a multilevel approach to single and multiobjective aerodynamic optimization an optimization platform with multilevel structure which is capable of efficiently solving design optimization problems in aerodynamics is proposed the multilevel structure relies on a two way regular exchange of information between successive search levels each level can be associated with a different evaluation software different problem parameterization and or different search tool for the minimization of the same objective function s the combination of some or all of the aforementioned strategies is possible although beyond the scope of this paper the basic optimization tool which is associated with at least one of the levels is a metamodel assisted evolutionary algorithm candidate solutions that have previously been examined are memorized and serve to train radial basis function networks operating as local surrogate evaluation models to handle multiobjective optimization problems with different search techniques at each level we hybridize an evolutionary algorithm that computes the front of pareto optimal solutions at the lower level s with a gradient based method at the upper level for the purpose of refinement through local search in aerodynamic optimization the adjoint method is used to compute the gradient of the objective function for the gradient method to apply to a front of solutions rather than a single individual an approximation of the spea 2 utility function gradient needs to be devised the multilevel platform is demonstrated on mathematical problems as well as the design of isolated and compressor cascade airfoils

resonant semilinear problems with nonlinear term depending on the derivative we study the existence of solution for a boundary value problem at resonance where the nonlinearity depends only on the derivative in a sense we can say that the problem considered is strongly resonant our proofs make use of the lyapunov schmidt reduction in so doing we are led with the asymptotic estimate of the corresponding bifurcation equation

relaxing convergence conditions for newton s method the classical kantorovich theorem on newton s method assumes that the first derivative of the operator involved satisfies a lipschitz condition 0 f x f y l x y in this paper we weaken this condition assuming that 0 f x f x0 x x0 for a given point x0

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

convergence analysis of the secant type methods in this paper under the hypothesis that derivative satisfies some kinds of weak lipschitz condition the radius estimates of the convergence balls of secant type methods for operator equations are established in banach space some applications and numerical experiments are also given

mysovskii type theorem for the secant method under holder continuous frechet derivative abstract the mysovskii type condition is considered in this study for the secant method in banach spaces to solve a nonlinear operator equation we suppose the inverse of divided difference of order one is bounded and the frechet derivative of the nonlinear operator is holder continuous by use of fibonacci generalized sequence a semilocal convergence theorem is established which matches with the convergence order of the method finally two simple examples are provided to show that our results apply where earlier ones fail

on newton s method under holder continuous derivative a mysovskii type theorem for newton s method under k p holder continuous derivative is considered in this paper for the application studied by others the new condition is weaker than ones in the literature also we prove that the optimal convergent order is p 1 for 0 p 1

evolutionary optimization of computationally expensive problems via surrogate modeling we present a parallel evolutionary optimization algorithm that leverages surrogate models for solving computationally expensive design problems with general constraints on a limited computational budget the essential backbone of our framework is an evolutionary algorithm coupled with a feasible sequential quadratic programming solver in the spirit of lamarckian learning we employ a trust region approach for interleaving use of exact modelsfortheobjectiveandconstraintfunctionswithcomputationallycheapsurrogatemodelsduringlocalsearch in contrastto earlier work we construct local surrogatemodels using radial basis functionsmotivated by theprinciple of transductive inference further the present approach retains the intrinsic parallelism of evolutionary algorithms and can hence be readily implemented on grid computing infrastructures experimental results are presented for some benchmark test functions and an aerodynamic wing design problem to demonstrate that our algorithm converges to good designs on a limited computational budget

extended newton methods for multiobjective optimization majorizing function technique and convergence analysis we consider the extended newton method for approaching a pareto optimum of a multiobjective optimization problem establish quadratic convergence criteria and estimate a radius of convergence ball

