scholarly journals Test Problem Construction for Single-Objective Bilevel Optimization

2014 ◽  
Vol 22 (3) ◽  
pp. 439-477 ◽  
Author(s):  
Ankur Sinha ◽  
Pekka Malo ◽  
Kalyanmoy Deb
Keyword(s):  
Evolutionary Algorithm ◽  
Optimization Algorithm ◽  
Test Problem ◽  
Bilevel Optimization ◽  
Standard Test ◽  
Test Problems ◽  
Problem Construction ◽  
Constrained Problems ◽  
Construction Procedure ◽  
Single Objective

In this paper, we propose a procedure for designing controlled test problems for single-objective bilevel optimization. The construction procedure is flexible and allows its user to control the different complexities that are to be included in the test problems independently of each other. In addition to properties that control the difficulty in convergence, the procedure also allows the user to introduce difficulties caused by interaction of the two levels. As a companion to the test problem construction framework, the paper presents a standard test suite of 12 problems, which includes eight unconstrained and four constrained problems. Most of the problems are scalable in terms of variables and constraints. To provide baseline results, we have solved the proposed test problems using a nested bilevel evolutionary algorithm. The results can be used for comparison, while evaluating the performance of any other bilevel optimization algorithm. The code related to the paper may be accessed from the website http://bilevel.org .

Research of Biogeography-Based Multi-Objective Evolutionary Algorithm

2011 ◽  
Vol 4 (2) ◽  
pp. 70-80 ◽  
Author(s):  
Hongwei Mo ◽  
Zhidan Xu
Keyword(s):  
Evolutionary Algorithm ◽  
Optimization Algorithm ◽  
Solution Set ◽  
Test Problems ◽  
Benchmark Test ◽  
Multi Objective ◽  
Simulation Results ◽  
Effectiveness And Efficiency ◽  
Evolutionary Operators ◽  
External Population

Biogeography-based optimization algorithm (BBO) is an optimization algorithm inspired by the migration of animals in nature. A new multi-objective evolutionary algorithm is proposed, which is called Biogeography-based multi-objective evolutionary algorithm (BBMOEA). The fitness assignment and the external population elitism of SPEA2 are adapted to ensure even distribution of the solution set. The population evolutionary operators of BBO are applied to the evolution of the external population to ensure the convergence of the solution set. Simulation results on benchmark test problems illustrate the effectiveness and efficiency of the proposed algorithm.

An Enhanced Memetic Algorithm for Single-Objective Bilevel Optimization Problems

2017 ◽  
Vol 25 (4) ◽  
pp. 607-642 ◽  
Author(s):  
Md Monjurul Islam ◽  
Hemant Kumar Singh ◽  
Tapabrata Ray ◽  
Ankur Sinha
Keyword(s):  
Local Search ◽  
Memetic Algorithm ◽  
Optimization Problems ◽  
Computational Cost ◽  
Real Life ◽  
Practical Interest ◽  
Bilevel Optimization ◽  
Standard Test ◽  
Computational Effort ◽  
Test Problems

Bilevel optimization, as the name reflects, deals with optimization at two interconnected hierarchical levels. The aim is to identify the optimum of an upper-level  leader problem, subject to the optimality of a lower-level follower problem. Several problems from the domain of engineering, logistics, economics, and transportation have an inherent nested structure which requires them to be modeled as bilevel optimization problems. Increasing size and complexity of such problems has prompted active theoretical and practical interest in the design of efficient algorithms for bilevel optimization. Given the nested nature of bilevel problems, the computational effort (number of function evaluations) required to solve them is often quite high. In this article, we explore the use of a Memetic Algorithm (MA) to solve bilevel optimization problems. While MAs have been quite successful in solving single-level optimization problems, there have been relatively few studies exploring their potential for solving bilevel optimization problems. MAs essentially attempt to combine advantages of global and local search strategies to identify optimum solutions with low computational cost (function evaluations). The approach introduced in this article is a nested Bilevel Memetic Algorithm (BLMA). At both upper and lower levels, either a global or a local search method is used during different phases of the search. The performance of BLMA is presented on twenty-five standard test problems and two real-life applications. The results are compared with other established algorithms to demonstrate the efficacy of the proposed approach.

