Dynamic Penalty Function Research Articles

A Multi-Objective PSO Algorithm for Energy-Efficient Scheduling

Based on the basic model of economic dispatch and energy-efficient power generation scheduling rules, the multi-objective optimization model of energy-efficient scheduling is given. The model is transformed into unconstrained optimization problem by means of penalty factor and the dynamic penalty function, and then solved by PSO algorithm. The results obtained demonstrate the rationality and effectiveness of the proposed model.

Multi-objective optimization of engineering systems using game theory and particle swarm optimization

This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.

The numerical condition of electron correlation theories when only active pairs of electrons are spin-unrestricted

The use of spin-unrestriction with high-quality correlation theory, such as coupled-cluster (CC) methods, is a common practice necessary to obtain high-quality potential energy surfaces. While this typically is a useful approach, we find that in the unrestricted limit of ROHF fragments (the unrestricted in active pair orbitals) the CC equations are singular if only the strongly correlated electrons are considered. Unstable amplitudes which do not represent the physics of the problem are easily found and could be unwittingly accepted without inspection. We use stability analysis and the condition number of the CC doubles Jacobian matrix to examine the problem, and present results for several molecular systems with a variety of unrestricted cluster models. Finally a regularization of the CC equations is proposed, using a dynamic penalty function, which allows us to apply CC, and Lagrangian gradient formulas even in the singular limit.

A genetic approach to automate preliminary design of gear drives

Determination of volume or weight of a gearbox is an important issue in preliminary design of power transmission applications. Trial and error procedure or some gear standards information sheets are commonly used in traditional design. The purpose of this paper is to automate preliminary design of gear drives by minimizing volume of gear trains. A stochastic approach Genetic Algorithm (GA) was applied to a parallel axis two stage helical gear trains problem. Static and dynamic penalty functions were introduced to the objective function for handling the design constraints. The results were compared with a deterministic design procedure developed. GA based approach produced quite well results promptly supplying preliminary design parameters of gear drives for different gear ratios to the designer.

A Robust Archived Differential Evolution Algorithm for Global Optimization Problems

A robust archived differential evolution algorithm is put forward by means of embedding a flexibility processing operator and an efficiency processing operator based on original DE and ADE. A special constraint-handling mechanism based on dynamic penalty functions and fitness calculation of individuals is adopted in the proposed method to deal with various constraints effectively, which is further extended by means of a flexibility processing operator so as to make it suitable for different type problems, including those with or without constraint(s) and those with continuous, discrete or mixed discrete-continuous variables. Furthermore, an archive of solutions is maintained during the evolutionary process so as to keep the useful information of previous solutions and local optima for the estimation of new solutions. Based on the archive of solutions, an iterative control operator and an efficiency processing operator are designed in the algorithm. The former guides the evolutionary process towards a promising search space, avoiding unnecessary and worthless search. The latter improves the local searching efficiency and the final searching quality. Experimental results based on a suite of six well-known optimization problems reveal that the proposed algorithm is robust, effective, efficient and suitable for different type global optimization problems.

Dynamic penalty function method for the side constrained traffic assignment problem

The traffic assignment problem (TAP) is traditionally defined without side constraints. Although introducing side constraints into the TAP is used as a way of refining applied assignment models, it brings some computational difficulties that prevent the refined model from being applicable in real-case problems. In this paper an iterative solution algorithm is developed for TAP with linear side constraints. This method is based on implicitly considering the side constraints by means of adding a dynamic penalty function (DPF) to the link travel times. Each of the iterations of the algorithm is reduced to an unconstrained assignment followed by updating the penalty functions. The unconstrained assignment is carried out using the linearization method that has been derived from the complementarity formulation of TAP. The implementation of the algorithm and computational experiments on some well-known networks are also presented for two types of side constraint, i.e. link and node capacity constraints. For the link-type side constraint, compared to the reported results of the existing alternative approaches, a DPF algorithm finds feasible solutions with better Beckmann’s objective functions in less number of iterations. For the node-type constraints, the algorithm also shows good performance.

Properties of a genetic algorithm equipped with a dynamic penalty function

A genetic algorithm aiming for finding the global minimum and multiple deep local minima of a function exhibiting a complex landscape is studied. A feedback dynamic penalty function is used as a means to direct the algorithm to look for new local minima. The penalty is applied in close vicinity of all minima found before the current search stage. The last one, where the population tends to be trapped in, is treated smoothly. The penalty becomes progressively active there causing that the population progressively transfers outside the trapping area. The method ascertains that, unlike in more classical approaches, after finding the global minimum and a number of local ones on the way, the algorithm continues the exploration and identifies new local minima. Performance tests are described for a task of indexing of a powder diffraction pattern. The presented way of constructing the penalty function is to some extent problem specific, but the applied scheme may be adapted to other global search and optimisation problems, in particular to those requiring identification of multiple deep local minima.

