Rituparna Datta

A fast and accurate solution of constrained optimization problems using a hybrid bi-objective and penalty function approach

Evolutionary algorithms are modified in various ways to solve constrained optimization problems. Of them, the use of a bi-objective evolutionary algorithm in which the minimization of the constraint violation is included as an additional objective, has received a significant attention. Classical penalty function approach is another common methodology which requires an appropriate knowledge of the associated penalty parameter. In this paper, we combine a bi-objective evolutionary approach with the penalty function methodology in a manner complementary to each other. The bi-objective optimization approach provides a good estimate of the penalty parameter, while the unconstrained penalty function approach using classical means provides the overall hybrid algorithm its convergence property. We demonstrate the working of the procedure on a two-variable problem and then solve a number of standard numerical test problems from the EA literature. In all cases, our proposed hybrid methodology is observed to take one or more orders of magnitude smaller number of function evaluations to find the constrained minimum solution accurately. To the best of our knowledge, no previous evolutionary constrained optimization algorithm has reported such a fast and accurate performance on the chosen problems.

A hybrid evolutionary multi-objective and SQP based procedure for constrained optimization

In this paper, we propose a hybrid reference-point based evolutionary multi-objective optimization (EMO) algorithm coupled with the classical SQP procedure for solving constrained single-objective optimization problems. The reference point based EMO procedure allows the procedure to focus its search near the constraint boundaries, while the SQP methodology acts as a local search to improve the solutions. The hybrid procedure is shown to solve a number of state-of-the-art constrained test problems with success. In some of the difficult problems, the SQP procedure alone is unable to find the true optimum, while the combined procedure solves them repeatedly. The proposed procedure is now ready to be tested on real-world optimization problems.

Hybrid evolutionary multi-objective optimization and analysis of machining operations

Evolutionary multi-objective optimization (EMO) has received significant attention in recent studies in engineering design and analysis due to its flexibility, wide-spread applicability and ability to find multiple trade-off solutions. Optimal machining parameter determination is an important matter for ensuring an efficient working of a machining process. In this article, the use of an EMO algorithm and a suitable local search procedure to optimize the machining parameters (cutting speed, feed and depth of cut) in turning operations is described. Thereafter, the efficiency of the proposed methodology is demonstrated through two case studies – one having two objectives and the other having three objectives. Then, EMO solutions are modified using a local search procedure to achieve a better convergence property. It has been demonstrated here that a proposed heuristics-based local search procedure in which the problem-specific heuristics are derived from an innovization study performed on the EMO solutions is a computationally faster approach than the original EMO procedure. The methodology adopted in this article can be used in other machining tasks or in other engineering design activities.

A bi-objective constrained optimization algorithm using a hybrid evolutionary and penalty function approach

Constrained optimization is a computationally difficult task, particularly if the constraint functions are nonlinear and non-convex. As a generic classical approach, the penalty function approach is a popular methodology which degrades the objective function value by adding a penalty proportional to the constraint violation. However, the penalty function approach has been criticized for its sensitivity to the associated penalty parameters. Since its inception, evolutionary algorithms have been modified in various ways to solve constrained optimization problems. Of them, the recent use of a bi-objective evolutionary algorithm in which the minimization of the constraint violation is included as an additional objective has received significant attention. In this article, a combination of a bi-objective evolutionary approach with the classical penalty function methodology is proposed, in a manner complementary to each other. The evolutionary approach provides an appropriate estimate of the penalty parameter, while the solution of an unconstrained penalized function by a classical method induces a convergence property to the overall hybrid algorithm. The working of the procedure on a number of standard numerical test problems and an engineering design problem is demonstrated. In most cases, the proposed hybrid methodology is observed to take one or more orders of magnitude fewer function evaluations to find the constrained minimum solution accurately than some of the best reported existing methodologies.

Hybrid gradient projection based Genetic Algorithms for constrained optimization

Genetic Algorithms (GAs) are a highly successful population based approach to solve global optimization problems. They have carved out a niche for themselves in solving optimization problems of varying difficulty levels involving single and multiple objectives. Most real-world optimization problems involve equality and / or inequality constraints and hence posed as constrained optimization problems. The most common approach to solve such problems using GAs is the method of penalty functions, which however suffers from the drawback of appropriate selection of penalty parameters for their optimal functioning. Given the nature of the problems at hand, we have used an adaptive mutation based Real-Coded GA (RGA), which uses a popular penalty parameter-less approach to handle constraints and search the feasible region effectively for the global best solution, and at the same time use an adaptive mutation strategy to maintain diversity in the population to enable creation of new solutions. We have coupled our RGA with ideas from the gradient projection method to specifically handle equality constraints. We have found our simple procedure working quite well in most of the test problems provided as part of the competition on Single-objective Constrained Real Parameter Optimization in CEC 2010 and hence simplicity remains the hallmark of our study here.

