on the effectiveness of immune inspired mutation operators in some discrete optimization problems abstract artificial immune systems have been widely applied to a variety of complex real world problems however theoretical studies on artificial immune system are still limited and there is a strong need for building a rigorous theoretical foundation to better understand these heuristics this paper contributes to a theoretical runtime analysis of immune inspired hypermutations on some discrete optimization problems in particular we are interested in the performance comparison among somatic contiguous hypermutations chm standard bit mutations sbm and local mutation we reveal that the immune inspired hypermutations can significantly outperform the standard bit mutation most often used in evolutionary algorithms on some well known pseudo boolean functions including trap and hierarchical if and only if functions and instances of two combinatorial optimization problems namely the max cut problem and the minimum s t cut problem the proofs give some insights into the relationships between the problem characteristics and algorithmic features the results of the analysis help strengthen the usefulness of artificial immune systems

a self stabilizing algorithm for cut problems in synchronous networks consider a synchronized distributed system where each node can only observe the state of its neighbors such a system is called self stabilizing if it reaches a stable global state in a finite number of rounds allowing two different states for each node induces a cut in the graph in each round every node decides whether it is locally satisfied with the current cut afterwards all unsatisfied nodes change sides independently with a fixed probability p using different notions of satisfaction enables the computation of maximal and minimal cuts respectively we analyze the expected time until such cuts are reached on several graph classes and study the impact of the parameter p and the initial cut

on the convergence rates of genetic algorithms this paper discusses the convergence rates of genetic algorithms by using the minorization condition in the markov chain theory we classify genetic algorithms into two kinds one with time invariant genetic operators another with time variant genetic operators for the former case we have obtained the bound on its convergence rate on the general state space for the later case we have bounded its convergence rate on the finite state space

randomized local search evolutionary algorithms and the minimum spanning tree problem randomized search heuristics among them randomized local search and evolutionary algorithms are applied to problems whose structure is not well understood as well as to problems in combinatorial optimization the analysis of these randomized search heuristics has been started for some well known problems and this approach is followed here for the minimum spanning tree problem after motivating this line of research it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms

towards a matrix theory for unbounded operator matrices sur l espace produit r l 2 r on considere l operateur 7b a a 0 ou d d dx est defini sur d a w 1 r 0 est la mesure de dirac en 0 c l 2 r est un operateur de r dans l 2 r et 7b a a un domaine d 7b a r w 1 r on introduit une fonction caracteristique non commutative donnant une facon efficace pour calculer le spectre de 7b a on obtient des resultats sur la generation de semi groupes par des matrices operateur

semi groupes sous markoviens engendres par des operateurs matriciels we characterize generators of sub markovian semigroups onlp by a version of kato s inequality this will be used to show under precise assumptions that the semigroup generated by a matrix operatora aij 1 i j n on lp n is sub markovian if and only if the semigroup generated by the sum of each rowai1 ain 1 i n is sub markovian the corresponding result on c0 x n characterizes dissipative operator matrices

extended hilbert space approach to few body problems a general formulation of the quantum scattering theory for a system of few particles which have an internal structure is given due to freezing out the internal degrees of freedom in the external channels a certain class of energy dependent potentials is generated by means of potential theory a modified faddeev equation is derived both in external and internal channels the fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem

a multiobjective evolutionary algorithm for approximating the efficient set abstract in this article a new framework for evolutionary algorithms for approximating the efficient set of a multiobjective optimization moo problem with continuous variables is presented the algorithm is based on populations of variable size and exploits new elite preserving rules for selecting alternatives generated by mutation and recombination together with additional assumptions on the considered moo problem and further specifications on the algorithm theoretical results on the approximation quality such as convergence in probability and almost sure convergence are derived

analysis of a multiobjective evolutionary algorithm on the 0 1 knapsack problem multiobjective evolutionary algorithms moeas are increasingly being used for effectively solving many real world problems and many empirical results are available however theoretical analysis is limited to a few simple toy functions in this work we select the well known knapsack problem for the analysis the multiobjective knapsack problem in its general form is np complete moreover the size of the set of pareto optimal solutions can grow exponentially with the number of items in the knapsack thus we formalize a 1 e approximate set of the knapsack problem and attempt to present a rigorous running time analysis of a moea to obtain the formalized set the algorithm used in the paper is based on a restricted mating pool with a separate archive to store the remaining population we call the algorithm a restricted evolutionary multiobjective optimizer remo we also analyze the running time of remo on a special bi objective linear function known as lotz leading ones trailing zeros whose pareto set is shown to be a subset of the knapsack an extension of the analysis to the simple evolutionary multiobjective optimizer semo is also presented a strategy based on partitioning of the decision space into fitness layers is used for the analysis

multivariable extremum seeking for pde dynamic systems in this article we address the design and analysis of multivariable newton based extremum seeking es for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations pdes multi input systems with distinct diffusion coefficients in each individual input channel are dealt with the phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation based averaging based estimate of the hessian s inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the diffusion pdes in addition the generalization of the results for different classes of parabolic reaction advection diffusion pdes wave equations and or first order hyperbolic transport dominated pdes is also discussed as for ordinary differential equations case the proposed newton approach removes the dependence of the algorithm s convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable unlike the gradient algorithm a numerical example illustrates the performance of our multivariable es for compensating pde based systems

a co evolutionary multi objective optimization algorithm based on direction vectors most real world multi objective problems mops have a complicated solution space facing such problems a direction vectors based co evolutionary multi objective optimization algorithm dvcmoa that introduces the decomposition idea from moea d to co evolutionary algorithms is proposed in this paper it is novel in the sense that dvcmoa applies the concept of direction vectors to co evolutionary algorithms dvcmoa first divides the entire population into several subpopulations on the basis of the initial direction vectors in the objective space then it solves mops through the co evolutionary interaction among the subpopulations in which individuals are classified according to their direction vectors finally it explores the less developed regions to maintain the relatively uniform distribution of the solution space in this way dvcmoa has advantages in convergence diversity and uniform distribution of the non dominated solution set which are explained through comparison with other state of the art multi objective optimization evolutionary algorithms moeas in this paper dvcmoa is shown to be effective on 6 multi objective 0 1 knapsack problems

