Solving the G-problems in less than 500 iterations: Improved efficient constrained optimization by surrogate modeling and adaptive parameter control

Constrained optimization of high-dimensional numerical problems plays an important role in many scientific and industrial applications. Function evaluations in many industrial applications are severely limited and no analytical information about objective function and constraint functions is available. For such expensive black-box optimization tasks, the constraint optimization algorithm COBRA was proposed, making use of RBF surrogate modeling for both the objective and the constraint functions. COBRA has shown remarkable success in solving reliably complex benchmark problems in less than 500 function evaluations. Unfortunately, COBRA requires careful adjustment of parameters in order to do so. In this work we present a new self-adjusting algorithm SACOBRA, which is based on COBRA and capable to achieve high-quality results with very few function evaluations and no parameter tuning. It is shown with the help of performance profiles on a set of benchmark problems (G-problems, MOPTA08) that SACOBRA consistently outperforms any COBRA algorithm with fixed parameter setting. We analyze the importance of the several new elements in SACOBRA and find that each element of SACOBRA plays a role to boost up the overall optimization performance. We discuss the reasons behind and get in this way a better understanding of high-quality RBF surrogate modeling

A modified differential evolution based solution technique for economic dispatch problems

Economic dispatch (ED) plays one of the major roles in power generation systems. The objective of economic dispatch problem is to find the optimal combination of power dispatches from different power generating units in a given time period to minimize the total generation cost while satisfying the specified constraints. Due to valve-point loading effects the objective function becomes nondifferentiable and has many local minima in the solution space. Traditional methods may fail to reach the global solution of ED problems. Most of the existing stochastic methods try to make the solution feasible or penalize an infeasible solution with penalty function method. However, to find the appropriate penalty parameter is not an easy task. Differential evolution is a population-based heuristic approach that has been shown to be very efficient to solve global optimization problems with simple bounds. In this paper, we propose a modified differential evolution based solution technique along with a tournament selection that makes pair-wise comparison among feasible and infeasible solutions based on the degree of constraint violation for economic dispatch problems. We reformulate the nonsmooth objective function to a smooth one and add nonlinear inequality constraints to original ED problems. We consider five ED problems and compare the obtained results with existing standard deterministic NLP solvers as well as with other stochastic techniques available in literature.Fundação para a Ciência e a Tecnologia (FCT

An artificial fish swarm filter-based Method for constrained global optimization

Ana Maria A.C. Rocha, M. Fernanda P. Costa and Edite M.G.P. Fernandes, An Artificial Fish Swarm Filter-Based Method for Constrained Global Optimization, B. Murgante, O. Gervasi, S. Mirsa, N. Nedjah, A.M. Rocha, D. Taniar, B. Apduhan (Eds.), Lecture Notes in Computer Science, Part III, LNCS 7335, pp. 57–71, Springer, Heidelberg, 2012.An artificial fish swarm algorithm based on a filter methodology for trial solutions acceptance is analyzed for general constrained global optimization problems. The new method uses the filter set concept to accept, at each iteration, a population of trial solutions whenever they improve constraint violation or objective function, relative to the current solutions. The preliminary numerical experiments with a wellknown benchmark set of engineering design problems show the effectiveness of the proposed method.Fundação para a Ciência e a Tecnologia (FCT

Composite Differential Evolution for Constrained Evolutionary Optimization

When solving constrained optimization problems (COPs) by evolutionary algorithms, the search algorithm plays a crucial role. In general, we expect that the search algorithm has the capability to balance not only diversity and convergence but also constraints and objective function during the evolution. For this purpose, this paper proposes a composite differential evolution (DE) for constrained optimization, which includes three different trial vector generation strategies with distinct advantages. In order to strike a balance between diversity and convergence, one of these three trial vector generation strategies is able to increase diversity, and the other two exhibit the property of convergence. In addition, to accomplish the tradeoff between constraints and objective function, one of the two trial vector generation strategies for convergence is guided by the individual with the least degree of constraint violation in the population, and the other is guided by the individual with the best objective function value in the population. After producing offspring by the proposed composite DE, the feasibility rule and the ϵ constrained method are combined elaborately for selection in this paper. Moreover, a restart scheme is proposed to help the population jump out of a local optimum in the infeasible region for some extremely complicated COPs. By assembling the above techniques together, a constrained composite DE is proposed. The experiments on two sets of benchmark test functions with various features, i.e., 24 test functions from IEEE CEC2006 and 18 test functions with 10 dimensions and 30 dimensions from IEEE CEC2010, have demonstrated that the proposed method shows better or at least competitive performance against other state-of-the-art methods