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

hybridization of a multi objective genetic algorithm a neural network and a classical optimizer for a complex design problem in fluid dynamics abstract this paper describes the combination of several optimization technologies that can be used to tackle challenging design problems the approach that uses a multi objective genetic algorithm a neural network and a gradient based optimizer is first outlined with the help of a computationally inexpensive mathematical test function then the methodology is applied to the design of a sailing yacht fin keel coupling the optimization codes to 3d navier stokes simulations to perform the multi objective optimization task a parallel computer is employed

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

a new semilocal convergence theorem for newton s method abstract a new semilocal convergence theorem for newton s method is established for solving a nonlinear equation f x 0 defined in banach spaces it is assumed that the operator f is twice frechet differentiable and f satisfies a lipschitz type condition results on uniqueness of solution and error estimates are also given finally these results are compared with those that use kantorovich conditions

convergence of gauss newton s method and uniqueness of the solution abstract in this paper we study the convergence of gauss newton s method for nonlinear least squares problems under the hypothesis that derivative satisfies some kinds of weak lipschitz condition we obtain the exact estimates of the radii of convergence ball of gauss newton s method and the uniqueness ball of the solution new results can be used to determinate approximation zero of gauss newton s method

existence of solutions to u u g t u u p t u 0 u π 0 abstract existence and multiplicity results for the boundary value problem formula are presented the proofs are based on the alternative method a connectedness result the contraction mapping principle and a detailed analysis of the bifurcation equation utilizing e g a generalization of the mean value theorem for integrals we shall obtain results with g bounded or unbounded having finite limits at or without limits thus extending some recent results in the literature the proofs offer a constructive way to find the bounds for p and to find numerically the number of solutions and the approximative solutions

two conditions concerning newton s method in this note we compare two conditions concerning newton s method and find out that even under some weaker condition one has the convergence of the newton s method and a better error bound than the relative strong condition

genetic local search for multi objective combinatorial optimization abstract the paper presents a new genetic local search gls algorithm for multi objective combinatorial optimization moco the goal of the algorithm is to generate in a short time a set of approximately efficient solutions that will allow the decision maker to choose a good compromise solution in each iteration the algorithm draws at random a utility function and constructs a temporary population composed of a number of best solutions among the prior generated solutions then a pair of solutions selected at random from the temporary population is recombined local search procedure is applied to each offspring results of the presented experiment indicate that the algorithm outperforms other multi objective methods based on gls and a pareto ranking based multi objective genetic algorithm ga on travelling salesperson problem tsp

aerodynamic shape design using evolutionary algorithms and new gradient assisted metamodels aerodynamic shape design and optimization problems based on evolutionary algorithms and surrogate evaluation tools i e the so called metamodels have recently found widespread use using metamodels trained either separately from or during the optimization loop a considerable reduction in the overall computing cost can be achieved to support metamodel based evolutionary algorithms a class of new metamodels which utilize both known responses and response gradients for their training is proposed the new gradient assisted metamodels are extensions of standard multi layer perceptrons and radial basis function networks to demonstrate the prediction capabilities of the proposed metamodels and investigate different implementation modes within search algorithms along with the relevant cpu cost a number of 2d and 3d aerodynamic shape namely airfoils and turbomachinery blades design problems are analyzed single and two objective problems aiming at designing shapes that reproduce known pressure distributions at specific operating points are considered the exact evaluation tool is a numerical solver of the compressible fluid flow equations the necessary gradient of the objective function is obtained by formulating and numerically solving adjoint equations

the guaranteed convergence of laguerre like method abstract the construction of initial conditions that provide a guaranteed convergence of zero finding methods has attracted a great deal of attention for many years in this paper we consider convergent properties of the laguerre like method of the fourth order for the simultaneous approximation of polynomial zeros using a procedure based on smale s point estimation theory and some recent results concerned with localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of practical importance

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

double resonance with landesman lazer conditions for planar systems of ordinary differential equations abstract we prove the existence of periodic solutions for first order planar systems at resonance the nonlinearity is indeed allowed to interact with two positively homogeneous hamiltonians both at resonance and some kind of landesman lazer conditions are assumed at both sides we are thus able to obtain as particular cases the existence results proposed in the pioneering papers by lazer and leach 1969 27 and by frederickson and lazer 1969 18 our theorem also applies in the case of asymptotically piecewise linear systems and in particular generalizes fabry s results in fabry 1995 10 for scalar equations with double resonance with respect to the dancer fucik spectrum

