on the generation of mono implicit runge kutta nystro m methods by mono implicit runge kutta methods mono implicit runge kutta methods can be used to generate implicit runge kutta nystrom irkn methods for the numerical solution of systems of second order differential equations the paper is concerned with the investigation of the conditions to be fulfilled by the mono implicit runge kutta mirk method in order to generate a mono implicit runge kutta nystrom method mirkn that is p stable one of the main theoretical results is the property that mirk methods in standard form cannot generate mirkn methods in standard form of order greater than 4 many examples of mirkn methods generated by mirk methods are presented

an adaptive boundary value runge kutta solver for first order boundary value problems the use of finite difference methods which combine features of both runge kutta processes and gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two point boundary conditions the order conditions for the coefficients of these processes are given a set of conditions by which these order conditions can be reduced in number are developed an eighth order a stable method which has second fourth and sixth order a stable methods embedded in it is given as an example a variable order variable step finite difference solver using embedded methods of this kind is described

disturbed exploitation compact differential evolution for limited memory optimization problems this paper proposes a novel and unconventional memetic computing approach for solving continuous optimization problems characterized by memory limitations the proposed algorithm unlike employing an explorative evolutionary framework and a set of local search algorithms employs multiple exploitative search within the main framework and performs a multiple step global search by means of a randomized perturbation of the virtual population corresponding to a periodical randomization of the search for the exploitative operators the proposed memetic computing approach is based on a populationless compact evolutionary framework which instead of processing a population of solutions handles its statistical model this evolutionary framework is based on a differential evolution which cooperatively employs two exploitative search operators the first is based on a standard differential evolution mutation and exponential crossover and the second is the trigonometric mutation these two search operators have an exploitative action on the algorithmic framework and thus contribute to the rapid convergence of the virtual population towards promising candidate solutions the action of these search operators is counterbalanced by a periodical stochastic perturbation of the virtual population which has the role of disturbing the excessively exploitative action of the framework and thus inhibits its premature convergence the proposed algorithm namely disturbed exploitation compact differential evolution is a simple and memory wise cheap structure that makes use of the memetic computing paradigm in order to solve complex optimization problems the proposed approach has been tested on a set of various test problems and compared with state of the art compact algorithms and with some modern population based meta heuristics numerical results show that disturbed exploitation compact differential evolution significantly outperforms all the other compact algorithms present in literature and reaches a competitive performance with respect to modern population algorithms including some memetic approaches and complex modern differential evolution based algorithms in order to show the potential of the proposed approach in real world applications disturbed exploitation compact differential evolution has been implemented for performing the control of a space robot by simulating the implementation within the robot micro controller numerical results show the superiority of the proposed algorithm with respect to other modern compact algorithms present in literature

an effective co evolutionary differential evolution for constrained optimization many practical problems can be formulated as constrained optimization problems due to the simple concept and easy implementation the penalty function method has been one of the most common techniques to handle constraints however the performance of this technique greatly relies on the setting of penalty factors which are usually determined by manual trial and error and the suitable penalty factors are often problem dependent and difficult to set in this paper a differential evolution approach based on a co evolution mechanism named cde is proposed to solve the constrained problems first a special penalty function is designed to handle the constraints second a co evolution model is presented and differential evolution de is employed to perform evolutionary search in spaces of both solutions and penalty factors thus the solutions and penalty factors evolve interactively and self adaptively and both the satisfactory solutions and suitable penalty factors can be obtained simultaneously simulation results based on several benchmark functions and three well known constrained design problems as well as comparisons with some existed methods demonstrate the effectiveness efficiency and robustness of the proposed method

bare bones differential evolution the barebones differential evolution bbde is a new almost parameter free optimization algorithm that is a hybrid of the barebones particle swarm optimizer and differential evolution differential evolution is used to mutate for each particle the attractor associated with that particle defined as a weighted average of its personal and neighborhood best positions the performance of the proposed approach is investigated and compared with differential evolution a von neumann particle swarm optimizer and a barebones particle swarm optimizer the experiments conducted show that the bbde provides excellent results with the added advantage of little almost no parameter tuning moreover the performance of the barebones differential evolution using the ring and von neumann neighborhood topologies is investigated finally the application of the bbde to the real world problem of unsupervised image classification is investigated experimental results show that the proposed approach performs very well compared to other state of the art clustering algorithms in all measured criteria