Dynamic penalty function as a strategy in solving water resources combinatorial optimization problems with honey-bee mating optimization (HBMO) algorithm

Because of the complexity of some optimization problems, evolutionary and meta-heuristic algorithms are sometimes more applicable than the traditional optimization methods. Some difficulties in solving design-operation problems in the field of engineering are due to the multi-modality of the solution region of these problems. Since the design variables usually are specified as discrete variables and other continuous decision variables have to be set according to the range of the discrete ones, the possibility of trapping the final solution into some local optimum increases. In such cases, the capability of both traditional and evolutionary algorithms decreases. Thus, the development of a strategy to overcome this problem is the subject of this paper. For water utilities, one of the greatest potential areas for energy cost-savings is the effective scheduling of daily pump operations. Optimum design operation of pumping stations is a potential problem in this area that performs a wide background of solutions to this problem with different methods. Computation in all methods is driven by an objective function that includes operating and capital costs subject to various performances and hydraulic constraints. This paper achieves the optimal control and operation of an irrigation pumping station system by one of the latest tools used in optimization problems, which is the honey-bees mating optimization (HBMO) algorithm and is tested with a practical design. The HBMO algorithm with dynamic penalty function is presented and compared with two other well-known optimization tools which are the Lagrange multipliers (LM) method and genetic algorithms (GA) as well as with the previous results of the HBMO algorithm with constant penalty function for the same problem. The LM, GA and HBMO approaches simultaneously determine the least total annual cost of the pumping station and its operation. The solution includes the selection of pump type, capacity and the number of units, as well as scheduling the operation of irrigation pumps that results in minimum design and operating cost for a set of water demand curves. In this paper, the HBMO algorithm is applied and the dynamic penalty function is tested to demonstrate the efficiency of this combination simultaneously. The results are very promising and prove the ability of combining the dynamic penalty function with the HBMO algorithm for solving combinatorial design–operation optimization problems. Application of all these models to a real-world project shows not only considerable savings in cost and energy but also highlights the efficiency and capability of the dynamic penalty function in the HBMO algorithm for solving complex problems of this type.

BASIC—A genetic algorithm for engineering problems solution

This paper introduces in details a genetic algorithm-called BASIC, which is designed to take advantage of well known genetic schemes so as to be able to deal with numerous optimization problems. BASIC GA follows all common steps of the genetic algorithms. It involves real representation schemes for both real and integer variables. Three biased selection schemes for reproduction; four for recombination and three for mutation are applied in it and a new selection scheme for replacement is approached. BASIC GA can be easy adjusted to the concrete problems by fitting its global and local parameters. It provides an opportunity to the genetic operators to be extended with new schemes. A range of various optimization problems has been solved to test its capability. To handle all sorts of constraints the static and dynamic penalty functions are used. The solutions obtained are commensurable with other genetic algorithms and solution techniques.

Genetic algorithm with adaptive and dynamic penalty functions for the selection of cleaner production measures: a constrained optimization problem

This paper presents a new approach of genetic algorithm (GA) to solve the constrained optimization problem. In a constrained optimization problem, feasible and infeasible regions occupy the search space. The infeasible regions consist of the solutions that violate the constraint. Oftentimes classical genetic operators generate infeasible or invalid chromosomes. This situation takes a turn for the worse when infeasible chromosomes alone occupy the whole population. To address this problem, dynamic and adaptive penalty functions are proposed for the GA search process. This is a novel strategy because it will attempt to transform the constrained problem into an unconstrained problem by penalizing the GA fitness function dynamically and adaptively. New equations describing these functions are presented and tested. The effects of the proposed functions developed have been investigated and tested using different GA parameters such as mutation and crossover. Comparisons of the performance of the proposed adaptive and dynamic penalty functions with traditional static penalty functions are presented. The result from the experiments show that the proposed functions developed are more accurate, efficient, robust and easy to implement. The algorithms developed in this research can be applied to evaluate environmental impacts from process operations.

