a trust region algorithm for heterogeneous multiobjective optimization this paper presents a trust region method for multiobjective heterogeneous optimization problems one of the objective functions is an expensive black box function given for example by a time co

multi objective retrospective optimization using stochastic zigzag search abstract we propose a new retrospective optimization ro method for multi objective simulation optimization moso problems ro algorithms generate a sequence of sample path sp problems and solve these sp problems iteratively using a nonlinear optimizer in this study a stochastic zigzag search algorithm is chosen in the ro framework to solve sp problems the key idea of zigzag search is searching around the pareto front by applying an efficient local search procedure using the gradients of the objective functions many continuous moso problems have smooth objective functions and their non dominated objective function values form a smooth surface in the image space this fact motivates developing the zigzag search method embedded in ro for such relatively well posed moso problems a numerical implementation of this method multi objective retrospective optimization using zigzag search morozs is presented particularly for continuous bi objective simulation optimization boso problems with well connected pareto optimal solutions morozs is designed for boso problems in which a simulation oracle returns both objective function values and gradients due to the local nature of zigzag search morozs can only guarantee the asymptotic convergence to local pareto optimality the efficiency of morozs is studied using three boso problems with noisy objective functions and is compared to that of genetic algorithms based nsga ii and a recently developed method mo compass

improving the mechanical performances of a multilayered plate with the orientations of its layers of fibers we consider a symmetric composite multilayered plate whose fiber orientation varies from a layer to another the plate model used is that of mindlin we are interested in determining the optimal fiber orientations that optimize in the same time two criteria minimizing the compliance and maximizing the smallest eigenfrequency of vibration or minimizing both of the compliance and the smallest eigenfrequency or maximizing both of the smallest eigenfrequency and the smallest buckling load in order to optimize one of the above criteria a metaheuristic algorithm of simulated annealing type is used while in the case of optimizing two objectives the pareto front method is used numerical results are presented for a rectangular eight layered plate

response surface approximation of pareto optimal front in multi objective optimization a systematic approach is presented to approximate the pareto optimal front pof by a response surface approximation the data for the pof is obtained by multi objective evolutionary algorithm improvements to address drift in the pof are also presented the approximated pof can help visualize and quantify trade offs among objectives to select compromise designs the bounds of this approximate pof are obtained using multiple convex hulls the proposed approach is applied to study trade offs among objectives of a rocket injector design problem where performance and life objectives compete the pof is approximated using a quintic polynomial the compromise region quantifies trade offs among objectives 2006 elsevier b v all rights reserved

mesh adaptive direct search algorithms for constrained optimization this paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available we introduce the mesh adaptive direct search mads class of algorithms which extends the generalized pattern search gps class by allowing local exploration called polling in an asymptotically dense set of directions in the space of optimization variables this means that under certain hypotheses including a weak constraint qualification due to rockafellar mads can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained the main gps convergence result is to identify limit points hat x where the clarke generalized derivatives are nonnegative in a finite set of directions called refining directions although in the unconstrained case nonnegative combinations of these directions span the whole space the fact that there can only be finitely many gps refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for gps the main result of this paper is that the general mads framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the clarke derivatives are nonnegative we propose an instance of mads for which the refining directions are dense in the hypertangent cone at hat x with probability 1 whenever the iterates associated with the refining directions converge to a single hat x the instance of mads is compared to versions of gps on some test problems we also illustrate the limitation of our results with examples

orthomads a deterministic mads instance with orthogonal directions the purpose of this paper is to introduce a new way of choosing directions for the mesh adaptive direct search mads class of algorithms the advantages of this new orthomads instantiation of mads are that the polling directions are chosen deterministically ensuring that the results of a given run are repeatable and that they are orthogonal to each other which yields convex cones of missed directions at each iteration that are minimal in a reasonable measure convergence results for orthomads follow directly from those already published for mads and they hold deterministically rather than with probability one as is the case for ltmads the first mads instance the initial numerical results are quite good for both smooth and nonsmooth and constrained and unconstrained problems considered here

required relationship between objective function and pareto frontier orders practical implications it is well known that the weighted sum aggregate objective function fails to capture pareto points that are located on a concave region of the pareto frontier conditions are established for a general aggregate objective function to capture points on a concave pareto frontier region specifically these conditions are developed in terms of the relative orders of the pareto frontier and aggregate objective function these critical conditions can be used in practice to allow an aggregate objective function to capture any potentially desirable pareto point conversely failure to satisfy these critical conditions can represent a pivotal failure point of the general application of design optimization because we may fail to obtain the design we seek

a method for constrained multiobjective optimization based on sqp techniques we propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an sqp type approach the proposed algorithm maintains a list of nondominated points that is improved both for spread along the pareto front and optimality by solving single objective constrained optimization problems these single objective problems are derived as sqp problems based on the given nondominated points under appropriate differentiability assumptions we discuss convergence to local optimal pareto points we provide numerical results for a set of unconstrained and constrained multiobjective optimization problems in the form of performance and data profiles where several performance metrics are used the numerical results confirm the superiority of the proposed algorithm against a state of the art multiobjective solver and a classical scalarization approach both in the quality of the approximated pareto front and in the computational effort necessary to compute the approximation

normal boundary intersection a new method for generating the pareto surface in nonlinear multicriteria optimization problems this paper proposes an alternate method for finding several pareto optimal points for a general nonlinear multicriteria optimization problem such points collectively capture the trade off among the various conflicting objectives it is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the pareto set given an evenly distributed set of parameters a property which the popular method of minimizing weighted combinations of objective functions lacks further this method can handle more than two objectives while retaining the computational efficiency of continuation type algorithms this is an improvement over continuation techniques for tracing the trade off curve since continuation strategies cannot easily be extended to handle more than two objectives

