On the convergence of augmented Lagrangian strategies for nonlinear programming

2021 ◽  
Author(s):  
Roberto Andreani ◽  
Ariel R Velazco ◽  
Alberto Ramos ◽  
Ademir A Ribeiro ◽  
Leonardo D Secchin
Keyword(s):  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Computational Cost ◽  
Optimization Methods ◽  
Stopping Criteria ◽  
Convergence Results ◽  
Sequential Optimality Conditions ◽  
New Strategy ◽  
Primal Dual

Abstract Augmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than previously stated in the literature. We show that a well-established safeguarded Powell–Hestenes–Rockafellar (PHR) AL algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a new primal–dual AL method capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the safeguarded PHR method when it tends to fail. We show by preliminary numerical tests that all the problems already successfully solved by the safeguarded PHR method remain unchanged, while others where the PHR method failed are now solved with an acceptable additional computational cost.

scholarly journals Strong complementary approximate Karush-Kuhn-Tucker conditions for multiobjective optimization problems

2021 ◽  
pp. 24-24
Author(s):  
Jitendra Maurya ◽  
Shashi Mishra
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Inequality Constraints ◽  
Multiobjective Optimization Problems ◽  
Sequential Optimality Conditions ◽  
Equality And Inequality Constraints ◽  
Optim Theory

In this paper, we establish strong complementary approximate Karush- Kuhn-Tucker (SCAKKT) sequential optimality conditions for multiobjective optimization problems with equality and inequality constraints without any constraint qualifications and introduce a weak constraint qualification which assures the equivalence between SCAKKT and the strong Karush-Kuhn-Tucker (J Optim Theory Appl 80 (3): 483{500, 1994) conditions for multiobjective optimization problems.

Augmented Lagrangian teaching–learning-based optimization for structural design

2017 ◽  
Vol 232 (12) ◽  
pp. 2195-2213
Author(s):  
Hong-Shuang Li ◽  
Qiao-Yue Dong ◽  
Jiao-Yang Yuan
Keyword(s):  
Design Optimization ◽  
Structural Design ◽  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Augmented Lagrangian Function ◽  
Lagrangian Multipliers ◽  
Structural Design Optimization ◽  
Teaching Learning Based Optimization ◽  
Teaching Learning

Stochastic optimization methods have been widely employed to find solutions to structural design optimization problems in the past two decades, especially for truss structures. The primary aim of this study is to introduce a design optimization method combining an augmented Lagrangian function and teaching–learning-based optimization for truss and nontruss structural design optimization. The augmented Lagrangian function serves as a constraint-handling tool in the proposed method and converts a constrained optimization problem into an unconstrained one. On the other hand, teaching–learning-based optimization is employed to resolve the transformed, unconstrained optimization problems. Since the proper values of the Lagrangian multipliers and penalty factors are unknown in advance, the proposed method is implemented in an iterative way to avoid the issue of selecting them, i.e. the Lagrangian multipliers and penalty factors are automatically updated according to the violation level of all constraints. To examine the performance of the proposed method, it is applied on a group of benchmark truss optimization problems and a group of nontruss optimization problems of aircraft wing structures. The computational results obtained by the proposed method are compared to the results produced by both other version of teaching–learning-based optimization and stochastic optimization methods.

Software suite for determining the tilts of tower-type structures according to terrestrial laser scanning

2020 ◽  
Vol 961 (7) ◽  
pp. 2-7
Author(s):  
A.V. Zubov ◽  
N.N. Eliseeva
Keyword(s):  
Laser Scanning ◽  
Optimization Problems ◽  
Optimization Methods ◽  
User Participation ◽  
Software Suite ◽  
Current State ◽  
Visual Basic For Applications ◽  
Operational Data ◽  
Theoretical Developments

The authors describe a software suite for determining tilt degrees of tower-type structures according to ground laser scanning indication. Defining the tilt of the pipe is carried out with a set of measured data through approximating the sections by circumferences. They are constructed using one of the simplest search engine optimization methods (evolutionary algorithm). Automatic filtering the scan of the current section from distorting data is performed by the method of assessing the quality of models constructed with that of least squares. The software was designed using Visual Basic for Applications. It contains several blocks (subprograms), with each of them performing a specific task. The developed complex enables obtaining operational data on the current state of the object with minimal user participation in the calculation process. The software suite is the result of practical implementing theoretical developments on the possibilities of using search methods at solving optimization problems in geodetic practice.

scholarly journals An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 225
Author(s):  
José García ◽  
Gino Astorga ◽  
Víctor Yepes
Keyword(s):  
Transfer Functions ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Set Covering ◽  
Covering Problem ◽  
Perturbation Operator ◽  
K Nearest Neighbors ◽  
Combinatorial Optimization Problems ◽  
Box Plots ◽  
Metaheuristic Techniques