the one dimensional ising model mutation versus recombination the investigation of genetic and evolutionary algorithms on ising model problems gives much insight into how these algorithms work as adaptation schemes the one dimensional ising model with periodic boundary conditions has been considered as a typical example with a clear building block structure suited well for two point crossover it has been claimed that gas based on recombination and appropriate diversity preserving methods by far outperform eas based on mutation only here a rigorous analysis of the expected optimization time proves that mutation based eas are surprisingly effective the 1 ea with an appropriate value is almost as efficient as typical gas moreover it is proved that specialized gas do even better and this holds for two point crossover as well as for one point crossover

on the analysis of the 1 1 evolutionary algorithm many experimental results are reported on all types of evolutionary algorithms but only few results have been proved a step towards a theory on evolutionary algorithms in particular the so called 1 1 evolutionary algorithm is performed linear functions are proved to be optimized in expected time o nlnn but only mutation rates of size 1 n can ensure this behavior for some polynomial of degree 2 the optimization needs exponential time the same is proved for a unimodal function both results were not expected by several other authors finally a hierarchy result is proved moreover methods are presented to analyze the behavior of the 1 1 evolutionary algorithm

runtime analysis of a multi objective evolutionary algorithm for obtaining finite approximations of pareto fronts previous theoretical analyses of evolutionary multi objective optimization emo mostly focus on obtaining approximations of pareto fronts however in practical applications an appropriate value of is critical but sometimes for a multi objective optimization problem mop with unknown attributes difficult to determine in this paper we propose a new definition for the finite representation of the pareto front the adaptive pareto front which can automatically accommodate the pareto front accordingly it is more practical to take the adaptive pareto front or its approximation termed the adaptive pareto front as the goal of an emo algorithm we then perform a runtime analysis of a m 1 multi objective evolutionary algorithm m 1 moea for three mops including a discrete mop with a polynomial pareto front denoted as a polynomial dmop a discrete mop with an exponential pareto front denoted as an exponential dmop and a simple continuous two objective optimization problem sctop by employing an estimator based update strategy in the m 1 moea we show that 1 for the polynomial dmop the whole pareto front can be obtained in the expected polynomial runtime by setting the population size m equal to the number of pareto vectors 2 for the exponential dmop the expected polynomial runtime can be obtained by keeping m increasing in the same order as that of the problem size n and 3 the diversity mechanism guarantees that in the expected polynomial runtime the moea can obtain an adaptive pareto front of sctop for any given precision theoretical studies and numerical comparisons with nsga ii demonstrate the efficiency of the proposed moea and should be viewed as an important step toward understanding the mechanisms of moeas

on the estimation of pareto fronts from the point of view of copula theory a probabilistic framework is applied to pareto front estimation a link between pareto front and level lines of a multivariate distribution is given parametric approximations of pareto front are provided for archimedean copulas consequences and limits of the archimedean hypothesis are discussed the full estimation methodology is analyzed on several examples given a first set of observations from a design of experiments sampled randomly in the design space the corresponding set of non dominated points usually does not give a good approximation of the pareto front we propose here to study this problem from the point of view of multivariate analysis introducing a probabilistic framework with the use of copulas this approach enables the expression of level lines in the objective space giving an estimation of the position of the pareto front when the level tends to zero in particular when it is possible to use archimedean copulas analytical expressions for pareto front estimators are available several case studies illustrate the interest of the approach which can be used at the beginning of the optimization when sampling randomly in the design space

runtime analysis of the 1 ant ant colony optimizer the runtime analysis of randomized search heuristics is a growing field where in the last two decades many rigorous results have been obtained first runtime analyses of ant colony optimization aco have been conducted only recently in these studies simple aco algorithms such as the 1 ant are investigated the influence of the evaporation factor in the pheromone update mechanism and the robustness of this parameter w r t the runtime behavior have been determined for the example function onemax this work puts forward the rigorous runtime analysis of the 1 ant on the example functions leadingones and binval with respect to evolutionary algorithms eas such analyses were essential to develop methods for the analysis on more complicated problems the proof techniques required for the 1 ant unfortunately differ significantly from those for eas which means that a new reservoir of methods has to be built up again the influence of the evaporation factor is analyzed rigorously and it is proved that its choice has a crucial impact on the runtime moreover the analyses provide insight into the working principles of aco algorithms our theoretical results are accompanied by experimental results that give us a more detailed impression of the 1 ant s performance furthermore the experiments also deal with the question whether using many ant solutions in one iteration can decrease the total runtime

convergence analysis and improvements of quantum behaved particle swarm optimization motivated by concepts in quantum mechanics and particle swarm optimization pso quantum behaved particle swarm optimization qpso was proposed as a variant of pso with better global search ability although it has been shown to perform well in finding optimal solutions for many optimization problems there has so far been little theoretical analysis on its convergence and performance this paper presents a convergence analysis and performance evaluation of the qpso algorithm and it also proposes two variants of the qpso algorithm first we investigate in detail the convergence of the qpso algorithm on a probabilistic metric space and prove that the qpso algorithm is a form of contraction mapping and can converge to the global optimum this is the first time that the theory of probabilistic metric spaces has been employed to analyze a stochastic optimization algorithm we provided a new definition for the convergence rate of a stochastic algorithm as well as definitions for three types of convergence according to the correlations between the convergence rate and the objective function values with these definitions the effectiveness of the qpso is evaluated by computing and analyzing the time complexity and the convergence rate of the algorithm then the qpso with random mean best position qpso rm and the qpso with ranking operator qpso ro are proposed as two improvements of the qpso algorithm finally some empirical studies on popular benchmark functions are performed in order to make a full performance evaluation and comparison between qpso qpso rm qpso ro and other variants of pso

sms emoa multiobjective selection based on dominated hypervolume abstract the hypervolume measure or s metric is a frequently applied quality measure for comparing the results of evolutionary multiobjective optimisation algorithms emoa the new idea is to aim explicitly for the maximisation of the dominated hypervolume within the optimisation process a steady state emoa is proposed that features a selection operator based on the hypervolume measure combined with the concept of non dominated sorting the algorithm s population evolves to a well distributed set of solutions thereby focussing on interesting regions of the pareto front the performance of the devised s metric selection emoa sms emoa is compared to state of the art methods on two and three objective benchmark suites as well as on aeronautical real world applications