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

approximate solutions for abstract inequality systems we consider conic inequality systems of the type f x ge k 0 with approximate solution x 0 associated to a parameter tau where f is a twice frechet differentiable function between hilbert spaces x and y and ge k is the partial order in y defined by a nonempty convex not necessarily closed cone k subseteq y we prove that under the suitable conditions the system f x ge k 0 is solvable and the ratio of the distance from x 0 to the solution set s over the distance from f x 0 to the cone k has an upper bound given explicitly in terms of tau and x 0 we show that the upper bound is sharp applications to analytic function inequality equality systems on euclidean spaces are given and the corresponding results of dedieu siam j optim 11 2000 pp 411 425 are extended and significantly improved

remarks on the range properties of certain semilinear problems of landesman lazer type abstract in the article we consider the resonant dirichlet problem u t u t g u t f t t 0 u 0 u 0 1 where g is a continuous function having finite limits at plus and minus infinity g g imposing g g s g for any s r we formulate a necessary condition our main result provides a characteristic of the set of functions f c 0 t such that 1 has a solution however to prove this we need that g is odd increasing and satisfies a certain asymptotic condition at infinity in addition the main tools used are lyapunov schmidt reduction and certain asymptotical methods

a note on the kantorovich theorem for newton iteration abstract in this paper a new theorem for the newton method convergence is obtained its condition is different from that of the kantorovich theorem and therefore it has theoretical and practical value

convergence of the family of the deformed euler halley iterations under the holder condition of the second derivative the convergence problem of the family of the deformed euler halley iterations with parameters for solving nonlinear operator equations in banach spaces is studied under the assumption that the second derivative of the operator satisfies the holder condition a convergence criterion of the order 2 p of the iteration family is established an application to a nonlinear hammerstein integral equation of the second kind is provided

existence and multiplicity of solutions for certain dirichlet problems with nonlinearity depending on the derivative abstract we study existence and multiplicity of solutions for the problem u t u t g u t f t u 0 u 0 where f c 0 and g c r has finite limits g lim u g u the main tools used are the lyapunov schmidt reduction principle and certain asymptotic methods

newton s method for solving generalized equations kantorovich s and smale s approaches abstract in this paper we study newton type methods for solving generalized equations involving set valued maps in banach spaces kantorovich type theorems both local and global versions are proved as well as the quadratic convergence of the newton sequence we also extend smale s classical theory to generalized equations these results are new and can be considered as an extension of many known ones in the literature for classical nonlinear equations our approach is based on tools from variational analysis the metric regularity concept plays an important role in our analysis

an improved convergence theorem for a class of secant like methods an improved semilocal convergence theorem for a class of secant like methods in banach spaces is provided for solving a nonlinear operator equation it is assumed that the first order divided difference of the nonlinear operator is holder continuous our convergence conditions strategically proposed are weaker than some existing ones and are easily satisfied finally two simple examples are provided to show that our results apply while earlier ones fail

a convergence theorem for the newton like methods under some kind of weak lipschitz conditions we provide sufficient conditions for the convergence of the newton like methods in the assumption that the derivative satisfies some kind of weak lipschitz conditions consequently some important convergence theorems follow from our main result in this paper