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

richardson extrapolation based sensitivity analysis in the multi objective optimization of analog circuits the feasible solutions provided by a multi objective evolutionary algorithm moea in the optimal sizing of analog integrated circuits ics can be very sensitive to process variations therefore to select the optimal sizes of metal oxide semiconductor field effect transistors mosfets but with low sensitivities we propose to perform multi parameter sensitivity analysis however since moeas generate feasible solutions without an explicit equation then we show the application of richardson extrapolation to approximate the partial derivatives associated to the sensitivities of the performances of an amplifier with respect to the sizes of every mosfet the proposed multi parameter sensitivity analysis is verified through the optimization of a recycled folded cascode rfc operational transconductance amplifier ota we show the behavior of the multi parameter sensitivity approach versus generations the final results show that the optimal sizes selected after executing the sensitivity approach guarantee the lowest sensitivities values while improving the performances of the rfc ota

approximating pareto frontier using a hybrid line search approach the aggregation of objectives in multiple criteria programming is one of the simplest and widely used approach but it is well known that this technique sometimes fail in different aspects for determining the pareto frontier this paper proposes a new approach for multicriteria optimization which aggregates the objective functions and uses a line search method in order to locate an approximate efficient point once the first pareto solution is obtained a simplified version of the former one is used in the context of pareto dominance to obtain a set of efficient points which will assure a thorough distribution of solutions on the pareto frontier in the current form the proposed technique is well suitable for problems having multiple objectives it is not limited to bi objective problems and require the functions to be continuous twice differentiable in order to assess the effectiveness of this approach some experiments were performed and compared with two recent well known population based metaheuristics namely parego and nsga ii when compared to parego and nsga ii the proposed approach not only assures a better convergence to the pareto frontier but also illustrates a good distribution of solutions from a computational point of view both stages of the line search converge within a short time average about 150ms for the first stage and about 20ms for the second stage apart from this the proposed technique is very simple easy to implement and use to solve multiobjective problems

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

box constrained multi objective optimization a gradient like method without a priori scalarization abstract the aim of this paper is the development of an algorithm to find the critical points of a box constrained multi objective optimization problem the proposed algorithm is an interior point method based on suitable directions that play the role of gradient like directions for the vector objective function the method does not rely on an a priori scalarization and is based on a dynamic system defined by a vector field of descent directions in the considered box the key tool to define the mentioned vector field is the notion of vector pseudogradient we prove that the limit points of the solutions of the system satisfy the karush kuhn tucker kkt first order necessary condition for the box constrained multi objective optimization problem these results allow us to develop an algorithm to solve box constrained multi objective optimization problems finally we consider some test problems where we apply the proposed computational method the numerical experience shows that the algorithm generates an approximation of the local optimal pareto front representative of all parts of optimal front

parallel space decomposition of the mesh adaptive direct search algorithm this paper describes a parallel space decomposition psd technique for the mesh adaptive direct search mads algorithm mads extends a generalized pattern search for constrained nonsmooth optimization problems the objective of the present work is to obtain good solutions to larger problems than the ones typically solved by mads the new method psd mads is an asynchronous parallel algorithm in which the processes solve problems over subsets of variables the convergence analysis based on the clarke calculus is essentially the same as for the mads algorithm a practical implementation is described and some numerical results on problems with up to 500 variables illustrate the advantages and limitations of psd mads

measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming one way of solving multiple objective mathematical programming problems is finding discrete representations of the efficient set a modified goal of finding good discrete representations of the efficient set would contribute to the practicality of vector maximization algorithms we define coverage uniformity and cardinality as the three attributes of quality of discrete representations and introduce a framework that includes these attributes in which discrete representations can be evaluated compared to each other and judged satisfactory or unsatisfactory by a decision maker we provide simple mathematical programming formulations that can be used to compute the coverage error of a given discrete representation our formulations are practically implementable when the problem under study is a multiobjective linear programming problem we believe that the interactive algorithms along with the vector maximization methods can make use of our framework and its tools

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

discrete representation of non dominated sets in multi objective linear programming abstract in this paper we address the problem of representing the continuous but non convex set of non dominated points of a multi objective linear programme by a finite subset of such points we prove that a related decision problem is np complete moreover we illustrate the drawbacks of the known global shooting normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples we propose a method which combines the global shooting and normal boundary intersection methods by doing so we overcome their limitations but preserve their advantages we prove that our method computes a set of evenly distributed non dominated points for which the coverage error and the uniformity level can be guaranteed we apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases finally we present numerical results on randomly generated examples

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

a non dominance based online stopping criterion for multi objective evolutionary algorithms a non dominance criterion based metric that tracks the growth of an archive of non dominated solutions over a few generations is proposed to generate a convergence curve for multi objective evolutionary algorithms moeas it was observed that similar to single objective optimization problems there were significant advances toward the pareto optimal front in the early phase of evolution while relatively smaller improvements were obtained as the population matured this convergence curve was used to terminate the moea search to obtain a good trade off between the computational cost and the quality of the solutions two analytical and two crashworthiness optimization problems were used to demonstrate the practical utility of the proposed metric

generalized pointwise bias error bounds for response surface approximations this paper proposes a generalized pointwise bias error bounds estimation method for polynomial based response surface approximations when bias errors are substantial a relaxation parameter is introduced to account for inconsistencies between the data and the assumed true model the method is demonstrated with a polynomial example where the model is a quadratic polynomial while the true function is assumed to be cubic polynomial the effect of relaxation parameter is studied it is demonstrated that when bias errors dominate the bias error bounds characterize the actual error field better than the standard error the bias error bound estimates also help to identify regions in the design space where the accuracy of the response surface approximations is inadequate it is demonstrated that this information can be utilized for adaptive sampling in order to improve accuracy in such regions copyright 2005 john wiley sons ltd

