulation for optimization of 2D-structure layouts submitted to mechanic and thermal shipments and applied an h-adaptive filter process which conduced to computational low spend and high definition structural layouts. The main goal of the formulation is to minimize the structure mass submitted to an effective state of stress of von Mises, with stability and lateral restriction variants. A criterion of global measurement was used for intents a parametric condition of stress fields. To avoid singularity problems was considerate a release on the stress restriction. On the optimization was used a material approach where the homogenized constructive equation was function of the material relative density. The intermediary density effective properties were represented for a SIMP-type artificial model. The problem was simplified by use of the method of finite elements of Galerkin using triangles with linear Lagrangian basis. On the solution of the optimization problem, was applied the augmented Lagrangian Method, that consists on minimum problem sequence solution with box-type restrictions, resolved by a 2nd orderprojection method which uses the method of the quasi-Newton without memory, during the problem process solution. This process reduces computational expends showing be more effective and solid. The results materialize more refined layouts with accurate topologic and shape of structure definitions. On the other hand formulation of mass minimization with global stress criterion provides to modeling ready structural layouts, with violation of the criterion of homogeneous distributed stress

Otimização de forma aplicando B-splines sob critério integral de tensões

This work proposes a computational methodology to solve problems of optimization in structural design. The application develops, implements and integrates methods for structural analysis, geometric modeling, design sensitivity analysis and optimization. So, the optimum design problem is particularized for plane stress case, with the objective to minimize the structural mass subject to a stress criterion. Notice that, these constraints must be evaluated at a series of discrete points, whose distribution should be dense enough in order to minimize the chance of any significant constraint violation between specified points. Therefore, the local stress constraints are transformed into a global stress measure reducing the computational cost in deriving the optimal shape design. The problem is approximated by Finite Element Method using Lagrangian triangular elements with six nodes, and use a automatic mesh generation with a mesh quality criterion of geometric element. The geometric modeling, i.e., the contour is defined by parametric curves of type B-splines, these curves hold suitable characteristics to implement the Shape Optimization Method, that uses the key points like design variables to determine the solution of minimum problem. A reliable tool for design sensitivity analysis is a prerequisite for performing interactive structural design, synthesis and optimization. General expressions for design sensitivity analysis are derived with respect to key points of B-splines. The method of design sensitivity analysis used is the adjoin approach and the analytical method. The formulation of the optimization problem applies the Augmented Lagrangian Method, which convert an optimization problem constrained problem in an unconstrained. The solution of the Augmented Lagrangian function is achieved by determining the analysis of sensitivity. Therefore, the optimization problem reduces to the solution of a sequence of problems with lateral limits constraints, which is solved by the Memoryless Quasi-Newton Method It is demonstrated by several examples that this new approach of analytical design sensitivity analysis of integrated shape design optimization with a global stress criterion purpose is computationally efficient