differential evolution with preferential crossover we study the mutation operation of the differential evolution algorithm in particular we study the effect of the scaling parameter of the differential vector in mutation we derive the probability density function of points generated by mutation and thereby identify some drawbacks of the scaling parameter we also visualize the drawbacks using simulation we then propose a crossover rule called the preferential crossover rule to reduce the drawbacks the preferential crossover rule uses points from an auxiliary population set we also introduce a variable scaling parameter in mutation motivations for these changes are provided a numerical study is carried out using 50 test problems many of which are inspired by practical applications numerical results suggest that the proposed modification reduces the number of function evaluations and cpu time considerably

boundary behavior of solutions of monge ampere equations with singular righthand sides abstract in this paper by karamata regular variation theory and constructing super and subsolutions we obtain the asymptotic boundary behavior of convex solutions of monge ampere equations with singular righthand sides

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

differential evolution with generalized differentials in this paper we study the mutation operation of the differential evolution de algorithm in particular we propose the differential of scaled vectors called the generalized differential as opposed to the existing scaled differential vector in the mutation of de we derive the probability distribution of points generated by the mutation with generalized differentials we incorporate a vector projection based exploratory method within the new mutation scheme the vector projection is not mandatory and it is only invoked if trial points continue to be unsuccessful an algorithm is then proposed which implements the mutation strategy based on the difference of the scaled vectors as well as the vector projection technique a numerical study is carried out using a set of 50 test problems many of which are inspired by practical applications numerical results suggest that the new algorithm is superior to de

differential evolution with dynamic stochastic selection for constrained optimization how much attention should be paid to the promising infeasible solutions during the evolution process is investigated in this paper stochastic ranking has been demonstrated as an effective technique for constrained optimization in stochastic ranking the comparison probability will affect the position of feasible solution after ranking and the quality of the final solutions in this paper the dynamic stochastic selection dss is put forward within the framework of multimember differential evolution firstly a simple version named dss mde is given where the comparison probability decreases linearly the algorithm dss mde has been compared with two state of the art evolution strategies and three competitive differential evolution algorithms for constrained optimization on 13 common benchmark functions dss mde is also evaluated on four well studied engineering design examples and the experimental results are significantly better than current available results secondly other dynamic settings of the comparison probability for dss mde are also designed and tested from the experimental results dss mde is effective for constrained optimization finally dss mde with a square root adjusted comparison probability is evaluated on the 22 benchmark functions in cec 06 and the experimental results on most functions are competitive

kernel induced fuzzy clustering of image pixels with an improved differential evolution algorithm a modified differential evolution de algorithm is presented for clustering the pixels of an image in the gray scale intensity space the algorithm requires no prior information about the number of naturally occurring clusters in the image it uses a kernel induced similarity measure instead of the conventional sum of squares distance use of the kernel function makes it possible to partition data that is linearly non separable and non hyper spherical in the original input space into homogeneous groups in a transformed high dimensional feature space a novel search variable representation scheme is adopted for selecting the optimal number of clusters from several possible choices extensive performance comparison over a test suite of 10 gray scale images and objective comparison with manually segmented ground truth indicates that the proposed algorithm has an edge over a few state of the art algorithms for automatic multi class image segmentation