structural model updating and prediction variability using pareto optimal models a multi objective identification method for structural model updating based on modal residuals is presented the method results in multiple pareto optimal structural models that are consistent with the experimentally measured modal data and the modal residuals used to measure the discrepancies between the measured and model predicted modal characteristics these pareto optimal models are due to uncertainties arising from model and measurement errors the relation between the multi objective identification method and the conventional single objective weighted modal residuals method for model updating is investigated using this relation an optimally weighted modal residuals method is also proposed to rationally select the most preferred model among the alternative multiple pareto optimal models for further use in structural model prediction studies computational issues related to the reliable solution of the resulting multi objective and single optimization problems are addressed the model updating methods are compared and their effectiveness is demonstrated using experimental results obtained from a three story laboratory structure tested at a reference and a mass modified configuration the variability of the pareto optimal models and their associated response prediction variability are explored using two structural model classes a simple 3 dof model class and a higher fidelity 546 dof finite element model class it is demonstrated that the pareto optimal structural models and the corresponding response and reliability predictions may vary considerably depending on the fidelity of the model class and the size of measurement errors

on min norm and min max methods of multi objective optimization this paper examines pareto optimality of solutions to multi objective problems scalarized in the min norm compromise programming generalized goal programming or unrestricted min max formulations issues addressed include among others uniqueness in solution or objective space penalization for over achievement of goals min max reformulation of goal programming inferiority in tchebycheff norm minimization strength and weakness of weighted bound optimization quasi satisficing decision making just attaining or even over passing the goals trading off by modifying weights or goals non convex pareto frontier new general necessary and sufficient conditions for both pareto optimality and weak pareto optimality are presented various formulations are compared in theoretical performance with respect to the goal point location ideas for advanced goal programming and interactive decision making are introduced

on the pareto optimum sensitivity analysis in multicriteria optimization to analyse the trade off relations among the set of criteria in multicriteria optimization pareto optimum sensitivity analysis is systematically studied in this paper original contributions cover two parts theoretical demonstrations are firstly made to validate the gradient projection method in pareto optimum sensitivity analysis it is shown that the projected gradient direction evaluated at a given pareto optimum in the design variable space rigorously corresponds to the tangent direction of the pareto curve surface at that point in the objective space this statement holds even for the change of the set of active constraints in the perturbed problem secondly a new active constraint updating strategy is proposed which permits the identification of the active constraint set change to determine the influence of this change upon the differentiability of the pareto curve and finally to compute directional derivatives in non differentiable cases this work will highlight some basic issues in multicriteria optimization some numerical problems are solved to illustrate these novelties

singular continuation generating piecewise linear approximations to pareto sets via global analysis we propose a strategy for approximating pareto optimal sets based on the global analysis framework proposed by smale global analysis and economics i pareto optimum and a generalization of morse theory in dynamical systems academic press new york 1973 pp 531 544 the method highlights and exploits the underlying manifold structure of the pareto sets approximating pareto optima by means of simplicial complexes the method distinguishes the hierarchy between singular set pareto critical set and stable pareto critical set and it can handle the problem of superposition of local pareto fronts occurring in the general nonconvex case furthermore a quadratic convergence result in a suitable setwise sense is proven and tested in a number of numerical examples

an adaptive scalarization method in multiobjective optimization this paper presents a new method for the numerical solution of nonlinear multiobjective optimization problems with an arbitrary partial ordering in the objective space induced by a closed pointed convex cone this algorithm is based on the well known scalarization approach by pascoletti and serafini and adaptively controls the scalarization parameters using new sensitivity results the computed image points give a nearly equidistant approximation of the whole pareto surface the effectiveness of this new method is demonstrated with various test problems and an applied problem from medicine

a radial boundary intersection aided interior point method for multi objective optimization abstract we propose a novel multi objective optimization technique combining non convex radial boundary intersection based decomposition with an interior point method which utilizes both line search and trust region steps suitable for non convex nonlinear optimization radial boundary intersection decomposes the multi objective optimization problem into subproblems which are concerned with finding the solutions closest to a reference point along equally spaced lines emanating radially outwards from the latter point the proposed approach is found to be able to generate good approximations of the pareto front including the periphery by generating a sufficiently diverse set of pareto optimal solutions the proposed method is extensively tested on a large number of recent benchmark problems and real world problems and the performance is found to be favorable in comparison to those of some of the cutting edge stochastic evolutionary optimization algorithms that are commonly used to solve non convex multi objective optimization problems

an evolutionary shape optimization for elastic contact problems subject to multiple load cases most structures in the real life are subject to multiple load cases this paper aims at extending the evolutionary structural optimization eso algorithm to optimal contact shape design for elastic bodies under the multiple load cases to evaluate the reference stresses of each contact node in a finite element framework an extreme stress criterion the worst case design and a weighted average criterion pareto design are presented in the extreme stress method the highest nodal contact stress under all load cases is adopted as the reference level in the weighted average method the weighted sum of nodal contact stresses over all the load cases is regarded as the reference it is found that these two criteria can produce different results in this paper the examples are presented to demonstrate some new features of contact shape optimization in the presence of the multiple load cases

a successive approach to compute the bounded pareto front of practical multiobjective optimization problems this work focuses on the generation of the pareto front for practical applications such as analog circuit sizing the pareto front is the solution to a multiobjective optimization problem it shows all possible optimal compromises between conflicting design objectives performances in technical applications the pareto front is usually bounded it is shown in this work that the boundary of the pareto front consists of so called trade off limits if certain criteria are fulfilled these trade off limits can be computed by solving a multiobjective optimization problem for a subset of the performances a novel approach to pareto optimization is presented which is based on the generation of the trade off limits the pareto front for any number of performances is built up systematically in a successive manner pareto optimal solutions on the boundary of a pareto front are generated first in the next step the inner part of the pareto front is populated with pareto optimal solutions this is the first approach which generates a discretized approximation that shows the total extent of a bounded pareto front without requiring to calculate suboptimal solutions it requires that the pareto front exists is bounded and connected with no gaps and that the set of weakly pareto optimal solutions and pareto optimal solutions are equal