Fault-tolerant embedded system design and optimization considering reliability estimation uncertainty

In this paper, we model embedded system design and optimization, considering component redundancy and uncertainty in the component reliability estimates. The systems being studied consist of software embedded in associated hardware components. Very often, component reliability values are not known exactly. Therefore, for reliability analysis studies and system optimization, it is meaningful to consider component reliability estimates as random variables with associated estimation uncertainty. In this new research, the system design process is formulated as a multiple-objective optimization problem to maximize an estimate of system reliability, and also, to minimize the variance of the reliability estimate. The two objectives are combined by penalizing the variance for prospective solutions. The two most common fault-tolerant embedded system architectures, N-Version Programming and Recovery Block, are considered as strategies to improve system reliability by providing system redundancy. Four distinct models are presented to demonstrate the proposed optimization techniques with or without redundancy. For many design problems, multiple functionally equivalent software versions have failure correlation even if they have been independently developed. The failure correlation may result from faults in the software specification, faults from a voting algorithm, and/or related faults from any two software versions. Our approach considers this correlation in formulating practical optimization models. Genetic algorithms with a dynamic penalty function are applied in solving this optimization problem, and reasonable and interesting results are obtained and discussed.

Novel orthogonal simulated annealing with fractional factorial analysis to solve global optimization problems

A classical simulated annealing (SA) method is a generic probabilistic and heuristic approach to solving global optimization problems. It uses a stochastic process based on probability, rather than a deterministic procedure, to seek the minima or maxima in the solution space. Although the classical SA method can find the optimal solution to most linear and nonlinear optimization problems, the algorithm always requires numerous numerical iterations to yield a good solution. The method also usually fails to achieve optimal solutions to large parameter optimization problems. This study incorporates well-known fractional factorial analysis, which involves several factorial experiments based on orthogonal tables to extract intelligently the best combination of factors, with the classical SA to enhance the numerical convergence and optimal solution. The novel combination of the classical SA and fractional factorial analysis is termed the orthogonal SA herein. This study also introduces a dynamic penalty function to handle constrained optimization problems. The performance of the proposed orthogonal SA method is evaluated by computing several representative global optimization problems such as multi-modal functions, noise-corrupted data fitting, nonlinear dynamic control, and large parameter optimization problems. The numerical results show that the proposed orthogonal SA method markedly outperforms the classical SA in solving global optimization problems with linear or nonlinear objective functions. Additionally, this study addressed two widely used nonlinear functions, proposed by Keane and Himmelblau to examine the effectiveness of the orthogonal SA method and the presented penalty function when applied to the constrained problems. Moreover, the orthogonal SA method is applied to two engineering optimization design problems, including the designs of a welded beam and a coil compression spring, to evaluate the capacity of the method for practical engineering design. The computational results show that the proposed orthogonal SA method is effective in determining the optimal design variables and the value of objective function.

Size/geometry optimization of trusses by the force method and genetic algorithm

AbstractIn this article size/geometry optimization of trusses is performed using the force method and genetic algorithm. A large number of design variables consisting of cross‐sectional areas and nodal coordinates are involved in such an optimization, and due to a large number of constraints, the dimensions of the design space are often numerous and in the case of discrete values for cross sections usually discontinuous. In order to avoid local optima, modified genetic algorithms are developed. Furthermore, the force method is employed to improve the speed of the optimization. In the first phase of the described method, the initial geometry of the truss is fixed and near optimum ranges for the cross‐section areas are obtained using the relationships from the force method. In the second phase, the geometry of the structure is altered with the aim of designing lower‐weight structures. Within the Genetic Algorithm a new dynamic penalty function is defined and a modified process of reproduction is presented. A contraction process is also employed for the design space using shorter substrings for the design variables.

Optimal control of fed-batch fermentation involving multiple feeds using Differential Evolution

Differential Evolution (DE), an exceptionally simple and robust evolutionary algorithm with Lagrangian like method, was used for solving optimal control and parameter selection problems of fed-batch fermentation involving general constraints on state variables. These infinite dimensional optimization problems were approximated into the finite dimensional optimization problems by control vector parameterization. Integration of the dynamic penalty functions was used to ensure the feasible solution of these dynamic optimization problems. State and end-point constraints were included in the formulation to reflect the operating objectives. The optimization strategy was able to accommodate these constraints in a relatively simple manner. The concept of non-uniform discretization in control vector parameterization was evaluated and shown to give superior results. Simple representative problems as well as a complex, multiple feed problem of simultaneous saccharification and fermentation (SSF) has been considered here to demonstrate the validity of the proposed methodology. The proposed methodology yielded an increase in productivity of approximately 20% in the case studies considered here.