order barriers and characterizations for continuous mono implicit runge kutta schemes the mono implicit runge kutta mirk schemes a subset of the family of implicit runge kutta irk schemes were originally proposed for the numerical solution of initial value odes more than fifteen years ago dur ing the last decade a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ode prob lems where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes recent work in this area has seen the development of some software packages for bound ary value odes based on these schemes unfortunately these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval while the collocation schemes provide a natural continuous solution approximation the availability of a continuous solution is important not only to the user of the software but also within the code itself for example in estimation of errors defect control mesh selection and the provision of initial solution estimates for new meshes an approach for the construction of a continuous solution approximation based on the mirk schemes is suggested by recent work in the area of continuous extensions for explicit runge kutta schemes for initial value odes in this paper we describe our work in the investigation of continuous versions of the mirk schemes i we give some lower bounds relating the stage order to the minimal number of stages for general continuous irk schemes ii we establish lower bounds on the number of stages needed to derive continuous mirk schemes of orders 1 through 6 and iii we provide characterizations of these schemes having a minimal number of stages for each of these orders

enhancing the search ability of differential evolution through orthogonal crossover differential evolution de is a class of simple yet powerful evolutionary algorithms for global numerical optimization binomial crossover and exponential crossover are two commonly used crossover operators in current popular de it is noteworthy that these two operators can only generate a vertex of a hyper rectangle defined by the mutant and target vectors therefore the search ability of de may be limited orthogonal crossover ox operators which are based on orthogonal design can make a systematic and rational search in a region defined by the parent solutions in this paper we have suggested a framework for using an ox in de variants and proposed oxde a combination of de rand 1 bin and ox extensive experiments have been carried out to study oxde and to demonstrate that our framework can also be used for improving the performance of other de variants

a numerical study of some modified differential evolution algorithms abstract modifications in mutation and localization in acceptance rule are suggested to the differential evolution algorithm for global optimization numerical experiments indicate that the resulting algorithms are considerably better than the original differential evolution algorithm therefore they offer a reasonable alternative to many currently available stochastic algorithms especially for problems requiring direct search type methods numerical study is carried out using a set of 50 test problems many of which are inspired by practical applications

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

a variable step runge kutta nystrom integrator for reversible systems of second order initial value problems a class of implicit runge kutta nystrom rkn integration formulae is developed for the numerical integration of reversible systems of second order ordinary differential equations of the form y f x y the formulae derived have high stage order provide a continuous solution are symmetric and have asymptotically correct local error estimates available at each step point the availability of these error estimates coupled with the symmetry of the algorithm allows the formulae to be implemented in a variable stepsize mode using the ideas of stoffer computing 55 1995 pp 1 22 numerical results presented for both fixed and variable stepsizes of integration show the power of the new methods

constrained optimization based on modified differential evolution algorithm this paper presents a novel constrained optimization based on modified differential evolution algorithm comde in the new algorithm a new directed mutation rule based on the weighted difference vector between the best and the worst individuals at a particular generation is introduced the new directed mutation rule is combined with the modified basic mutation strategy de rand 1 bin where only one of the two mutation rules is applied with the probability of 0 5 the proposed mutation rule is shown to enhance the local search ability of the basic differential evolution de and to get a better trade off between convergence rate and robustness two new scaling factors are introduced as uniform random variables to improve the diversity of the population and to bias the search direction additionally a dynamic non linear increased crossover probability is utilized to balance the global exploration and local exploitation comde also includes a modified constraint handling technique based on feasibility and the sum of constraints violations a new dynamic tolerance technique to handle equality constraints is also adopted the effectiveness and benefits of the new directed mutation strategy and modified basic strategy used in comde has been experimentally investigated the effect of the parameters of the crossover probability function and the parameters of the dynamic tolerance equation on the performance of comde have been analyzed and evaluated by different experiments numerical experiments on 13 well known benchmark test functions and five engineering design problems have shown that the new approach is efficient effective and robust the comparison results between the comde and the other 28 state of the art evolutionary algorithms indicate that the proposed comde algorithm is competitive with and in some cases superior to other existing algorithms in terms of the quality efficiency convergence rate and robustness of the final solution