high accuracy approximation techniques for analytic functions a generalization of the familiar mesh point technique for numerical approximation of functions is presented high accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points in particular derivative operators are represented by highly nonlocal matrices but this is no drawback when one has computing machines to perform the algebraic manipulations some examples are given from familiar quantum mechanical problems

a derivative free trust region method for biobjective optimization we consider unconstrained black box biobjective optimization problems in which analytic forms of the objective functions are not available and function values can be obtained only through computationally expensive simulations we propose a new algorithm to approximate the pareto optimal solutions of such problems based on a trust region approach at every iteration we identify a trust region then sample and evaluate points from it to determine nondominated solutions in the trust region we employ a scalarization method to convert the two objective functions into one we construct and optimize quadratic regression models for the two original objectives and the converted single objective we then remove dominated points from the current pareto approximation and construct a new trust region around the most isolated point in order to explore areas that have not been visited we prove convergence of the method under general regularity conditions and present numerical results suggesting that the method effic

necessary and sufficient karush kuhn tucker conditions for multiobjective markov chains optimality the solution concepts proposed in this paper follow the karush kuhn tucker kkt conditions for a pareto optimal solution in finite time ergodic and controllable markov chains multi objective programming problems in order to solve the problem we introduce the tikhonov s regularizator for ensuring the objective function is strict convex then we consider the c variable method for introducing equality constraints that guarantee the result belongs to the simplex and satisfies ergodicity constraints lastly we restrict the cost functions allowing points in the pareto front to have a small distance from one another the computed image points give a continuous approximation of the whole pareto surface the constraints imposed by the c variable method make the problem computationally tractable and the restriction imposed by the small distance change ensures the continuation of the pareto front we transform the multi objective nonlinear problem into an equivalent nonlinear programming problem by introducing the lagrange function multipliers as a result we obtain that the objective function is strict convex the inequality constraints are continuously differentiable and the equality constraint is an affine function under these settings the kkt optimality necessary and sufficient conditions are elicited naturally a numerical example is solved for providing the basic techniques to compute the pareto optimal solutions by resorting to kkt conditions

selection mechanisms based on the maximin fitness function to solve multi objective optimization problems in this paper we study three selection mechanisms based on the maximin fitness function and we propose another one these selection mechanisms give rise to the following moeas mc moea md moea mh moea and mah moea we validated them using standard test functions taken from the specialized literature having from three up to ten objective functions we compare these four moeas among them and also with respect to moea d which is based on decomposition and to sms emoa which is based on the hypervolume indicator our preliminary results indicate that md moea and mah moea are promising alternatives for solving mops with either low or high dimensionality

a convex hull approach for the reliability based design optimization of nonlinear transient dynamic problems nonlinear problems such as transient dynamic problems exhibit structural responses that can be discontinuous due to numerous bifurcations this hinders gradient based or response surface based optimization this paper proposes a novel approach to split the design space into regions where the response is continuous this makes traditional optimization viable a convex hull approach is adopted to isolate the points corresponding to unwanted bifurcations in the design space the proposed approach is applied to a tube impacting a rigid wall representing a transient dynamic problem since nonlinear behavior is highly sensitive to small variations in design reliability based design optimization is performed the proposed method provides the designer an optimal design with a prescribed dynamic behavior

conical regularization for multi objective optimization problems abstract this paper focuses on multi objective optimization problem with linear infinite dimensional constraints the main novelty of the contribution is that we do not assume that the ordering cone has a nonempty interior and consequently there might not exist kkt system to compute the solution numerically we devise a regularization framework which consists of replacing the ordering cone by a family of solid cones we provide general features of the original problem and the family of regularized problems using tools from the set valued analysis we prove that the weak outer limit of regularized solutions is the set of minimal elements of the original problem we scalarize the original problem as well as the regularized analogs and establish that the corresponding scalarized solutions parametrize the regularized minimal elements we demonstrate the convergence in the kuratowski mosco sense of the regularized solutions to the efficient set of the original problem we prove an a priori holder continuity estimate for the regularized efficient set valued map when the original problem is regular and a verifiable norm boundedness property of the multipliers holds we present detailed numerical examples to show the efficacy of the developed framework

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

a new selection metric for multiobjective hydrologic model calibration a novel selection metric called convex hull contribution chc is introduced for solving multiobjective mo optimization problems with pareto fronts that can be accurately approximated by a convex curve the hydrologic model calibration literature shows that many biobjective calibration problems with a proper setup result in such pareto fronts the chc selection approach identifies a subset of archived nondominated solutions whose map in the objective space forms convex approximation of the pareto front the optimization algorithm can sample solely from these solutions to more accurately approximate the convex shape of the pareto front it is empirically demonstrated that chc improves the performance of pareto archived dynamically dimensioned search pa dds when solving mo problems with convex pareto fronts this conclusion is based on the results of several benchmark mathematical problems and several hydrologic model calibration problems with two or three objective functions the impact of chc on pa dds performance is most evident when the computational budget is somewhat limited it is also demonstrated that 1 000 solution evaluations limited budget in this study is sufficient for pa dds with chc based selection to achieve very high quality calibration results relative to the results achieved after 10 000 solution evaluations

normal constraint method with guarantee of even representation of complete pareto frontier multiobjective optimization is rapidly becoming an invaluable tool in engineering design a particular class of solutions to the multiobjective optimization problem is said to belong to the pareto frontier a pareto solution the set of which comprises the pareto frontier is optimal in the sense that any improvement in one design objective can only occur with the worsening of at least one other accordingly the pareto frontier plays an important role in engineering design it characterizes the tradeoffs between conflicting design objectives some optimization methods can be used to automatically generate a set of pareto solutions from which a final design is subjectively chosen by the designer for this approach to be successful the generated pareto set must be truly representative of the complete optimal design space pareto frontier in other words the set must not overrepresent one region of the design space or neglect others some commonly used methods comply with this requirement whereas others do not this paper offers a new phase in the development of the normal constraint method which is a simple approach for generating pareto solutions that are evenly distributed in the design space of an arbitrary number of objectives the even distribution of the generated pareto solutions can facilitate the process of developing an analytical expression for the pareto frontier in n dimension an even distribution of pareto solutions also facilitates the task of choosing the most desirable final design from among the set of pareto solutions the normal constraint method bears some similarities to the normal boundary intersection and constraint methods importantly the developments presented in this paper define its critical distinction namely the ability to generate a set of evenly distributed pareto solutions over the complete pareto frontier examples are provided that show the normal constraint method to perform favorably under the new developments when compared with the normal boundary intersection method as well as with the original normal constraint method