markov chain analysis of evolutionary algorithms on onemax function from coupon collector s problem to 1 1 ea abstract the theoretical investigation of evolutionary algorithms eas has increased our understanding of the computational mechanism of algorithms onemax is a test function most frequently and deeply studied in the field of eas in this work a method is presented for describing the runtime properties of 1 1 ea on onemax this method is motivated by the work of erdos and renyi treating the coupon collector s problem they showed that the success probability of the coupon collector s problem is given by a function of double exponential form and that the number of uncollected coupons follows the poisson distribution today the double exponential function is called gumbel function which is one of three fundamental functions in extreme value statistics we introduce an algorithm that is a variant of the 1 1 ea first order evolutionary algorithm fo ea fo ea takes into account only the effect of single bit mutations in the 1 1 ea which in general includes multiple bit mutations we modified the method of erdos and renyi to apply fo ea we apply the gumbel distribution for calculating the success probability of the 1 1 ea on onemax this method turns out to give a sufficiently reliable estimation for success probabilities even in the tail region

stochastic runtime analysis of a cross entropy algorithm for traveling salesman problems abstract this article analyzes the stochastic runtime of a cross entropy algorithm mimicking an max min ant system with iteration best reinforcement it investigates the impact of magnitude of the sample size on the runtime to find optimal solutions for tsp instances for simple tsp instances that have a 1 n valued distance function and a unique optimal solution we show that sample size n ln n results in a stochastically polynomial runtime and n o ln n results in a stochastically exponential runtime where stochastically means with a probability of 1 n 1 and n represents number of cities in particular for n ln n we prove a stochastic runtime of o n n 6 with the vertex based random solution generation and a stochastic runtime of o n n 3 ln n with the edge based random solution generation these runtimes are very close to the best known expected runtime for variants of max min ant system with best so far reinforcement by choosing a small n ln n they are obtained for the stronger notion of stochastic runtime and analyze the runtime in most cases we also inspect more complex instances with n vertices positioned on an m m grid when the n vertices span a convex polygon we obtain a stochastic runtime of o n 4 m 5 with the vertex based random solution generation and a stochastic runtime of o n 3 m 5 for the edge based random solution generation when there are k o 1 many vertices inside a convex polygon spanned by the other n k vertices we obtain a stochastic runtime of o n 4 m 5 n 6 k 1 m with the vertex based random solution generation and a stochastic runtime of o n 3 m 5 n 3 k m with the edge based random solution generation these runtimes are better than the expected runtime for the so called ea reported in a recent article and again obtained for the stronger notion of stochastic runtime

evolutionary algorithms with preference polyhedron for interval multi objective optimization problems multi objective optimization problems mops with interval parameters are ubiquitous in real world applications existing evolutionary optimization methods however aim to obtain a set of well converged and evenly distributed pareto optimal solutions in this paper we presented a novel evolutionary algorithm ea that interacts with a decision maker dm during the optimization process to obtain the most preferred solution the theory of preference polyhedron for the optimization problem with interval parameters was developed first an interactive evolutionary algorithm iea for mops with interval parameters was then proposed based on the theory of preference polyhedron the algorithm periodically provides a set of non dominated solutions to the dm a preference polyhedron is constructed in the objective space by taking the worst solution chosen by the dm as the vertex the solutions with the same rank are sorted based on the polyhedron constructed finally our method was empirically evaluated on several mops with interval parameters where the value functions were used to simulate the dm s responses the numerical results indicated that our method is simpler and more efficient than the a posteriori method

min max minimal hypersurface in manifolds with convex boundary and hbox ric ge 0 let m n 1 partial m g be a compact manifold with non negative ricci curvature convex boundary and 2 le n le 6 we show that the min max minimal hypersurface with respect to one parameter families of hypersurfaces in m partial m is orientable of index one and multiplicity one

min max theory for free boundary minimal hypersurfaces ii general morse index bounds and applications for any smooth riemannian metric on an n 1 dimensional compact manifold with boundary m partial m where 3 le n 1 le 7 we establish general upper bounds for the morse index of free boundary minimal hypersurfaces produced by min max theory in the almgren pitts setting we apply our morse index estimates to prove that for almost every in the c infty baire sense riemannan metric the union of all compact properly embedded free boundary minimal hypersurfaces is dense in m if partial m is further assumed to have a strictly mean convex point we show the existence of infinitely many compact properly embedded free boundary minimal hypersurfaces whose boundaries are non empty our results prove a conjecture of yau for generic metrics in the free boundary setting

removal of the resolvent like energy dependence from interactions and invariant subspaces of a total hamiltonian the spectral problem a v z z is considered where the main hamiltonian a is a self adjoint operator of sufficiently arbitrary nature the perturbation v z b a z 1b depends on the energy z as resolvent of another self adjoint operator a the latter is usually interpreted as a hamiltonian describing an internal structure of physical system the operator b is assumed to have a finite hilbert schmidt norm the conditions are formulated when one can replace the perturbation v z with an energy independent potential w such that the hamiltonian h a w has the same spectrum more exactly a part of spectrum and the same eigenfunctions as the initial spectral problem the hamiltonian h is constructed as a solution of the non linear operator equation h a v h it is established that this equation is closely connected with the problem of searching for invariant subspaces of the hamiltonian h b aa b the orthogonality and expansion theorems are proved for eigenfunction systems of the hamiltonian h a w

performance analysis of randomised search heuristics operating with a fixed budget when for a difficult real world optimisation problem no good problem specific algorithm is available often randomised search heuristics are used they are hoped to deliver good solutions in acceptable time the theoretical analysis usually concentrates on the average time needed to find an optimal or approximately optimal solution this matches neither the application in practice nor the empirical analysis since usually optimal solutions are not known and even if found cannot be recognised more often the algorithms are stopped after some time this motivates a theoretical analysis to concentrate on the quality of the best solution obtained after a pre specified number of function evaluations called budget using this perspective two simple randomised search heuristics random local search and the 1 1 evolutionary algorithm are analysed on some well known example problems upper and lower bounds on the expected quality of a solution for a fixed budget of function evaluations are proven the analysis shows novel and challenging problems in the study of randomised search heuristics it demonstrates the potential of this shift in perspective from expected run time to expected solution quality

statistical dynamics of the royal road genetic algorithm abstract metastability is a common phenomenon many evolutionary processes both natural and artificial alternate between periods of stasis and brief periods of rapid change in their behavior in this paper an analytical model for the dynamics of a mutation only genetic algorithm ga is introduced that identifies a new and general mechanism causing metastability in evolutionary dynamics the ga s population dynamics is described in terms of flows in the space of fitness distributions the trajectories through fitness distribution space are derived in closed form in the limit of infinite populations we then show how finite populations induce metastability even in regions where fitness does not exhibit a local optimum in particular the model predicts the occurrence of fitness epochs periods of stasis in population fitness distributions at finite population size and identifies the locations of these fitness epochs with the flow s hyperbolic fixed points this enables exact predictions of the metastable fitness distributions during the fitness epochs as well as giving insight into the nature of the periods of stasis and the innovations between them all these results are obtained as closed form expressions in terms of the ga s parameters an analysis of the jacobian matrices in the neighborhood of an epoch s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space s topological structure more general quantitative features of the dynamics fitness fluctuation amplitudes epoch stability and speed of the innovations are also determined from the jacobian eigenvalues the analysis shows how quantitative predictions for a range of dynamical behaviors that are specific to the finite population dynamics can be derived from the solution of the infinite population dynamics the theoretical predictions are shown to agree very well with statistics from ga simulations we also discuss the connections of our results with those from population genetics and molecular evolution theory