Research of Biogeography-Based Multi-Objective Evolutionary Algorithm

2013 ◽  
pp. 125-135 ◽  
Author(s):  
Hongwei Mo ◽  
Zhidan Xu
Keyword(s):  
Evolutionary Algorithm ◽  
Optimization Algorithm ◽  
Solution Set ◽  
Test Problems ◽  
Benchmark Test ◽  
Multi Objective ◽  
Simulation Results ◽  
Effectiveness And Efficiency ◽  
Evolutionary Operators ◽  
External Population

Biogeography-based optimization algorithm (BBO) is an optimization algorithm inspired by the migration of animals in nature. A new multi-objective evolutionary algorithm is proposed, which is called Biogeography-based multi-objective evolutionary algorithm (BBMOEA). The fitness assignment and the external population elitism of SPEA2 are adapted to ensure even distribution of the solution set. The population evolutionary operators of BBO are applied to the evolution of the external population to ensure the convergence of the solution set. Simulation results on benchmark test problems illustrate the effectiveness and efficiency of the proposed algorithm.

DIFFERENTIAL EVOLUTION FOR SOLVING MULTIOBJECTIVE OPTIMIZATION PROBLEMS

2004 ◽  
Vol 21 (02) ◽  
pp. 225-240 ◽  
Author(s):  
RUHUL SARKER ◽  
HUSSEIN A. ABBASS
Keyword(s):  
Differential Evolution ◽  
Evolutionary Algorithm ◽  
Optimization Problems ◽  
Pareto Frontier ◽  
Population Based ◽  
Standard Test ◽  
Evolutionary Strategies ◽  
Test Problems ◽  
Strength Pareto Evolutionary Algorithm ◽  
Multiobjective Optimization Problems

The use of evolutionary strategies (ESs) to solve problems with multiple objectives [known as vector optimization problems (VOPs)] has attracted much attention recently. Being population-based approaches, ESs offer a means to find a set of Pareto-optimal solutions in a single run. Differential evolution (DE) is an ES that was developed to handle optimization problems over continuous domains. The objective of this paper is to introduce a novel Pareto-frontier differential evolution (PDE) algorithm to solve VOPs. The solutions provided by the proposed algorithm for two standard test problems, outperform the "strength Pareto evolutionary algorithm", one of the state-of-the-art evolutionary algorithm for solving VOPs.

Multimodal Optimization Using a Bi-Objective Evolutionary Algorithm

2012 ◽  
Vol 20 (1) ◽  
pp. 27-62 ◽  
Author(s):  
Kalyanmoy Deb ◽  
Amit Saha
Keyword(s):  
Evolutionary Algorithm ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Search Space ◽  
Test Problems ◽  
Multimodal Optimization ◽  
Optimal Solutions ◽  
Optimization Task ◽  
Pareto Optimal Set ◽  
Single Objective

In a multimodal optimization task, the main purpose is to find multiple optimal solutions (global and local), so that the user can have better knowledge about different optimal solutions in the search space and as and when needed, the current solution may be switched to another suitable optimum solution. To this end, evolutionary optimization algorithms (EA) stand as viable methodologies mainly due to their ability to find and capture multiple solutions within a population in a single simulation run. With the preselection method suggested in 1970, there has been a steady suggestion of new algorithms. Most of these methodologies employed a niching scheme in an existing single-objective evolutionary algorithm framework so that similar solutions in a population are deemphasized in order to focus and maintain multiple distant yet near-optimal solutions. In this paper, we use a completely different strategy in which the single-objective multimodal optimization problem is converted into a suitable bi-objective optimization problem so that all optimal solutions become members of the resulting weak Pareto-optimal set. With the modified definitions of domination and different formulations of an artificially created additional objective function, we present successful results on problems with as large as 500 optima. Most past multimodal EA studies considered problems having only a few variables. In this paper, we have solved up to 16-variable test problems having as many as 48 optimal solutions and for the first time suggested multimodal constrained test problems which are scalable in terms of number of optima, constraints, and variables. The concept of using bi-objective optimization for solving single-objective multimodal optimization problems seems novel and interesting, and more importantly opens up further avenues for research and application.

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

2020 ◽  
Vol 35 (6) ◽  
pp. 355-366
Author(s):  
Vladimir V. Shashkin ◽  
Gordey S. Goyman
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Time Integration ◽  
Standard Test ◽  
Integration Scheme ◽  
Test Problems ◽  
Cubed Sphere ◽  
Lagrangian Scheme ◽  
Inertia Gravity Waves

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

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