a generalization of the banach contraction principle with high order of convergence of successive approximations in this paper we present the following generalization of the banach contraction principle let t d x xt d x x be a continuous operator on a complete metric space x d x d satisfying d tx t2x d x tx for all x d tx d with d x tx j where j r j r is an interval containing 0 j j j j is a gauge function of order r 1r 1 on jj in the sense that 0 0 t tr 0 0 t tr is nondecreasing on j 0 j 0 and t t t t on j 0 j 0 suppose that x0 dx0 d is an initial point such that d x0 tx0 jd x0 tx0 j and all of the iterates xn 1 txn n 0 1 belong to dd then the iterative sequence xn 1 txnxn 1 txn converges to a fixed point of tt with error estimates d xn sn r 1 rnd x0 tx0 andd xn d xn xn 1 1 d xn xn 1 where sn r 1 r rn 1 d x0 tx0 sn r 1 r rn 1 d x0 tx0 and is a nonnegative and nondecreasing function on jj such that t t t this fixed point theorem includes as special cases some results of mysovskih i p mysovskih on the convergence of l v kantorovic s method of solution of functional equations and its applications dokl acad nauk sssr 70 1950 565 568 in russian rheinboldt w c rheinboldt a unified convergence theory for a class of iterative processes siam j numer anal 5 1968 42 63 gel man a gel man a certain fixed point principle dokl akad nauk sssr 198 1971 506 508 in russian soviet math dokl 12 1971 813 816 engl transl huang z d huang on newton s method under holder continuous derivative j math anal appl 270 2002 332 339 and others

proper analytic free maps this paper concerns analytic free maps these maps are free analogs of classical analytic functions in several complex variables and are defined in terms of non commuting variables amongst which there are no relations they are free variables analytic free maps include vector valued polynomials in free non commuting variables and form a canonical class of mappings from one non commutative domain d in say g variables to another non commutative domain d in g variables as a natural extension of the usual notion an analytic free map is proper if it maps the boundary of d into the boundary of d assuming that both domains contain 0 we show that if f d d is a proper analytic free map and f 0 0 then f is one to one moreover if also g g then f is invertible and f 1 is also an analytic free map these conclusions on the map f are the strongest possible without additional assumptions on the domains d and d

on newton s method under mild differentiability conditions and applications in this study we are concerned with the problem of approximating a locally unique solution of a nonlinear operator equation in a banach space we examine the convergence of the newton kantorovich method by assuming that the frechet derivative of the nonlinear operator involved is only holder continuous many results exist already in the literature to cover the stronger case when the operator is lipschitz continuous moreover we use this study to counter some claims made by galperin and waksman on related work of ours furthermore we use our results to solve a two point boundary value problem appearing in kinetic theory of gasses elasticity and other applied areas

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

a new fourth order family of simultaneous methods for finding polynomial zeros using a suitable fixed point relation with a complex parameter a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented some numerical results for various values of the parameter are given

the convergence of a family of parallelzero finding methods abstract one of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence in the spirit of smale s point estimation theory introduced in 1981 we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of significant practical importance

viscous inviscid analysis of transonic and low reynolds number airfoils a method of accurately calculating transonic and low reynolds number airfoil flows implemented in the viscous inviscid design analysis code ises is presented the euler equations are discretized on a conservative streamline grid and are strongly coupled to a two equation integral boundary layer formulation using the displacement thickness concept a transition prediction formulation of the e type is derived and incorporated into the viscous formulation the entire discrete equation set including the viscous and transition formulations is solved as a fully coupled nonlinear system by a global newton method this is a rapid and reliable method for dealing with strong viscous inviscid interactions which invariably occur in transonic and low reynolds number airfoil flows the results presented demonstrate the ability of the ises code to predict transitioning separation bubbles and their associated losses the rapid airfoil performance degradation with decreasing reynolds number is thus accurately predicted also presented is a transonic airfoil calculation involving shock induced separation showing the robustness of the global newton solution procedure good agreement with experiment is obtained further demonstrating the performance of the present integral boundary layer formulation

convergence ball of a modified secant method with convergence order 1 839 abstract a local convergence theorem on a modified secant method with convergence order 1 839 for solving nonlinear equations is established under the hypotheses that the second order and third order derivative of the function involved are bounded an estimate of the radius of the convergence ball of this method is obtained moreover an error estimate is provided which matches the convergence order of the method