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

illustration of fairness in evolutionary multi objective optimization it is widely assumed that evolutionary algorithms for multi objective optimization problems should use certain mechanisms to achieve a good spread over the pareto front in this paper we examine such mechanisms from a theoretical point of view and analyze simple algorithms incorporating the concept of fairness this mechanism tries to balance the number of offspring of all individuals in the current population we rigorously analyze the runtime behavior of different fairness mechanisms and present illustrative examples to point out situations where the right mechanism can speed up the optimization process significantly we also indicate drawbacks for the use of fairness by presenting instances where the optimization process is slowed down drastically

convergence of multi objective evolutionary algorithms to a uniformly distributed representation of the pareto front in evolutionary multi objective optimization emo the convergence to the pareto set of a multi objective optimization problem mop and the diversity of the final approximation of the pareto front are two important issues in the existing definitions and analyses of convergence in multi objective evolutionary algorithms moeas convergence with probability is easily obtained because diversity is not considered however diversity cannot be guaranteed by combining the convergence with diversity this paper presents a new definition for the finite representation of a pareto set the b pareto set and a convergence metric for moeas based on a new archive updating strategy the convergence of one such moea to the b pareto sets of mops is proved numerical results show that the obtained b pareto front is uniformly distributed along the pareto front when according to the new definition of convergence the algorithm is convergent

choosing selection pressure for wide gap problems abstract to exploit an evolutionary algorithm s performance to the full extent the selection scheme should be chosen carefully empirically it is commonly acknowledged that low selection pressure can prevent an evolutionary algorithm from premature convergence and is thereby more suitable for wide gap problems however there are few theoretical time complexity studies that actually give the conditions under which a high or a low selection pressure is better in this paper we provide a rigorous time complexity analysis showing that low selection pressure is better for the wide gap problems with two optima

complete second order linear differential operator equations in hilbert space and applications in hydrodynamics we study the cauchy problem for a complete second order linear differential operator equation in a hilbert space h of the form d 2 u dt 2 f ik du dt bu f u 0 u 0 u 1 0 u 1 problems of this kind arise e g in hydrodynamics where the coefficients f k and b are unbounded selfadjoint operators it is assumed that f is the dominating operator in the cauchy problem above i e d f d b d f d k we also suppose that f and b are bounded from below but the operator coefficients are not assumed to commute the main results concern the existence of strong solutions to the stated cauchy problem and applications of these results to the cauchy problem associated with small motions of some hydrodynamical systems

runtime analysis of a binary particle swarm optimizer we investigate the runtime of a binary particle swarm optimizer pso for optimizing pseudo boolean functions f 0 1 n r the binary pso maintains a swarm of particles searching for good solutions each particle consists of a current position from 0 1 n its own best position and a velocity vector used in a probabilistic process to update its current position the velocities for a particle are then updated in the direction of its own best position and the position of the best particle in the swarm we present a lower bound for the time needed to optimize any pseudo boolean function with a unique optimum to prove upper bounds we transfer a fitness level argument that is well established for evolutionary algorithms eas to pso this method is applied to estimate the expected runtime for the class of unimodal functions a simple variant of the binary pso is considered in more detail for the test function onemax showing that there the binary pso is competitive to eas an additional experimental comparison reveals further insights

eventual norm continuity for neutral semigroups on banach spaces in this paper we prove under suitable hypotheses eventually norm continuity and compactness of the solution semigroups of certain neutral differential equations in banach spaces our approach is based on a general perturbation theorem obtained from closed loop systems of infinite dimensional control systems with unbounded control and observation operators in fact the solution semigroup can be viewed as the semigroup of an appropriate closed loop system in product state spaces

a proof of the multiplicity 1 conjecture for min max minimal surfaces in arbitrary codimension given any admissible k dimensional family of immersions of a given closed oriented surface into an arbitrary closed riemannian manifold we prove that the corresponding min max width for the area is achieved by a smooth possibly branched immersed minimal surface with multiplicity 1 and morse index bounded by k

drift analysis and average time complexity of evolutionary algorithms the computational time complexity is an important topic in the theory of evolutionary algorithms eas this paper reports some new results on the average time complexity of eas based on drift analysis some useful drift conditions for deriving the time complexity of eas are studied including conditions under which an ea will take no more than polynomial time in problem size to solve a problem and conditions under which an ea will take at least exponential time in problem size to solve a problem the paper first presents the general results and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of eas while previous work only considered 1 1 eas without any crossover the eas considered in this paper are fairly general which use a finite population crossover mutation and selection 2001 elsevier science b v all rights reserved

a framework for co optimization algorithm performance and its application to worst case optimization traditional black box optimization searches a set of potential solutions for those optimizing the value of a function whose analytical or algebraic form is unknown or inexistent but whose value can be queried for any input co optimization is a generalization of this setting in which fully evaluating a potential solution may require querying some function more than once typically a very large number of times when that s the case co optimization poses unique difficulties to designing and assessing algorithms a generally applicable approach is to judge co optimization algorithm performance via an aggregate over all possible functions in the problem domain we establish formal definitions of such aggregate performance and then investigate the following questions concerning algorithm design 1 are some algorithms strictly better than others i e is there free lunch 2 do optimal algorithms exist and 3 if so are they practical we formally define free lunch and aggregate optimality of co optimization algorithms and derive generic conditions for their existence we review and explain prior no free lunch results from the perspective of these conditions we also show how this framework can be used to bridge several fields of research by allowing formalization of their various problems and views on performance we then apply and extend the generic results in a context involving a particular type of co optimization called worst case optimization in this context we show that there exist algorithms that are aggregately optimal for any budget allowed number of function calls and any starting point set of previously uncovered function call outcomes and also non trivially strictly optimal for many budgets and starting points moreover we formalize the operation of such optimal algorithms and show that for certain domains budgets and starting points this operation is equivalent to a simple procedure with tractable implementation a first of its kind result for co optimization