parallel memetic structures memetic computing mc structures are algorithms composed of heterogeneous operators memes for solving optimization problems in order to address these problems this study investigates and proposes a simple yet extremely efficient structure namely parallel memetic structure pms pms is a single solution optimization algorithm composed of tree operators the first one being a stochastic global search which explores the entire decision space searching for promising regions in analogy with electrical networks downstream of the global search component there is a parallel of two alternative elements i e two local search algorithms with different features in terms of search logic whose purpose is to refine the search in the regions detected by the upstream element the first local search explores the space along the axes while the second performs diagonal movements in the direction of the estimated gradient the pms algorithm despite its simplicity displays a respectable performance compared to that of popular meta heuristics and modern optimization algorithms representing the state of the art in the field thanks to its simple structure pms appears to be a very flexible algorithm for various problem features and dimensionality values unlike modern complex algorithm that are specialized for some benchmarks and some dimensionality values pms achieves solutions with a high quality in various and diverse contexts for example both on low dimensional and large scale problems an application example in the field of magnetic sensors further proves the potentials of the proposed approach this study confirms the validity of the ockham s razor in mc efficiently designed simple structures can perform as well as if not better than complex algorithms composed of many parts

adaptive memetic differential evolution with global and local neighborhood based mutation operators differential evolution de is one of the most popular optimization methods for real valued problems and a large number of its variants have been proposed so far however bringing together different ideas that already led to successful de versions is rare in the literature in the present paper a novel de algorithm is proposed in which three among the most efficient concepts already applied separately within de framework are gathered together namely 1 the adaptation of algorithm control parameters and probabilities of using different mutation strategies 2 the use of nelder mead algorithm as a local search method hybridized with de and 3 the splitting mutation into global and local models when local mutation model is based on the concept of neighborhood of individuals organized on a ring topology the performance of the novel algorithm called adaptive memetic de with global and local neighborhood based mutation operators is compared with 13 different de variants on a set of 25 popular problems which include rotated shifted and hybrid composition functions it is found that although none de algorithm outperforms all the others for the majority of problems on average the proposed approach perform better than all 13 de algorithms selected for comparison the proposed algorithm is another heuristic approach developed to solve optimization problems the question may arise whether proposing novel methods is useful as no free lunch theorems for optimization state that the expected performance of all possible heuristics on all possible problems is equal in the last section of the paper the limitations and implications of no free lunch theorems are discussed based on rich but unfortunately frequently neglected literature a very simple continuous and differentiable minimization problem is proposed for which it is empirically verified that each among considered 14 de algorithms perform poorer than random sampling it is also empirically shown that when all considered de algorithms search for the maximum of the proposed problem they found lower minimum than de algorithms searching for the minimum such result is not unexpected according to no free lunch theorems and should be considered as a precaution from generalization of good performance of heuristic optimization methods

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

multistride l stable fourth order methods for the numerical solution of odes recently chawla et al 1 have obtained a one parameter family of double stride l stable methods of fourth order by the coupling of three linear multistep methods lmms here we present an alternative derivation technique based on mono implicit runge kutta mirk methods the mirk method provides a framework within which all families of double and triple stride l stable methods of fourth order can be expressed the well established theory for runge kutta methods can then be used to derive these families in a straightforward manner

starting algorithms for a class of rk methods for index 2 daes when semiexplicit differential algebraic equations are solved with implicit runge kutta methods rk the computational effort is dominated by the cost of solving the nonlinear systems and therefore it is important to have good starting values to begin the iterations for semiexplicit index 2 daes starting algorithms without additional cost for rk methods with regular matrix coefficient were studied in a previous paper however the regularity condition on the matrix coefficient excludes some interesting methods like lobatto iiia and esdirk methods in this paper we study starting algorithms without additional computational cost for a class of runge kutta methods in the case of index 2 daes

order results for mono implicit runge kutta methods the mono implicit runge kutta methods are a subclass of the well known implicit runge kutta methods and have application in the efficient numerical solution of systems of initial and boundary value ordinary differential equations although the efficiency and stability properties of this class of methods have been studied in a number of papers the specific question of determining the maximum order of an s stage mono implicit runge kutta method has not been dealt with in addition to the complete characterization of some subclasses of these methods having a number of stages s leq 5 a main result of this paper is a proof that the order of an s stage mono implicit runge kutta method is at most s 1