patching and weak approximation in isometry groups let r be a semilocal principal ideal domain two algebraic objects over r in which scalar extension makes sense e g quadratic spaces are said to be of the same genus if they become isomorphic after extending scalars to all completions of r and its fraction field we prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over not necessarily commutative r orders is always a finite power of 2 and under further assumptions e g that the order is hereditary this number is 1 the same result is also shown for related objects e g systems of sesquilinear forms a key ingredient in the proof is a weak approximation theorem for groups of isometries which is valid over any topological base field and even over semilocal base rings

solving multiobjective mixed integer convex optimization problems multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the varia

optimizations under uncertainty using gradients hessians and surrogate models in this article a first order moment method and a kriging surrogate model are used for optimizations under uncertainty applied to two bar truss designs and two dimensional lift constrained drag minimizations given uncertainties in statistically independent random normally distributed input variables the two approaches are used to propagate these uncertainties through the mathematical model and to approximate output statistics of interest to assess the validity of the approximations the results are compared with full monte carlo simulations when using first order moment methods for robust optimizations first order sensitivity derivatives appear in the objective function and system constraints therefore second order sensitivity derivatives are needed for gradient based optimization approaches when the kriging surrogate model is used to calculate the objective function value and system constraints it will be shown that a gradient predictor for the kriging model can be successfully used for gradi

multiobjective bilevel optimization in this work nonlinear non convex multiobjective bilevel optimization problems are discussed using an optimistic approach it is shown that the set of feasible points of the upper level function the so called induced set can be expressed as the set of minimal solutions of a multiobjective optimization problem this artificial problem is solved by using a scalarization approach by pascoletti and serafini combined with an adaptive parameter control based on sensitivity results for this problem the bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated for the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable an algorithm is presented for the first time and applied to two problems a theoretical example and a problem arising in applications

direct multisearch for multiobjective optimization in practical applications of optimization it is common to have several conflicting objective functions to optimize frequently these functions are subject to noise or can be of black box type preventing the use of derivative based techniques we propose a novel multiobjective derivative free methodology calling it direct multisearch dms which does not aggregate any of the objective functions our framework is inspired by the search poll paradigm of direct search methods of directional type and uses the concept of pareto dominance to maintain a list of nondominated points from which the new iterates or poll centers are chosen the aim of our method is to generate as many points in the pareto front as possible from the polling procedure itself while keeping the whole framework general enough to accommodate other disseminating strategies in particular when using the here also optional search step dms generalizes to multiobjective optimization moo all direct search methods of directional type

variational collocation on finite intervals in this paper we study a set of functions defined on an interval of finite width which are orthogonal and which reduce to the sinc functions when the appropriate limit is taken we show that these functions can be used within a variational approach to obtain accurate results for a variety of problems we have applied them to the interpolation of functions on finite domains and to the solution of the schrodinger equation and we have compared the performance of the present approach with others

constraint propagation using dominance in interval branch bound for nonlinear biobjective optimization abstract constraint propagation has been widely used in nonlinear single objective optimization inside interval branch bound algorithms as an efficient way to discard infeasible and non optimal regions of the search space on the other hand when considering two objective functions constraint propagation is uncommon it has mostly been applied in combinatorial problems inside particular methods the difficulty is in the exploitation of dominance relations in order to discard the so called non pareto optimal solutions inside a decision domain which complicates the design of complete and efficient constraint propagation methods exploiting dominance relations in this paper we present an interval branch bound algorithm which integrates dominance contractors constraint propagation mechanisms that exploit an upper bound set using dominance relations this method discards from the decision space values yielding solutions dominated by some solutions from the upper bound set the effectiveness of the approach is shown on a sample of benchmark problems

concept selection using s pareto frontiers we introduce the notion of s pareto optimality and show how it can be used to improve concept selection in engineering design specie c design alternatives are classie ed as s pareto optimal when there are no other alternatives from the same or any other general design concept that exhibit improvement in all design objectives further we say that the set of s pareto design alternatives comprises the s pareto frontier under the proposed approach the s pareto frontier plays a paramount role in the concept selection process as it is used to dee ne and classify concept dominance thes pareto frontier based concept selection method can becharacterized as onethat capitalizes on the benee ts of computational optimization during the conceptual phase of design before a general design concept has been chosen an introduction to s pareto optimality and a method for generating s pareto frontiersaredeveloped anapproachforusing s paretofrontierstoperformconceptselectionisalsopresented the methods proposed can effectively aid in the elimination of dominated design concepts keep competitive concepts and ultimately choose a specie cdesign alternative from theselected design concept a truss design problem is used to illustrate the usefulness of the method nomenclature g vector of inequality constraints h vector of equality constraints j aggregate objective function mi number of points along ni ni ith vector dee ning the utopia plane n number of design objectives nx number of design variables t k relaxation slack variable for concept k x p generic point on the utopia plane x vector of design variables increment by which feasible space is reduced vector of design objectives or design metrics i ith anchor point k optimum design objective value for concept k si s anchor point for the ith objective