plateaus can be harder in multi objective optimization abstract in recent years a lot of progress has been made in understanding the behavior of evolutionary computation methods for single and multi objective problems our aim is to analyze the diversity mechanisms that are implicitly used in evolutionary algorithms for multi objective problems by rigorous runtime analyses we show that even if the population size is small the runtime can be exponential where corresponding single objective problems are optimized within polynomial time to illustrate this behavior we analyze a simple plateau function in a first step and extend our result to a class of instances of the well known setcover problem

population size versus runtime of a simple evolutionary algorithm evolutionary algorithms eas find numerous applications and practical knowledge on eas is immense in practice sophisticated population based eas employing selection mutation and crossover are applied in contrast theoretical analysis of eas often concentrates on very simple algorithms such as the 1 1 ea where the population size equals 1 in this paper the question is addressed whether the use of a population by itself can be advantageous a population based ea that neither makes use of crossover nor any diversity maintaining operator is investigated on an example function it is shown that an increase of the population size by a constant factor decreases the expected runtime from exponential to polynomial thereby the best gap known so far is improved from superpolynomial vs polynomial to exponential vs polynomial moreover it is proved that the exponential and polynomial runtime bounds occur with a probability exponentially close to one if the population size is a constant resp a small polynomial finally a second example function where only a small population leads to a polynomial runtime and a hierarchy result on the appropriate population size are presented the analyses show formally how the population size can lead to different attractors in the search space

a modification to moea d de for multiobjective optimization problems with complicated pareto sets this paper presents an algorithm named the uniform design multiobjective differential algorithm based on decomposition umode d for optimizing multiobjective problems the algorithm is a modification to the new version of moea d based on differential evolution de i e moea d de proposed by li and zhang 2009 20 its distinguishing features include 1 the uniform design method is applied to generate the aggregation coefficient vectors so that the decomposed scalar optimization subproblems are uniformly distributed and therefore the algorithm could explore uniformly the region of interest from the initial iteration 2 the simplified quadratic approximation with three best points is employed to improve the local search ability and the accuracy of the minimum scalar aggregation function value umode d is compared with the original moea d de and nsga ii by solving a wide set of problems with complicated pareto set shapes in this paper experimental results indicate that umode d significantly outperforms moea d de and nsga ii on these test problems two sets of experiments are carried out to illustrate the efficiency of the uniform design method and the simplified quadratic approximation separately in addition umode d is tested on cec 2009 problems and combinatorial knapsack problems experimental results show that the proposed algorithm performs better than the further improved moea d for almost all the cec 2009 problems and the results obtained are very competitive when comparing umode d with some other algorithms on these multiobjective knapsack problems

analysis of an iterated local search algorithm for vertex cover in sparse random graphs recently various randomized search heuristics have been studied for the solution of the minimum vertex cover problem in particular for sparse random instances according to the g n c n model where c 0 is a constant methods from statistical physics suggest that the problem is easy if c

a comparison of simulated annealing with a simple evolutionary algorithm on pseudo boolean functions of unitation the development in the area of randomized search heuristics has shown the importance of a rigorous theoretical analysis of the performance of these heuristics unfortunately the analysis of the expected optimization time of a specific algorithm has in general no implications on the behaviour of other algorithms even if they differ only in some aspects indeed small differences may imply large differences in the optimization time hence it is an important issue to compare fundamental heuristics and to find out for which problems they behave in such a similar way that results on one heuristic can be transferred to the other one and to describe problems where they behave quite differently such an approach is performed here to the simple and well known 1 1 ea which is based on elitist selection and a global search operator and simulated annealing which is based on nonelitist selection and a local search operator

wellposedness and wave propagation for a class of integrodifferential equations in banach space in recent years there has been an increasing interest in the mathematical treatment of problems which originate in the theory of viscoelastic materials in such problems the constitutive relations for these materials incorporate the history of the strain or the strain rate and in the linear theory lead to linear volterra integrodifferential equations with infinite memory in a banach space x the purpose of this paper is to examine the problems of wellposedness and wave propagation for one such equation in particular we are concerned with equations of the form x t 4 x r j f t s x s ds al

asymptotic stability of the solution of the m mb 1 queueing model by using positive c 0 semigroup theory we study the asymptotic stability of the solution of a bulk queueing model m m b 1

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

on the approximation ability of evolutionary optimization with application to minimum set cover evolutionary algorithms eas are heuristic algorithms inspired by natural evolution they are often used to obtain satisficing solutions in practice in this paper we investigate a largely underexplored issue the approximation performance of eas in terms of how close the solution obtained is to an optimal solution we study an ea framework named simple ea with isolated population seip that can be implemented as a single or multi objective ea we analyze the approximation performance of seip using the partial ratio which characterizes the approximation ratio that can be guaranteed specifically we analyze seip using a set cover problem that is np hard we find that in a simple configuration seip efficiently achieves an h n approximation ratio the asymptotic lower bound for the unbounded set cover problem we also find that seip efficiently achieves an h k k 18k 9 approximation ratio the currently best achievable result for the k set cover problem moreover for an instance class of the k set cover problem we disclose how seip using either one bit or bit wise mutation can overcome the difficulty that limits the greedy algorithm

on nonautonomous functional differential equations abstract in this paper we prove a theorem on boundary perturbation of nonautonomous cauchy problems and then apply this result to show the existence and uniqueness of classical solutions of the nonautonomous banach space valued functional differential equation formula

a mixed discrete continuous fragmentation model abstract motivated by the occurrence of shattering mass loss observed in purely continuous fragmentation models this work concerns the development and the mathematical analysis of a new class of hybrid discrete continuous fragmentation models once established the model which takes the form of an integro differential equation coupled with a system of ordinary differential equations is subjected to a rigorous mathematical analysis using the theory and methods of operator semigroups and their generators most notably by applying the theory relating to the kato voigt perturbation theorem honest substochastic semigroups and operator matrices the existence of a unique differentiable solution to the model is established this solution is also shown to preserve nonnegativity and conserve mass