A surrogate-assisted evolution strategy for constrained multi-objective optimization

New surrogate-assisted ES for constrained multi-objective optimization is developed.Surrogates are used to identify the most promising among many trial offspring.A radial basis function (RBF) model is used to implement the method.Method is tested on benchmark problems and manufacturing and robotics applications.Proposed method generally outperforms an ES and NSGA-II on the problems used. In many real-world optimization problems, several conflicting objectives must be achieved and optimized simultaneously and the solutions are often required to satisfy certain restrictions or constraints. Moreover, in some applications, the numerical values of the objectives and constraints are obtained from computationally expensive simulations. Many multi-objective optimization algorithms for continuous optimization have been proposed in the literature and some have been incorporated or used in conjunction with expert and intelligent systems. However, relatively few of these multi-objective algorithms handle constraints, and even fewer, use surrogates to approximate the objective or constraint functions when these functions are computationally expensive. This paper proposes a surrogate-assisted evolution strategy (ES) that can be used for constrained multi-objective optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. Such an algorithm can be incorporated into an intelligent system that finds approximate Pareto optimal solutions to simulation-based constrained multi-objective optimization problems in various applications including engineering design optimization, production management and manufacturing. The main idea in the proposed algorithm is to generate a large number of trial offspring in each generation and use the surrogates to predict the objective and constraint function values of these trial offspring. Then the algorithm performs an approximate non-dominated sort of the trial offspring based on the predicted objective and constraint function values, and then it selects the most promising offspring (those with the smallest predicted ranks from the non-dominated sort) to become the actual offspring for the current generation that will be evaluated using the expensive objective and constraint functions. The proposed method is implemented using cubic radial basis function (RBF) surrogate models to assist the ES. The resulting RBF-assisted ES is compared with the original ES and to NSGA-II on 20 test problems involving 2-15 decision variables, 2-5 objectives and up to 13 inequality constraints. These problems include well-known benchmark problems and application problems in manufacturing and robotics. The numerical results showed that the RBF-assisted ES generally outperformed the original ES and NSGA-II on the problems used when the computational budget is relatively limited. These results suggest that the proposed surrogate-assisted ES is promising for computationally expensive constrained multi-objective optimization.

A classical-cum-Evolutionary Multi-objective optimization for optimal machining parameters

Optimal machining parameters are very important for every machining process. This paper presents an Evolutionary Multi-objective Genetic Algorithm based optimization technique to optimize the machining parameters (cutting speed, feed and depth of cut) in a turning process. The effect of these parameters on production time, production cost and surface roughness (which are conflicting to each other) are mathematically formulated. The non-dominated sorting genetic algorithm (NSGA-II) is used to get a Pareto-optimal front of the machining problem. The Pareto-optimal points are checked using ε-constraint single objective GA as well as using a classical optimization (SQP) method. An analysis of the obtained points is carried out to find the useful relation between the objective function and variable values.

Multi-objective design and analysis of robot gripper configurations using an evolutionary-classical approach

This paper is concerned with the determination of optimum forces extracted by robot grippers on the surface of a grasped rigid object -- a matter which is crucial to guarantee the stability of the grip without causing defect or damage to the grasped object. A multi-criteria optimization of robot gripper design problem is solved with two different configurations involving two conflicting objectives and a number of constraints. The objectives involve minimization of the difference between maximum and minimum gripping forces and simultaneous minimization of the transmission ratio between the applied gripper actuator force and the force experienced at the gripping ends. Two different configurations of the robot gripper are designed by a state-of-the-art algorithm (NSGA-II) and the obtained results are compared with a previous study. Due to presence of geometric constraints, the resulting optimization problem is highly non-linear and multi-modal. For both gripper configurations, the proposed methodology outperforms the results of the previous study. The Pareto-optimal solutions are thoroughly investigated to establish some meaningful relationships between the objective functions and variable values. In addition, it is observed that one of the gripper configurations completely outperforms the other one from the point of view of both objectives, thereby establishing a complete bias towards the use of one of the configurations in practice.

Evolutionary Constrained Optimization

This book makes available a self-contained collection of modern research addressing the general constrained optimization problems using evolutionary algorithms. Broadly the topics covered include constraint handling for single and multi-objective optimizations; penalty function based methodology; multi-objective based methodology; new constraint handling mechanism; hybrid methodology; scaling issues in constrained optimization; design of scalable test problems; parameter adaptation in constrained optimization; handling of integer, discrete and mix variables in addition to continuous variables; application of constraint handling techniques to real-world problems; and constrained optimization in dynamic environment. There is also a separate chapter on hybrid optimization, which is gaining lots of popularity nowadays due to its capability of bridging the gap between evolutionary and classical optimization. The material in the book is useful to researchers, novice, and experts alike. The book will also be useful for classroom teaching and future research.

An adaptive normalization based constrained handling methodology with hybrid bi-objective and penalty function approach

A hybrid adaptive normalization based constraint handling approach is proposed in the present study. In most constrained optimization problems, constraints may be of different scale. Normalization of constraints is crucial for the efficient performance of a constraint handling algorithm. A growing number of researchers have proposed different strategies using bi-objective methodologies. Classical penalty function approach is another common method among both evolutionary and classical optimization research communities due to its simplicity and ease of implementation. In the present study, we propose a hybrid approach of both bi-objective method and the penalty function approach where constraints are normalized adaptively during the optimization process. The proposed bi-objective evolutionary method estimates the penalty parameter and the starting solution needed for the penalty function approach. We test and compare our algorithm on seven mathematical test problems and two engineering design problems taken from the literature. We compare our obtained results with our previous studies in terms of function evaluations and solution accuracy. The obtained optima are also compared with those of other standard algorithms. In many cases, our proposed methodology perform better than all algorithms considered in this study. Results are promising and motivate further application of the proposed adaptive normalization strategy.