The optimization methods and, in particular, metaheuristics must be constantly improved to reduce execution times, improve the results, and thus be able to address broader instances. In particular, addressing combinatorial optimization problems is critical in the areas of operational research and engineering. In this work, a perturbation operator is proposed which uses the k-nearest neighbors technique, and this is studied with the aim of improving the diversification and intensification properties of metaheuristic algorithms in their binary version. Random operators are designed to study the contribution of the perturbation operator. To verify the proposal, large instances of the well-known set covering problem are studied. Box plots, convergence charts, and the Wilcoxon statistical test are used to determine the operator contribution. Furthermore, a comparison is made using metaheuristic techniques that use general binarization mechanisms such as transfer functions or db-scan as binarization methods. The results obtained indicate that the KNN perturbation operator improves significantly the results.

scholarly journals Assessing the Performance of Interactive Multiobjective Optimization Methods

2021 ◽  
Vol 54 (4) ◽  
pp. 1-27
Author(s):  
Bekir Afsar ◽  
Kaisa Miettinen ◽  
Francisco Ruiz
Keyword(s):  
Multiobjective Optimization ◽  
Decision Maker ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Interactive Methods ◽  
Value Functions ◽  
Technical Aspects ◽  
Preference Information ◽  
Multiobjective Optimization Problems ◽  
Human Decision

Interactive methods are useful decision-making tools for multiobjective optimization problems, because they allow a decision-maker to provide her/his preference information iteratively in a comfortable way at the same time as (s)he learns about all different aspects of the problem. A wide variety of interactive methods is nowadays available, and they differ from each other in both technical aspects and type of preference information employed. Therefore, assessing the performance of interactive methods can help users to choose the most appropriate one for a given problem. This is a challenging task, which has been tackled from different perspectives in the published literature. We present a bibliographic survey of papers where interactive multiobjective optimization methods have been assessed (either individually or compared to other methods). Besides other features, we collect information about the type of decision-maker involved (utility or value functions, artificial or human decision-maker), the type of preference information provided, and aspects of interactive methods that were somehow measured. Based on the survey and on our own experiences, we identify a series of desirable properties of interactive methods that we believe should be assessed.

scholarly journals Concurrent material and structure optimization of multiphase hierarchical systems within a continuum micromechanics framework

2021 ◽  
Author(s):  
Tarun Gangwar ◽  
Dominik Schillinger
Keyword(s):  
Optimization Problems ◽  
Computational Cost ◽  
Structure Optimization ◽  
Material Behavior ◽  
Hierarchical Systems ◽  
Material Optimization ◽  
Continuum Micromechanics ◽  
Benchmark Tests ◽  
Constraint Optimization Problems ◽  
First Time

AbstractWe present a concurrent material and structure optimization framework for multiphase hierarchical systems that relies on homogenization estimates based on continuum micromechanics to account for material behavior across many different length scales. We show that the analytical nature of these estimates enables material optimization via a series of inexpensive “discretization-free” constraint optimization problems whose computational cost is independent of the number of hierarchical scales involved. To illustrate the strength of this unique property, we define new benchmark tests with several material scales that for the first time become computationally feasible via our framework. We also outline its potential in engineering applications by reproducing self-optimizing mechanisms in the natural hierarchical system of bamboo culm tissue.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

scholarly journals Optimization of injection molding process for car fender in consideration of energy efficiency and product quality

2014 ◽  
Vol 1 (4) ◽  
pp. 256-265 ◽  
Author(s):  
Hong Seok Park ◽  
Trung Thanh Nguyen
Keyword(s):  
Energy Efficiency ◽  
Energy Consumption ◽  
Injection Molding ◽  
Product Quality ◽  
Process Parameters ◽  
Optimization Problems ◽  
Computational Cost ◽  
Injection Molding Process ◽  
Molding Process ◽  
Nsga Ii

Abstract Energy efficiency is an essential consideration in sustainable manufacturing. This study presents the car fender-based injection molding process optimization that aims to resolve the trade-off between energy consumption and product quality at the same time in which process parameters are optimized variables. The process is specially optimized by applying response surface methodology and using nondominated sorting genetic algorithm II (NSGA II) in order to resolve multi-object optimization problems. To reduce computational cost and time in the problem-solving procedure, the combination of CAE-integration tools is employed. Based on the Pareto diagram, an appropriate solution is derived out to obtain optimal parameters. The optimization results show that the proposed approach can help effectively engineers in identifying optimal process parameters and achieving competitive advantages of energy consumption and product quality. In addition, the engineering analysis that can be employed to conduct holistic optimization of the injection molding process in order to increase energy efficiency and product quality was also mentioned in this paper.

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