from black box complexity to designing new genetic algorithms black box complexity theory recently produced several surprisingly fast black box optimization algorithms in this work we exhibit one possible reason these black box algorithms often profit from solutions inferior to the previous best in contrast evolutionary approaches guided by the survival of the fittest paradigm often ignore such solutions we use this insight to design a new crossover based genetic algorithm it uses mutation with a higher than usual mutation probability to increase the exploration speed and crossover with the parent to repair losses incurred by the more aggressive mutation a rigorous runtime analysis proves that our algorithm for many parameter settings is asymptotically faster on the onemax test function class than all what is known for classic evolutionary algorithms a fitness dependent choice of the offspring population size provably reduces the expected runtime further to linear in the dimension our experimental analysis on several test function classes shows advantages already for small problem sizes and broad parameter ranges also a simple self adaptive choice of these parameters gives surprisingly good results

optimal parameters for search using a barrier tree markov model the performance on a given problem of search heuristics such as simulated annealing and descent with variable mutation can be described as a function of and optimised over the parameters of the heuristic e g the annealing or mutation schedule we describe heuristics as markov processes the search for optimal parameters is then rendered feasible by the use of level accessible barrier trees for state amalgamation results are presented for schedules minimising where you are and best so far cost over binary perceptron spin glass and max sat problems we also compute first passage time for several toy heuristics including constant temperature annealing and fixed rate mutation search

existence of minimal hypersurfaces in complete manifolds of finite volume we prove that every complete non compact manifold of finite volume contains a possibly non compact minimal hypersurface of finite volume the main tool is the following result of independent interest if a region u can be swept out by a family of hypersurfaces of volume at most v then it can be swept out by a family of mutually disjoint hypersurfaces of volume at most v varepsilon

towards an analytic framework for analysing the computation time of evolutionary algorithms in spite of many applications of evolutionary algorithms in optimisation theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few it is still unclear when an evolutionary algorithm is expected to solve an optimisation problem efficiently or otherwise this paper gives a general analytic framework for analysing first hitting times of evolutionary algorithms the framework is built on the absorbing markov chain model of evolutionary algorithms the first step towards a systematic comparative study among different eas and their first hitting times has been made in the paper

a large population size can be unhelpful in evolutionary algorithms the utilization of populations is one of the most important features of evolutionary algorithms eas there have been many studies analyzing the impact of different population sizes on the performance of eas however most of such studies are based on computational experiments except for a few cases the common wisdom so far appears to be that a large population would increase the population diversity and thus help an ea indeed increasing the population size has been a commonly used strategy in tuning an ea when it did not perform as well as expected for a given problem he and yao 2002 8 showed theoretically that for some problem instance classes a population can help to reduce the runtime of an ea from exponential to polynomial time this paper analyzes the role of population further in eas and shows rigorously that large populations may not always be useful conditions under which large populations can be harmful are discussed in this paper although the theoretical analysis was carried out on one multimodal problem using a specific type of eas it has much wider implications the analysis has revealed certain problem characteristics which can be either the problem considered here or other problems that lead to the disadvantages of large population sizes the analytical approach developed in this paper can also be applied to analyzing eas on other problems

predicting monthly streamflow using data driven models coupled with data preprocessing techniques 1 in this paper the accuracy performance of monthly streamflow forecasts is discussed when using data driven modeling techniques on the streamflow series a crisp distributed support vectors regression cdsvr model was proposed for monthly streamflow prediction in comparison with four other models autoregressive moving average arma k nearest neighbors knn artificial neural networks anns and crisp distributed artificial neural networks cdann with respect to distributed models of cdsvr and cdann the fuzzy c means fcm clustering technique first split the flow data into three subsets low medium and high levels according to the magnitudes of the data and then three single svrs or anns were fitted to three subsets this paper gives a detailed analysis on reconstruction of dynamics that was used to identify the configuration of all models except for arma to improve the model performance the data preprocessing techniques of singular spectrum analysis ssa and or moving average ma were coupled with all five models some discussions were presented 1 on the number of neighbors in knn 2 on the configuration of ann and 3 on the investigation of effects of ma and ssa two streamflow series from different locations in china xiangjiaba and danjiangkou were applied for the analysis of forecasting forecasts were conducted at four different horizons 1 3 6 and 1 2 month ahead forecasts the results showed that models fed by preprocessed data performed better than models fed by original data and cdsvr outperformed other models except for at a 6 month ahead horizon for danjiangkou for the perspective of streamflow series the ssa exhibited better effects on danjingkou data because its raw discharge series was more complex than the discharge of xiangjiaba the ma considerably improved the performance of ann cdann and cdsvr by adjusting the correlation relationship between input components and output of models it was also found that the performance of cdsvr deteriorated with the increase of the forecast horizon

the choice of the offspring population size in the 1 λ evolutionary algorithm we extend the theory of non elitist evolutionary algorithms eas by considering the offspring population size in the 1 l ea we establish a sharp threshold at l log e e 1n 5log 1 0n between exponential and polynomial running times on onemax for any smaller value the 1 l ea needs exponential time on every function that has only one global optimum we also consider arbitrary unimodal functions and show that the threshold can shift towards larger offspring population sizes in particular for the function leadingones there is a sharp threshold at l 2log e e 1n 10log 1 0n finally we investigate the relationship between the offspring population size and arbitrary mutation rates on onemax we get sharp thresholds for l that decrease with the mutation rate this illustrates the balance between selection and mutation

the use of tail inequalities on the probable computational time of randomized search heuristics for the purpose of analyzing the time cost of evolutionary algorithms eas or other types of randomized search heuristics rshs with certain requirements on the probability of obtaining a target solution this paper proposes a new index called the probable computational time pct which complements expected running time analysis using simple tail inequalities such as markov s inequality and chebyshev s inequality we also provide basic properties of pct explicitly exhibiting the general relationships between the expected running time and the pct to present deeper estimations of the pct for specific rshs and problems we demonstrate a new inequality that is based on the general form of the chernoff inequality and previous methods such as fitness based partitions and potential functions which have been used to analyze the expected running time of rshs the precondition of the new inequality is that the total running time can be described as the sum of a linear combination of some independent geometrically distributed variables and a constant term the new inequality always provides meaningful upper bounds for the pct under such circumstances some applications of the new inequality for simple eas ant colony optimization aco and particle swarm optimization pso algorithms on simple pseudo boolean functions are illustrated in this paper