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an ε -global minimizer is proved. At each iteration k, the framework requires the ε(k) -global minimizer of a subproblem, where ε(k)→ε . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an ε(k) -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the ε(k)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ci ˜ encia e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

A review of optimization techniques in spacecraft flight trajectory design

For most atmospheric or exo-atmospheric spacecraft flight scenarios, a well-designed trajectory is usually a key for stable flight and for improved guidance and control of the vehicle. Although extensive research work has been carried out on the design of spacecraft trajectories for different mission profiles and many effective tools were successfully developed for optimizing the flight path, it is only in the recent five years that there has been a growing interest in planning the flight trajectories with the consideration of multiple mission objectives and various model errors/uncertainties. It is worth noting that in many practical spacecraft guidance, navigation and control systems, multiple performance indices and different types of uncertainties must frequently be considered during the path planning phase. As a result, these requirements bring the development of multi-objective spacecraft trajectory optimization methods as well as stochastic spacecraft trajectory optimization algorithms. This paper aims to broadly review the state-of-the-art development in numerical multi-objective trajectory optimization algorithms and stochastic trajectory planning techniques for spacecraft flight operations. A brief description of the mathematical formulation of the problem is firstly introduced. Following that, various optimization methods that can be effective for solving spacecraft trajectory planning problems are reviewed, including the gradient-based methods, the convexification-based methods, and the evolutionary/metaheuristic methods. The multi-objective spacecraft trajectory optimization formulation, together with different class of multi-objective optimization algorithms, is then overviewed. The key features such as the advantages and disadvantages of these recently-developed multi-objective techniques are summarised. Moreover, attentions are given to extend the original deterministic problem to a stochastic version. Some robust optimization strategies are also outlined to deal with the stochastic trajectory planning formulation. In addition, a special focus will be given on the recent applications of the optimized trajectory. Finally, some conclusions are drawn and future research on the development of multi-objective and stochastic trajectory optimization techniques is discussed

Application of Genetic Algorithm in Multi-objective Optimization of an Indeterminate Structure with Discontinuous Space for Support Locations

In this thesis, an indeterminate structure was developed with multiple competing objectives including the equalization of the load distribution among the supports while maximizing the stability of the structure. Two different coding algorithms named “Continuous Method” and “Discretized Method” were used to solve the optimal support locations using Genetic Algorithms (GAs). In continuous method, a continuous solution space was considered to find optimal support locations. The failure of this method to stick to the acceptable optimal solution led towards the development of the second method. The latter approach divided the solution space into rectangular grids, and GAs acted on the index number of the nodal points to converge to the optimality. The average value of the objective function in the discretized method was found to be 0.147 which was almost onethird of that obtained by the continuous method. The comparison based on individual components of the objective function also proved that the proposed method outperformed the continuous method. The discretized method also showed faster convergence to the optima. Three circular discontinuities were added to the structure to make it more realistic and three different penalty functions named flat, linear and non-linear penalty were used to handle the constraints. The performance of the two methods was observed with the penalty functions while increasing the radius of the circles by 25% and 50% which showed no significant difference. Later, the discretized method was coded to eliminate the discontinuous area from the solution space which made the application of the penalty functions redundant. A paired t-test (α=5%) showed no statistical difference between these two methods. Finally, to make the proposed method compatible with irregular shaped discontinuous areas, “FEA Integrated Coded Discretized Method (FEAICDM)” was developed. The manual elimination of the infeasible areas from the candidate surface was replaced by the nodal points of the mesh generated by Solid Works. A paired t-test (α=5%) showed no statistical difference between these two methods. Though FEAICDM was applied only to a class of problem, it can be concluded that FEAICDM is more robust and efficient than the continuous method for a class of constrained optimization problem