Reliability optimization of series-parallel systems using a genetic algorithm

A problem-specific genetic algorithm (GA) is developed and demonstrated to analyze series-parallel systems and to determine the optimal design configuration when there are multiple component choices available for each of several k-out-of-n:G subsystems. The problem is to select components and redundancy-levels to optimize some objective function, given system-level constraints on reliability, cost, and/or weight. Previous formulations of the problem have implicit restrictions concerning the type of redundancy allowed, the number of available component choices, and whether mixing of components is allowed. GA is a robust evolutionary optimization search technique with very few restrictions concerning the type or size of the design problem. The solution approach was to solve the dual of a nonlinear optimization problem by using a dynamic penalty function. GA performs very well on two types of problems: (1) redundancy allocation originally proposed by Fyffe, Hines, Lee, and (2) randomly generated problem with more complex k-out-of-n:G configurations.

Adaptive Penalty Methods for Genetic Optimization of Constrained Combinatorial Problems

The application of genetic algorithms (GA) to constrained optimization problems has been hindered by the inefficiencies of reproduction and mutation when feasibility of generated solutions is impossible to guarantee and feasible solutions are very difficult to find. Although several authors have suggested the use of both static and dynamic penalty functions for genetic search, this paper presents a general adaptive penalty technique which makes use of feedback obtained during the search along with a dynamic distance metric. The effectiveness of this method is illustrated on two diverse combinatorial applications: (1) the unequal-area, shape-constrained facility layout problem and (2) the series-parallel redundancy allocation problem to maximize system reliability given cost and weight constraints. The adaptive penalty function is shown to be robust with regard to random number seed, parameter settings, number and degree of constraints, and problem instance.

A dynamic penalty function method for the solution of structural optimization problems

This paper presents an adaptation of an existing dynamic trajectory method for unconstrained minimization to handle constrained optimization problems. This is done by the application of a dynamic penalty parameter procedure to allow for the constraints. The method is applied to structural optimization problems that involve the determination of minimum weight structures of trusses and frames, subject to stress, displacement, and frequency constraints, under various prescribed load conditions. Because structural problems, in general, require detailed finite-element analyses to evaluate the constraint functions, the direct application of the trajectory method, requiring updated information at each step along the path, would be expensive. This problem is overcome by the successive application of the trajectory method to approximate quadratic subproblems that can be solved economically. The comprehensive new approach is called the DYNAMIC-Q method. The method is successfully applied to a number of truss and frame problems and is found to be both reliable and easy to use.

Dynamic programming and penalty functions

One manipulates the optimization problems with a constraint (or constraints) related to three basic inequalities: the arithmetic and geometric inequality, the Holder inequality, and the Minkowski inequality. One reveales notably the versatility, adaptability and flexibility of the dynamic programming technique in conjunction with the method of penalty functions

Optimization of a Cyclic Steam Injection Process Using Penalty Functions

Abstract A penalty-function technique is described that can be used to solve constrained nonlinear optimization problems. The method is based upon the strategy of transforming the original constrained optimization problem into an equivalent unconstrained problem, and then solving a sequence of unconstrained optimization problems. This strategy is particularly attractive because unconstrained optimization problems are generally much easier to solve than constrained problems. The method is applied to a cyclic steam injection process that has been studied previously using process that has been studied previously using other techniques. Detailed solutions are presented and the characteristics of the method are discussed The method works, and it exhibits several favorable characteristics, such as applicability to a wide variety of nonlinear problems and freedom from convergence difficulties or computer storage limitations. Moreover, the method can be understood and implemented on a digital computer with ease. Introduction It is difficult for the analyst to recognize the methods that are best suited to a particular problem. Recent texts on optimization theory have not aided the situation because little is presented about which techniques are most applicable to different problems. Thus, the difficulties of identifying particular optimization techniques for various specialized applications remain largely unresolved. This paper presents a penalty-function optimization algorithm is suited to a variety of nonlinear engineering optimization problems. The method is applied to a problem that has been previously optimization of a cyclic steam injection previously optimization of a cyclic steam injection process. This example has a serial structure and may, process. This example has a serial structure and may, therefore, appear to be an unlikely application for a penalty-function optimizer. There are, however, several significant advantages to the use of a penalty-function algorithm rather than dynamic penalty-function algorithm rather than dynamic programming or the maximum principle. programming or the maximum principle.