convergence of set based multi objective optimization indicators and deteriorative cycles multi objective optimization deals with the task of computing a set of solutions that represents possible trade offs with respect to a given set of objective functions set based approaches such as evolutionary algorithms are very popular for solving multi objective optimization problems convergence of set based approaches for multi objective optimization is essential for their success we take an order theoretic view on the convergence of set based multi objective optimization and examine how the use of indicator functions can help to direct the search towards pareto optimal sets in doing so we point out that set based multi objective optimization working on the dominance relation of search points has to deal with a cyclic behavior that may lead to worsening with respect to the pareto dominance relation defined on sets later on we show in which situations well known binary and unary indicators can help to avoid this cyclic behavior and therefore guarantee convergence of the algorithm we also study the impact of deteriorative cycles on the runtime behavior and give an example in which they provably slow down the optimization process

optimal parameter choices via precise black box analysis abstract it has been observed that some working principles of evolutionary algorithms in particular the influence of the parameters cannot be understood from results on the asymptotic order of the runtime but only from more precise results in this work we complement the emerging topic of precise runtime analysis with a first precise complexity theoretic result our vision is that the interplay between algorithm analysis and complexity theory becomes a fruitful tool also for analyses more precise than asymptotic orders of magnitude as particular result we prove that the unary unbiased black box complexity of the onemax benchmark function class is n ln n c n o n for a constant c which is between 0 2539 and 0 2665 this runtime can be achieved with a simple 1 1 type algorithm using a fitness dependent mutation strength when translated into the fixed budget perspective our algorithm finds solutions which are roughly 13 closer to the optimum than those of the best previously known algorithms to prove our results we formulate several new versions of the variable drift theorems which also might be of independent interest

spectral enclosures for a class of block operator matrices abstract we prove new spectral enclosures for the non real spectrum of a class of 2 2 block operator matrices with self adjoint operators a and d on the diagonal and operators b and b as off diagonal entries one of our main results resembles gershgorin s circle theorem the enclosures are applied to j frame operators

optimizing linear functions with the 1 λ evolutionary algorithm different asymptotic runtimes for different instances abstract we analyze how the 1 evolutionary algorithm ea optimizes linear pseudo boolean functions we prove that it finds the optimum of any linear function within an expected number of o 1 n log n n iterations we also show that this bound is sharp for some linear functions e g the binary value function since previous works shows an asymptotically smaller runtime for the special case of onemax it follows that for the 1 ea different linear functions may have run times of different asymptotic order the proof of our upper bound heavily relies on a number of classic and recent drift analysis methods in particular we show how to analyze a process displaying different types of drifts in different phases our work corrects a wrongfully claimed better asymptotic runtime in an earlier work 13 we also use our methods to analyze the runtime of the 1 ea on the onemax test function and obtain a new upper bound of o n log log log for the case that is larger than o log n log log n log log log n this is the cut off point where a linear speed up ceases to exist while our results are mostly spurred from a theory driven interest they also show that choosing the right size of the offspring population can be crucial for both the binary value and the onemax test function we observe that once a linear speed up ceases to exist in fact the speed up from a larger reduces to sub logarithmic still at the price of a linear increase of the cost of each generation

algorithmic analysis of a basic evolutionary algorithm for continuous optimization in practical optimization applying evolutionary algorithms has nearly become a matter of course their theoretical analysis however is far behind practice so far theorems on the runtime are limited to discrete search spaces results for continuous search spaces are limited to convergence theory or even rely on validation by experiments which is unsatisfactory from a theoretical point of view the simplest or most basic evolutionary algorithms use a population consisting of only one individual and use random mutations as the only search operator here the so called 1 1 evolution strategy for minimization in r n is investigated when it uses isotropically distributed mutation vectors in particular so called gaussian mutations are analyzed when the so called 1 5 rule is used for their adaptation obviously a reasonable analysis must respect the function to be minimized and furthermore the runtime must be measured with respect to the approximation error a first algorithmic analysis of how the runtime depends on n the dimension of the search space is presented this analysis covers all unimodal functions that are monotone with respect to the distance from the optimum it turns out that in the scenario considered gaussian mutations in combination with the 1 5 rule indeed ensure asymptotically optimal runtime namely q n steps function evaluations are necessary and sufficient to halve the approximation error

an efficient adaptive parameter variation scheme for metaheuristics based on the epsilon constraint method abstract this paper discusses methods for generating or approximating the pareto set of multiobjective optimization problems by solving a sequence of constrained single objective problems the necessity of determining the constraint value a priori is shown to be a serious drawback of the original epsilon constraint method we therefore propose a new adaptive scheme to generate appropriate constraint values during the run a simple example problem is presented where the running time measured by the number of constrained single objective sub problems to be solved of the original epsilon constraint method is exponential in the problem size number of decision variables although the size of the pareto set grows only linearly we prove that independent of the problem or the problem size the time complexity of the new scheme is o km 1 where k is the number of pareto optimal solutions to be found and m the number of objectives simulation results for the example problem as well as for different instances of the multiobjective knapsack problem demonstrate the behavior of the method and links to reference implementations are provided

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

generalized decomposition and cross entropy methods for many objective optimization decomposition based algorithms for multi objective optimization problems have increased in popularity in the past decade although convergence to the pareto optimal front pf for such algorithms can often be superior to that of pareto based alternatives the problem of effectively distributing pareto optimal solutions in a high dimensional space has not been solved in this work we introduce a novel concept which we call generalized decomposition generalized decomposition provides a framework with which the decision maker dm can guide the underlying search algorithm toward specific regions of interest or the entire pareto front with the desired distribution of pareto optimal solutions the method simplifies many objective problems by unifying the three performance objectives of an a posteriori multi objective optimizer convergence to the pf evenly distributed pareto optimal solutions and coverage of the entire front to only one that of convergence a framework established on generalized decomposition and an estimation of distribution algorithm eda based on low order statistics namely the cross entropy method is created to illustrate the benefits of the proposed concept for many objective problems the algorithm mace gd is shown to be highly competitive with the existing best in class decomposition based algorithm moea d and a more elaborate eda method rm meda

non existence of linear universal drift functions drift analysis is a powerful tool to prove upper and lower bounds on the runtime of randomized search heuristics its most famous application is a simple proof for the classical problem how the 1 1 evolutionary algorithm ea optimizes linear pseudo boolean functions a relatively simple potential function allows to track the progress of the ea optimizing any linear function in this work we show that such beautiful proofs cease to exist if the mutation probability is slightly larger than the standard value of 1 n

