Sufficient conditions for exact penalty in constrained optimization
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Abstract In this paper we use the penalty approach in order to study constrained minimization problems. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. We discuss very simple sufficient conditions for the exact penalty property. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)Keywords:
Penalty Method
Minification
Constrained optimization
Constrained optimization problem
Penalty Method
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Constrained optimization
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Due to significant improvements in the performance of computers, Genetic Algorithms are actively applied to actual engineering problems. Most engineering problems are constrained optimization problems that are used to optimize objective functions under many constraints. The penalty method is well-known solution to such constrained optimization problems. However, the benchmarks of constrained optimization problems have only a small number of constraints. Thus, the effectiveness of the penalty method has not been investigated in numerous constrained problems. In many constrained optimization problems, penalty method has a risk whereby all constraints cannot be satisfied. This paper proposes the stepwise satisfaction method of constraints to satisfy conditions that involve many constraints. In the proposed method, the priority of constraints to be satisfied is defined based on the initial population and the objective functions are optimized after the satisfaction of all constraints. Furthermore, this paper studies the effects of classifying constraints into difficult and easy ones as well as combining the proposed method with the penalty method. In the experiment, the performance of the proposed method and the penalty method was compared in two problems with more than 50 constraints.
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Artificial Tribe Algorithm (ATA) is a novel optimization algorithm. This paper presents the comparison results on the performance of the ATA for solving constrained optimization problems. The penalty function method and non-parameter penalty method are applied to a set of constrained problems. The simulation results show that ATA is an efficient algorithm for constrained optimization problems.
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Constrained optimization problem
Constrained optimization
Tribe
Function optimization
Optimization algorithm
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Interior point method
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In this paper, a novel penalty function approach is proposed for constrained optimization problems with linear and nonlinear constraints. It is shown that by using a mapping function to "wrap" up the constraints, a constrained optimization problem can be converted to an unconstrained optimization problem. It is also proved mathematically that the best solution of the converted unconstrained optimization problem will approach the best solution of the constrained optimization problem if the tuning parameter for the wrapping function approaches zero. A tailored genetic algorithm incorporating an adaptive tuning method is then used to search for the global optimal solutions of the converted unconstrained optimization problems. Four test examples were used to show the effectiveness of the approach.
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The penalty function is one of the most commonly used approaches for constrained optimization problems. However, it often leads to additional parameters and the parameters are not easy for the users to select. A new way without additional parameters to deal the constrained optimizations was proposed. Firstly, a new penalty function was defined using the constrained functions without additional parameters. Secondly, combining the penalty function and the original objective function, a new objective function without any constrained conditions was got. Then Differential Evolution algorithm was used to solve the non-constrained optimization problem. The numerical experiments show its advantage over the other existing method.
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Constrained optimization problem
Differential Evolution
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In this paper, we combined Langrage Multiplier, Penalty Function and Conjugate Gradient Methods (CPLCGM), to enable Conjugate Gradient Method (CGM) to be employed for solving constrained optimization problems. In the year past, Langrage Multiplier Method (LMM) has been used extensively to solve constrained optimization problems likewise Penalty Function Method (PFM). However, with some special features in CGM, which makes it unique in solving unconstrained optimization problems, we see that this features we be advantageous to solve constrained optimization problems if it can be properly amended. This, then call for the CPLCGM that is aimed at taking care of some constrained optimization problems, either with equality or inequality constraint but in this paper, we focus on equality constraints. The authors of this paper desired that, with the construction of the new Algorithm, it will circumvent the difficulties undergone using only LMM and as well as PFM to solve constrained optimization problems and its application will further improve the result of the Conjugate Gradient Method in solving this class of optimization problem. We applied the new algorithm to some constrained optimization problems and compared the results with the LMM and PFM.
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Augmented Lagrangian method
Constrained optimization problem
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Constrained optimization problems in mechanical engineering are very difficult for the optimization algorithm. In 2013, an improved version of constrained differential evolution, named ArATM-ICDE was proposed to optimize the constrained optimization problem. An archiving-based adaptive trade-off model (ArATM) was constructed to handle the constraints; resulting in an algorithm referred to as ArATM-ICDE. This paper applies ArATM-ICDE to solve constraint optimization problems in mechanical engineering. We also combine the penalty technique for constraint handling into the ICDE, named Penalty-ICDE; which compares the abilities of the constraint handling techniques. Our experiments were conducted on ten widely used constraint engineering optimization problems. The experiment results proved the ArATM-ICDE to be more reliable than the Penalty-ICDE. Additionally, ArATM-ICDE consumed a lesser number of function calls than Penalty-ICDE. This paper further compared the effectiveness of ArATM-ICDE and Penalty-ICDE with eight state-of-the-art algorithms, which revealed that ArATM-ICDE and Penalty-ICDE produced solutions of higher quality than those produced by the comparative algorithms. The ArATM-ICDE also consumed less effort in its process.
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Constrained optimization
Constrained optimization problem
Differential Evolution
Engineering optimization
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Penalty Method
Constrained optimization
Feasible region
Convexity
Constrained optimization problem
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