scholarly journals A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Wilhelm P. Freire ◽  
Afonso C. C. Lemonge ◽  
Tales L. Fonseca ◽  
Hernando J. R. Franco
Keyword(s):  
Genetic Algorithm ◽  
Constrained Optimization ◽  
Augmented Lagrangian ◽  
Dual Problem ◽  
Optimization Problems ◽  
Lagrangian Duality ◽  
Dual Function ◽  
Constrained Optimization Problems ◽  
Nonconvex Constrained Optimization ◽  
Nonsmooth And Nonconvex Optimization

The Interior Epigraph Directions (IED) method for solving constrained nonsmooth and nonconvex optimization problem via Generalized Augmented Lagrangian Duality considers the dual problem induced by a Generalized Augmented Lagrangian Duality scheme and obtains the primal solution by generating a sequence of iterates in the interior of the epigraph of the dual function. In this approach, the value of the dual function at some point in the dual space is given by minimizing the Lagrangian. The first version of the IED method uses the Matlab routine fminsearch for this minimization. The second version uses NFDNA, a tailored algorithm for unconstrained, nonsmooth and nonconvex problems. However, the results obtained with fminsearch and NFDNA were not satisfactory. The current version of the IED method, presented in this work, employs a Genetic Algorithm, which is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as GA, is applied alone to solve constrained optimization problems. Two sets of constrained optimization problems from mathematics and mechanical engineering were solved and compared with literature. It is shown that the proposed hybrid algorithm is able to solve problems where fminsearch and NFDNA fail.

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 A Genetic Algorithm Using Infeasible Solutions for Constrained Optimization Problems

2014 ◽  
Vol 8 (1) ◽  
pp. 904-912 ◽  
Author(s):  
Yalong Zhang ◽  
Hisakazu Ogura ◽  
Xuan Ma ◽  
Jousuke Kuroiwa ◽  
Tomohiro Odaka
Keyword(s):  
Genetic Algorithm ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Constrained Optimization Problems

scholarly journals Improvement Analysis and Application of Real-Coded Genetic Algorithm for Solving Constrained Optimization Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Jiquan Wang ◽  
Zhiwen Cheng ◽  
Okan K. Ersoy ◽  
Panli Zhang ◽  
Weiting Dai ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Local Search ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Global Search ◽  
Single Mutation ◽  
Constrained Optimization Problems ◽  
Fitness Value ◽  
Steering Mechanism ◽  
Real Coded Genetic Algorithm

An improved real-coded genetic algorithm (IRCGA) is proposed to solve constrained optimization problems. First, a sorting grouping selection method is given with the advantage of easy realization and not needing to calculate the fitness value. Secondly, a heuristic normal distribution crossover (HNDX) operator is proposed. It can guarantee the cross-generated offsprings to locate closer to the better one among the two parents and the crossover direction to be very close to the optimal crossover direction or to be consistent with the optimal crossover direction. In this way, HNDX can ensure that there is a great chance of generating better offsprings. Thirdly, since the GA in the existing literature has many iterations, the same individuals are likely to appear in the population, thereby making the diversity of the population worse. In IRCGA, substitution operation is added after the crossover operation so that the population does not have the same individuals, and the diversity of the population is rich, thereby helping avoid premature convergence. Finally, aiming at the shortcoming of a single mutation operator which cannot simultaneously take into account local search and global search, this paper proposes a combinational mutation method, which makes the mutation operation take into account both local search and global search. The computational results with nine examples show that the IRCGA has fast convergence speed. As an example application, the optimization model of the steering mechanism of vehicles is formulated and the IRCGA is used to optimize the parameters of the steering trapezoidal mechanism of three vehicle types, with better results than the other methods used.

scholarly journals Error Bounds and Finite Termination for Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wenling Zhao ◽  
Daojin Song ◽  
Bingzhuang Liu
Keyword(s):  
Constrained Optimization ◽  
Error Bound ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Merit Function ◽  
Global Error Bound ◽  
Constrained Optimization Problems ◽  
Nonconvex Constrained Optimization ◽  
Necessary And Sufficient

We present a global error bound for the projected gradient of nonconvex constrained optimization problems and a local error bound for the distance from a feasible solution to the optimal solution set of convex constrained optimization problems, by using the merit function involved in the sequential quadratic programming (SQP) method. For the solution sets (stationary points set andKKTpoints set) of nonconvex constrained optimization problems, we establish the definitions of generalized nondegeneration and generalized weak sharp minima. Based on the above, the necessary and sufficient conditions for a feasible solution of the nonconvex constrained optimization problems to terminate finitely at the two solutions are given, respectively. Accordingly, the results in this paper improve and popularize existing results known in the literature. Further, we utilize the global error bound for the projected gradient with the merit function being computed easily to describe these necessary and sufficient conditions.

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