first hitting times under drift abstract for the last ten years almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory making it the arguably most important tool in this domain its success is due to its ease of use and its powerful result drift theory allows the user to derive bounds on the expected first hitting time of a random process by bounding expected local changes of the process the drift this is usually far easier than bounding the expected first hitting time directly due to the widespread use of drift theory it is of utmost importance to have the best drift theorems possible we improve the fundamental additive multiplicative and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary our additive drift theorem for upper bounds only requires the process to be lower bounded that is we remove unnecessary restrictions like a finite discrete or bounded state space as corollaries the same is true for our upper bounds in the case of variable and multiplicative drift by bounding the step size of the process we derive new lower bounding multiplicative and variable drift theorems last we also state theorems that are applicable when the process has a drift of 0 by using a drift on the variance of the process

a new approach to estimating the expected first hitting time of evolutionary algorithms evolutionary algorithms ea have been shown to be very effective in solving practical problems yet many important theoretical issues of them are not clear the expected first hitting time is one of the most important theoretical issues of evolutionary algorithms since it implies the average computational time complexity in this paper we establish a bridge between the expected first hitting time and another important theoretical issue i e convergence rate through this bridge we propose a new general approach to estimating the expected first hitting time using this approach we analyze eas with different configurations including three mutation operators with without population a recombination operator and a time variant mutation operator on a hard problem the results show that the proposed approach is helpful for analyzing a broad range of evolutionary algorithms moreover we give an explanation of what makes a problem hard to eas and based on the recognition we prove the hardness of a general problem

how to analyse evolutionary algorithms many variants of evolutionary algorithms have been designed and applied the experimental knowledge is immense the rigorous analysis of evolutionary algorithms is difficult but such a theory can help to understand design and teach evolutionary algorithms in this survey first the history of attempts to analyse evolutionary algorithms is described and then new methods for continuous as well as discrete search spaces are presented and discussed

the time complexity of maximum matching by simulated annealing the random heuristic search algorithm called simulated annealing is considered for the problem of finding the maximum cardinality matching in a graph it is shown that neither a basic form of the algorithm nor any other algorithm in a fairly large related class of algorithms can find maximum cardinality matchings such that the average time required grows as a polynomial in the number of nodes of the graph in contrast it is also shown for arbitrary graphs that a degenerate form of the basic annealing algorithm obtained by letting temperature be a suitably chosen constant produces matchings with nearly maximum cardinality in polynomial average time

the impact of parametrization in memetic evolutionary algorithms memetic evolutionary algorithms integrate local search into the search process of evolutionary algorithms as computational resources have to be spread adequately among local and evolutionary search one has to care about when to apply local search and how much computational effort to devote to local search often local search is called with a fixed frequency and run for a fixed number of iterations the local search depth there is empirical evidence that these parameters have a significant impact on performance but a theoretical understanding as well as concrete design guidelines are missing we initiate the rigorous theoretical analysis of memetic algorithms to this end we consider a simple memetic algorithm for pseudo boolean optimization that captures basic working principles of memetic algorithms the interplay of genetic operators like mutation and selection with local search we present function classes where even small changes of the parametrization have a strong impact on performance for almost every reasonable parameter setting we construct a function that with high probability can be optimized in polynomial time however changing the local search depth by a small additive term in any direction yields a superpolynomial optimization time with high probability for another class of functions altering the local search frequency by a factor of 2 even yields exponential optimization times our results show exemplarily that parametrizing memetic evolutionary algorithms can be extremely hard moreover this work yields insights into the dynamic behavior of memetic algorithms and contributes to a theoretical foundation of hybrid metaheuristics

the effect of multiple optima on the simple ga run time complexity genetic algorithms are stochastic search algorithms that have been applied to optimization problems in this paper we analyze the run time complexity of a genetic algorithm when we are interested in one of a set of distinguished solutions one such case occurs when multiple optima exist we define the worst case scenario and derive a probabilistic worst case bound on the number of iterations required to find one of these multiple solutions of interest

the spectrum of unbounded operator matrices with non diagonal domain abstract systems of linear evolution equations involving non diagonal boundary conditions yield operator matrices having non diagonal domains in products of banach spaces we show how to compute the spectrum of these matrices by using virtual matrix elements and characteristic operator functions

polynomial operator matrices as semigroup generators the 2 2 case on considere des systemes d equations d evolution non lineaires ecrits sous la forme u t au t ou u est une fonction a valeurs dans un espace produit e n et a a ij n m est une matrice dont les a ij sont des operateurs lineaires sur e on etudie le cas ou les a ij sont des polynomes p ij a par rapport a un seul operateur a sur e et on se restreint au cas des matrices 2 2

more effective crossover operators for the all pairs shortest path problem the all pairs shortest path problem is the first non artificial problem for which it was shown that adding crossover can significantly speed up a mutation only evolutionary algorithm recently the analysis of this algorithm was refined and it was shown to have an expected optimization time w r t the number of fitness evaluations of q n 3 2 5 logn 0 2 5 in contrast to this simple algorithm evolutionary algorithms used in practice usually employ refined recombination strategies in order to avoid the creation of infeasible offspring we study extensions of the basic algorithm by two such concepts which are central in recombination namely repair mechanisms and parent selection we show that repairing infeasible offspring leads to an improved expected optimization time of o n 3 2 logn 0 2 as a second part of our study we prove that choosing parents that guarantee feasible offspring results in an optimization time of o n 3logn both results show that already simple adjustments of the recombination operator can asymptotically improve the runtime of evolutionary algorithms

global existence for semilinear parabolic systems on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme u t a t u f t x u du d 2m 1 u dans t 0 b t u 0 sur t 0 u t 0 u 0 sur ou est un domaine de r n

on the spectrum of evolution operators generated by hyperbolic systems abstract we prove that the asymptotic behavior of semigroups generated by mixed initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation the proof is based on the following reduction principle we construct a reduced system which is much simpler than the given one such that the difference of the semigroups is a compact operator the construction of the reduced system is presented in the non autonomous case because it has implifications in the study of systems which are time periodic

matrices of operators and regularized semigroups where p is a matr ix of polynomials in n variables and ia1 ia is a family of c o m m u t i n g generators of bounded linear strongly cont inuous groups of operators intuitively one wants to take the exponent ia l of the matr ix of opera to r s p a since the spec t rum of ak is real for all k this means we want the complex matr ix pi j x to be contained in a left half plane in some sense for all x in r the strongest sense is to have the numerical range of the complex matr ix conta ined in a half plane r e z 0 xer a sense that is easier to calculate or at least estimate is to have the spectrum of p d x conta ined in a left half plane for all x r the system 1 1 is then said to be petrowsky correct see 9 the intuit ion we just described is not quite true in general wi thout some