nondomination level coordinate system for parameterizing and exploring pareto frontiers this paper presents a method for defining a barycentric coordinate system on a k objective pareto frontier that is constructed from nondomination levels calculated over subsets of the objectives taken k 1 at a time unlike past approaches these nondomination level coordinates are not limited by the dimensionality convexity or curvature of the frontier and they have an inherent meaning as relative preferences for the competing objectives this intuitive behavior makes the coordinates particularly useful as the basis for parametric models of the frontier and for conducting sensitivity studies and interactive trade space exploration the method is demonstrated on three mathematical example problems exhibiting different geometric properties and on a conceptual wing sizing problem

generalized inverse multiobjective optimization with application to cancer therapy we generalize the standard method of solving inverse optimization problems to allow for the solution of inverse problems that would otherwise be ill posed or infeasible in multiobjective linear optimization given a solution that is not a weakly efficient solution to the forward problem our method generates objective function weights that make the given solution a near weakly efficient solution our generalized inverse optimization model specializes to the standard model when the given solution is weakly efficient and retains the complexity of the underlying forward problem we provide a novel interpretation of our inverse formulation as the dual of the well known benson s method and by doing so develop a new connection between inverse optimization and pareto surface approximation techniques we apply our method to prostate cancer data obtained from princess margaret cancer centre in toronto canada we demonstrate that clinically acceptable treatments can be generated using a small number of objective

a combined scalarizing method for multiobjective programming problems abstract in this paper a new general scalarization technique for solving multiobjective optimization problems is presented after studying the properties of this formulation two problems as special cases of this general formula are considered it is shown that some well known methods such as the weighted sum method the constraint method the benson method the hybrid method and the elastic constraint method can be subsumed under these two problems then considering approximate solutions some relationships between e weakly properly efficient points of a general without any convexity assumption multiobjective optimization problem and optimal solutions of the introduced scalarized problem are achieved

a steepest descent method for vector optimization in this work we propose a cauchy like method for solving smooth unconstrained vector optimization problems when the partial order under consideration is the one induced by the nonnegative orthant we regain the steepest descent method for multicriteria optimization recently proposed by fliege and svaiter we prove that every accumulation point of the generated sequence satisfies a certain first order necessary condition for optimality which extends to the vector case the well known gradient equal zero condition for real valued minimization finally under some reasonable additional hypotheses we prove global convergence to a weak unconstrained minimizer as a by product we show that the problem of finding a weak constrained minimizer can be viewed as a particular case of the so called abstract equilibrium problem

comparison of design concepts in multi criteria decision making using level diagrams in this work we address the evaluation of design concepts and the analysis of multiple pareto fronts in multi criteria decision making using level diagrams such analysis is relevant when two or more design concepts with different design alternatives lie in the same objective space but describe different pareto fronts therefore the problem can be stated as a pareto front comparison between two or more design concepts that only differ in their relative complexity implementation issues or the theory applied to solve the problem at hand such analysis will help the decision maker obtain a better insight of a conceptual solution and be able to decide if the use of a complex concept is justified instead of a simple concept the approach is validated in a set of multi criteria decision making benchmark problems

a method for generating a well distributed pareto set in nonlinear multiobjective optimization a method is presented for generating a well distributed pareto set in nonlinear multiobjective optimization the approach shares conceptual similarity with the physical programming based method the normal boundary intersection and the normal constraint methods in its systematic approach investigating the objective space in order to obtain a well distributed pareto set the proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space it is shown that the proposed modification allows the method to generate an even representation of the entire pareto surface the generation is performed for both convex and nonconvex pareto frontiers a simple algorithm has been proposed to remove local pareto solutions the suggested approach has been verified by several test cases including the generation of both convex and concave pareto frontiers

optimal linear precoding strategies for wideband noncooperative systems based on game theory part i nash equilibria in this two part paper we propose a decentralized strategy based on a game theoretic formulation to find out the optimal precoding multiplexing matrices for a multipoint to multipoint communication system composed of a set of wideband links sharing the same physical resources i e time and bandwidth we assume as optimality criterion the achievement of a nash equilibrium and consider two alternative optimization problems 1 the competitive maximization of mutual information on each link given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies and 2 the competitive maximization of the transmission rate using finite order constellations under the same constraints as above plus a constraint on the average error probability in this first part of the paper we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies then we prove that the optimal precoding multiplexing scheme for both games leads to a channel diagonalizing structure so that both matrix valued problems can be recast in a simpler unified vector power control game with no performance penalty thus we study this simpler game and derive sufficient conditions ensuring the uniqueness of the nash equilibrium interestingly although derived under stronger constraints incorporating for example spectral mask constraints our uniqueness conditions have broader validity than previously known conditions finally we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a pareto optimal centralized scheme to reach the nash equilibria of the game in part ii we propose alternative distributed algorithms along with their convergence conditions

a numerical method for constructing the pareto front of multi objective optimization problems in this paper a new numerical method is presented for constructing an approximation of the pareto front of multi objective optimization problems this method is based on the well known scalarization approach by pascoletti and serafini the proposed method is applied to four test problems that illustrate specific difficulties encountered in multi objective optimization problems such as nonconvex disjoint and local pareto fronts the effectiveness of the proposed method is demonstrated by comparing it with the nsga ii algorithm and the normal constraint method

a mesh adaptive direct search algorithm for multiobjective optimization abstract this work studies multiobjective optimization mop of nonsmooth functions subject to general constraints we first present definitions and optimality conditions as well as some single objective formulations of mop parameterized with respect to some reference point in the space of objective functions next we propose a new algorithm called m ulti m ads multiobjective mesh adaptive direct search for mop m ulti m ads generates an approximation of the pareto front by solving a series of single objective formulations of mop generated using the nbi natural boundary intersection framework these single objective problems are solved using the m ads mesh adaptive direct search algorithm for constrained nonsmooth optimization the pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the clarke calculus m ulti m ads is then tested on problems from the literature with different pareto front landscapes and on a styrene production process simulation problem from chemical engineering