deferred correction with mono implicit runge kutta methods for first order ivps abstract to reach a high order of accuracy for numerical solutions of ivps with mono implicit runge kutta mirk methods the technique of deferred correction is used special attention is paid to the possible increase of the order and the stability of such schemes several schemes are given

mono implicit runge kutta schemes for the parallel solution of initial value odes among the numerical techniques commonly considered for the efficient solution of stiff initial value ordinary differential equations are the implicit runge kutta irk schemes the calculation of the stages of the irk method involves the solution of a nonlinear system of equations usually employing some variant of newton s method since the costs of the linear algebra associated with the implementation of newton s method generally dominate the overall cost of the computation many subclasses of irk schemes such as diagonally implicit runge kutta schemes singly implicit runge kutta schemes and mono implicit mirk schemes have been developed to attempt to reduce these costs in this paper we are concerned with the design of mirk schemes that are inherently parallel in that smaller systems of equations are apportioned to concurrent processors this work builds on that of an earlier investigation in which a special subclass of the mirk formulas were implemented in parallel while suitable parallelism was achieved the formulas were limited to some extent because they all had only stage order 1 this is of some concern since in the application of a runge kutta method to a system of stiff odes the phenomenon of order reduction can arise the irk method can behave as if its order were only its stage order or its stage order plus one regardless of its classical order the formulas derived in the current paper represent an improvement over the previous investigation in that the full class of mirk formulas is considered and therefore it is possible to derive efficient parallel formulas of orders 2 3 and 4 having stage orders 2 or 3

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

superconvergent deferred correction methods for first order systems of nonlinear two point boundary value problems iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2 in this paper we consider the construction of mono implicit runge kutta mirk methods where an increase of four orders of accuracy is obtained for each deferred correction we develop a very powerful yet rather straightforward theory which allows us to identify the appropriate runge kutta formulae for inclusion in such schemes in particular we will focus on the construction of pairs of mirk formulae of order 4 and 8 which will allow this superconvergence to be realized we will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available

an effective memetic differential evolution algorithm based on chaotic local search this paper proposes an effective memetic differential evolution de algorithm or decls that utilizes a chaotic local search cls with a shrinking strategy the cls helps to improve the optimizing performance of the canonical de by exploring a huge search space in the early run phase to avoid premature convergence and exploiting a small region in the later run phase to refine the final solutions moreover the parameter settings of the decls are controlled in an adaptive manner to further enhance the search ability to evaluate the effectiveness and efficiency of the proposed decls algorithm we compared it with four state of the art de variants and the ipop cma es algorithm on a set of 20 selected benchmark functions results show that the decls is significantly better than or at least comparable to the other optimizers in terms of convergence performance and solution accuracy besides the decls has also shown certain advantages in solving high dimensional problems

free lunches on the discrete lipschitz class the no free lunch theorem states that there does not exist a genuine general purpose optimizer because all algorithms have the identical performance on average over all functions however such a result does not imply that search heuristics or optimization algorithms are futile if we are more cautious with the applicability of these methods and the search space in this paper within the no free lunch framework we firstly introduce the discrete lipschitz class by transferring the lipschitz functions i e functions with bounded slope as a measure to fulfill the notion of continuity in discrete functions we then investigate the properties of the discrete lipschitz class generalize an algorithm called subthreshold seeker for optimization and show that the generalized subthreshold seeker outperforms random search on this class finally we propose a tractable sampling test scheme to empirically demonstrate the superiority of the generalized subthreshold seeker under practical configurations this study concludes that there exist algorithms outperforming random search on the discrete lipschitz class in both theoretical and practical aspects and indicates that the effectiveness of search heuristics may not be universal but still general in some broad sense