diversity maximization approach for multiobjective optimization one of the most common approaches for multiobjective optimization is to generate the whole or partial efficient frontier and then decide about the preferred solution in a higher level decision making process in this paper a new method for generating the efficient frontier for multiobjective problems is developed called the diversity maximization approach dma this approach is capable of solving mixed integer and combinatorial problems the dma finds pareto optimal solutions by maximizing a proposed diversity measure and guarantees generating the complete set of efficient points given a subset of the efficient frontier dma finds the next pareto optimal solution which combined with the existing ones yields the most diversified subset of efficient points this solution is defined as the most diverse solution in fact it aims to maximize the distance between the new efficient point and the closest point in the given subset of the efficient frontier the proposed approach can be applied to any problem that can be solved for the single objective case we can use the dma by solving directly a modified version of the mixed integer programming mip formulation of the multiobjective problem in this case the pareto optimal solutions are found sequentially in an iterative way consequently as we terminate the procedure before completion a partial efficient frontier is available the diversity measure assures that in every stage of the procedure the partial efficient frontier is well diversified this partial efficient frontier can be perceived as a filtered set of the complete efficient frontier and can be used by the decision maker in case the complete efficient frontier contains too many points an additional way of using dma is by incorporating it in a problem oriented branch and bound algorithm detailed examples of both approaches are given

ability of objective functions to generate points on nonconvex pareto frontiers new ground is broken in our understanding of objective functions ability to capture pareto solutions for multi objective design optimization problems it is explained why widely used objective functions fail to capture pareto solutions when the pareto frontier is not convex in objective space and the means to avoid this limitation when possible is provided these conditions are developed and presented in the general context ofn dimensional objective space and numerical examples are provided an important point is that most objective function structures can be made to generate nonconvex pareto frontier solutions if the curvature of the objective function can be varied by setting one or more parameters because the occurrence of nonconvex efficient frontiers is common in practice the results are of direct practical usefulness

on finding multiple pareto optimal solutions using classical and evolutionary generating methods in solving multi objective optimization problems evolutionary algorithms have been adequately applied to demonstrate that multiple and well spread pareto optimal solutions can be found in a single simulation run in this paper we discuss and put together various different classical generating methods which are either quite well known or are in oblivion due to publication in less accessible journals and some of which were even suggested before the inception of evolutionary methodologies these generating methods specialize either in finding multiple pareto optimal solutions in a single simulation run or specialize in maintaining a good diversity by systematically solving a number of scalarizing problems most classical generating methodologies are classified into four groups mainly based on their working principles and one representative method from each group is chosen in the present study for a detailed discussion and for its performance comparison with a state of the art evolutionary method on visual comparisons of the efficient frontiers obtained for a number of two and three objective test problems the results bring out interesting insights about the strengths and weaknesses of these approaches the results should motivate researchers to design hybrid multi objective optimization algorithms which may be better than each of the individual methods

a new scalarization technique and new algorithms to generate pareto fronts we propose a new scalarization technique for nonconvex multiobjective optimization problems and establish its theoretical properties by combining our new scalarization approach with existing grid

pareto adaptive penalty based boundary intersection method for multi objective optimization abstract penalty based boundary intersection pbi method is a frequently used scalarizing method in decomposition based multi objective evolutionary algorithms moeas it works well when a proper penalty value is provided however the determination of a suitable penalty value depends on the problem itself more precisely the pareto optimal front pf shape as the penalty value increases the pbi method becomes less effective in terms of convergence but is more capable of handling various pf shapes in this study a simple yet effective method called pareto adaptive pbi pap is proposed by which a suitable penalty value can be adaptively identified which therefore can maintain fast convergence speed meanwhile leading to a good approximation of the pf the pap strategy integrated into the state of the art decomposition algorithm moea d denoted as moea d pap is examined on a set of multi objective benchmarks with different pf shapes experimental results show that the pap strategy is more effective than the weighted sum the weighted tchebycheff and the pbi method with representative fixed penalty values in general in addition the moea d pap is examined on a real world problem multi objective optimization of a hybrid renewable energy system whose pf is unknown the outcome of the experiment further confirms its feasibility and superiority

anharmonic oscillator and the optimized basis expansion we introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one dimensional anharmonic oscillators we present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions we show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy also using the harmonic oscillator basis functions we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity

a variational sinc collocation method for strong coupling problems we have devised a variational sinc collocation method vscm which can be used to obtain accurate numerical solutions to many strong coupling problems sinc functions with an optimal grid spacing are used to solve the linear and nonlinear schrodinger equations and a lattice 4 model in 1 1 our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision

a multiobjective evolutionary algorithm based on decomposition with normal boundary intersection for traffic grooming in optical networks abstract this paper presents a multiobjective optimization algorithm for solving tri objective optimization problems with objectives of very different scales as a version of the multiobjective evolutionary algorithm based on decomposition moea d it adopts the normal boundary intersection nbi based tchebycheff approach to decompose a multiobjective optimization problem into a number of single objective subproblems particular attention has been paid to set the weight vectors of these subproblems for handling disparately scaled objectives we have applied our proposed algorithm to a traffic grooming problem in the telecommunication fields we also propose to use an indirect encoding approach to handle this very complicated problem our experimental studies show that our proposed method is able to produce very promising results for this real world telecommunication problem

nonsmooth multiobjective programming with quasi newton methods this paper proposes a new algorithm to solve nonsmooth multiobjective programming the algorithm is a descent direction method to obtain the critical point a necessary condition for pareto optimality we analyze both global and local convergence results under some assumptions numerical tests are also given

many objective particle swarm optimization many objective problems refer to the optimization problems containing more than three conflicting objectives to obtain a representative set of well distributed non dominated solutions close to pareto front in the objective space remains a challenging problem many papers have proposed different multi objective evolutionary algorithms to solve the lack of the convergence and diversity in many objective problems one of the more promising approaches uses a set of reference points to discriminate the solutions and guide the search process however this approach was incorporated mainly in multi objective evolutionary algorithms and there are just some few promising adaptations of particle swarm optimization approaches for effectively tackling many objective problems regarding convergence and diversity thus this paper proposes a practical and efficient many objective particle swarm optimization algorithm for solving many objective problems our proposal uses a set of reference points dynamically determined according to the search process allowing the algorithm to converge to the pareto front but maintaining the diversity of the pareto front our experimental results demonstrate superior or similar performance when compared to other state of art algorithms

bilevel programming for generating discrete representations in multiobjective optimization the solution to a multiobjective optimization problem consists of the nondominated set that portrays all relevant trade off information the ultimate goal is to identify a decision maker s most preferred solution without generating the entire set of nondominated solutions we propose a bilevel programming formulation that can be used to this end the bilevel program is capable of delivering an efficient solution that maps into a given set provided that one exits if the decision maker s preferences are known a priori they can be used to specify the given set alternatively we propose a method to obtain a representation of the nondominated set when the decision maker s preferences are not available this requires a thorough search of the outcome space the search can be facilitated by a partitioning scheme similar to the ones used in global optimization since the bilevel programming formulation either finds a nondominated solution in a given partition element or determines that there is none a representation with a specified coverage error level can be found in a finite number of iterations while building a discrete representation the algorithm also generates an approximation of the nondominated set within the specified error factor we illustrate the algorithm on the multiobjective linear programming problem

norm based approximation in multicriteria programming abstract based on new theoretical results on norms heuristic algorithms to approximate the nondominated set of multicriteria programs are proposed by automatically adapting to the problem s structure and scaling the approximation is constructed objectively without interaction with the decision maker as the algorithms extend the results obtained for bicriteria programs difficulties encountered when dealing with more than two criteria are discussed

multiobjective optimization through a series of single objective formulations this work deals with bound constrained multiobjective optimization mop of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited or are absent typical situations arise when the functions are computed as the result of a computer simulation we first present definitions and optimality conditions as well as two families of single objective formulations of mop next we propose a new algorithm called for the biobjective optimization bop problem i e mop with two objective functions the property that pareto points may be ordered in bop and not in mop is exploited by our algorithm generates an approximation of the pareto front by solving a series of single objective formulations of bop these single objective problems are solved using the recent mesh adaptive direct search algorithm for nonsmooth optimization the pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the clarke calculus finally is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization such as a nonconvex or disjoint pareto front local pareto fronts or a nonuniform pareto front

mathematical and pragmatic perspectives of physical programming physical programming pp is an emerging multiobjective and design optimization method that has been applied successfully in diverse areas of engineering and operations research the application of pp calls for the designer to express preferences by defining ranges of differing degrees of desirability for each design metric although this approach works well in practice it has never been shown that the resulting optimal solution is not unduly sensitive to these numerical range definitions pp is shown to be numerically well conditioned and its sensitivity to designer input with respect to optimal solution is compared with that of other popular methods the important proof is provided that all solutions obtained through pp are pareto optimal and the notion of pareto optimality is extended to one of pragmatic implication the important notion of p dominance that extends the concept of pareto optimality beyond the cases minimize and maximize is introduced p dominance is shown to lead to the important concept of generalized pareto optimality numerical results are provided that illustrate the favorable numerical properties of physical programming

scalarizing functions for generating the weakly efficient solution set in convex multiobjective problems a method that uses normal directions of convex sets is proposed to construct a sequence of scalarizing functions which generates all weakly efficient solutions of a convex multiobjective optimization problem the convergence of the method is proven and some small scale numerical examples are given

representing sampled pareto frontiers as parameterized continuous manifolds using self organizing maps this paper describes a method for defining a coordinate system to parameterize a sampled pareto frontier of a continuous multi attribute design problem using a modified self organizing map by defining such a coordinate system the design problem may be reformulated from y f x to y x g where x is a vector of design variables y is a vector of attributes and is a vector of barycentric coordinates exploration of the design problem using as the independent variables has the following desirable properties 1 every vector corresponds to a pareto efficient design and every pareto efficient design has a corresponding 2 the number of coordinates is equal to the number of attributes regardless of the number of design variables 3 each attribute yi has a corresponding coordinate i such that the sign of yi i is positive if the objective is to maximize yi and negative if the objective is to minimize yi i e the attributes improve monotonically as their corresponding coordinates increase

response surface techniques for diffuser shape optimization the design of incompressible diffusers for maximum pressure recovery is used to demonstrate the utility of response surface approximations for design optimization of eow devices two examples involving two and eve design variables are treated with the diffuser wall shapes described by polynomials and b splines in both cases monotonicity conditions drastically reduce the design space in this irregularly shaped space a pool of designs is selected by a d optimality criterion and analyzed by a e nite volume computational euid dynamics cfd code quadratic polynomial response surfaces are then e tted to the pressure recovery coefe cients to improve the prediction accuracy uncertain regressor terms and possible outlier design points are excluded based on statistical tests a standard optimization algorithm is used to e nd the optimal diffuser design from the response surface approximations the optimum diffusers exhibit minimal e ow separation and yield similar wall shapes for the two parameterizations a main asset of the response surface optimization approach lies in the smoothing of noisy response functions therefore theissueofnumericalnoiseincfd results basedon theuseof two different analysis codes is addressed

newton s method for multiobjective optimization we propose an extension of newton s method for unconstrained multiobjective optimization multicriteria optimization this method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions newton s direction at each iterate is obtained by minimizing the max ordering scalarization of the variations on the quadratic approximations of the objective functions the objective functions are assumed to be twice continuously differentiable and locally strongly convex under these hypotheses the method as in the classical case is locally superlinear convergent to optimal points again as in the scalar case if the second derivatives are lipschitz continuous the rate of convergence is quadratic our convergence analysis uses a kantorovich like technique as a byproduct existence of optima is obtained under semilocal assumptions