a note on teaching learning based optimization algorithm teaching learning based optimization tlbo seems to be a rising star from amongst a number of metaheuristics with relatively competitive performances it is reported that it outperforms some of the well known metaheuristics regarding constrained benchmark functions constrained mechanical design and continuous non linear numerical optimization problems such a breakthrough has steered us towards investigating the secrets of tlbo s dominance this paper reports our findings on tlbo qualitatively and quantitatively through code reviews and experiments respectively our findings have revealed three important mistakes regarding tlbo 1 at least one unreported but important step 2 incorrect formulae on a number of fitness function evaluations and 3 misconceptions about parameter less control additionally unfair experimental settings conditions were used to conduct experimental comparisons e g different stopping criteria the experimental results for constrained and unconstrained benchmark functions under fairly equal conditions failed to validate its performance supremacy the ultimate goal of this paper is to provide reminders for metaheuristics researchers and practitioners in order to avoid similar mistakes regarding both the qualitative and quantitative aspects and to allow fair comparisons of the tlbo algorithm to be made with other metaheuristic algorithms

superconvergent interpolants for the collocation solution of boundary value ordinary differential equations a long standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation obtained at the mesh points which subdivide the problem interval it is well known that the use of collocation at gauss points leads to solution approximations at the mesh points for which the global error is o h2k where k is the number of collocation points used per subinterval and h is the subinterval size this discrete solution is said to be superconvergent the collocation solution also yields a c0 continuous solution approximation that has a global error of o hk 1 in this paper we show how to efficiently augment the superconvergent discrete collocation solution to obtain c1 continuous superconvergent interpolants whose global errors are o h2k the key ideas are to use the theoretical framework of continuous runge kutta schemes and to augment the collocation solution with inexpensive monoimplicit runge kutta stages specific schemes are derived for k 1 2 3 and 4 numerical results are provided to support the theoretical analysis

cultured differential evolution for constrained optimization a cultural algorithm with a differential evolution population is proposed in this paper this cultural algorithm uses different knowledge sources to influence the variation operator of the differential evolution algorithm in order to reduce the number of fitness function evaluations required to obtain competitive results comparisons are provided with respect to three techniques that are representative of the state of the art in the area the results obtained by our algorithm are similar in quality to those obtained by the other approaches with respect to which it was compared however our approach requires a lower number of fitness function evaluations than the others

optimization with randomized search heuristics the a nfl theorem realistic scenarios and difficult functions the no free lunch nfl theorem due to wolpert and macready ieee trans evol comput 1 1 1997 67 has led to controversial discussions on the usefulness of randomized search heuristics in particular evolutionary algorithms here a short and simple proof of the nfl theorem is given to show its elementary character moreover the proof method leads to a generalization of the nfl theorem afterwards realistic complexity theoretical based scenarios for black box optimization are presented and it is argued why nfl theorems are not possible in such situations however an almost no free lunch anfl theorem shows that for each function which can be optimized efficiently by a search heuristic there can be constructed many related functions where the same heuristic is bad as a consequence search heuristics use some idea how to look for good points and can be successful only for functions giving the right hints the consequences of these theoretical considerations for some well known classes of functions are discussed

an improved differential evolution algorithm with fitness based adaptation of the control parameters differential evolution de is arguably one of the most powerful stochastic real parameter optimization algorithms of current interest de operates through the similar computational steps as employed by a standard evolutionary algorithm ea however unlike the traditional eas the de variants perturb the current generation population members with the scaled differences of randomly selected and distinct population members therefore no separate probability distribution has to be used which makes the scheme self organizing in this respect scale factor f and crossover rate cr are two very important control parameters of de since the former regulates the step size taken while mutating a population member in de and the latter controls the number of search variables inherited by an offspring from its parent during recombination this article describes a very simple yet very much effective adaptation technique for tuning both f and cr on the run without any user intervention the adaptation strategy is based on the objective function value of individuals in the de population comparison with the best known and expensive variants of de over fourteen well known numerical benchmarks and one real life engineering problem reflects the superiority of proposed parameter tuning scheme in terms of accuracy convergence speed and robustness