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

multiple collinear cracks in a piezoelectric material abstract in this paper an exact solution of multiple collinear cracks in piezoelectric material is obtained the permittivity of air environment is considered two cases have been studied in the first case the permittivity of air is far less than that of piezoelectric material therefore the electric induction in the air is negligible in the second case the permittivity of air is comparable with that of piezoelectric material the problem is deduced into riemann hilbert problem and solved by the way the electric boundary conditions are discussed this result demonstrates that the consideration of air in crack gap reduces the stress intensity factor

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

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

complex power distribution analysis in plates covered with passive constrained layer damping patches the vibration of a plate partially covered with a passive constrained layer damping pcld patch is studied from an energetic point of view the damped plate is excited by an acoustic plane wave the study is done with a numerical two dimensional multilayer plate model results of the present model are compared to those obtained with three dimensional finite element models it is shown that the present model gives accurate results even for the layer s inner behavior it is less expansive in terms of computational cost hence it can simulate efficiently the structure for higher frequencies mathematical formulas for complex mechanical power are presented and the link with strain and kinetic energies and dissipated power is detailed both local and global complex power balance are established and corresponding expressions for the discretized problem are formulated conservative and dissipative powers are studied for the pcld damped plate after a global balance analysis versus frequency a local study has been carried out in order to quantify the relative contribution of the components of strain and stress tensors to the total strain energy and dissipated power the individual layer s contributions is also investigated the in plane distributions of powers are mapped showing the location where dissipative phenomenon occurs and where strain energy is stored finally three criteria based on the previous power quantities are proposed in order to quantify the mechanical damping efficiency of the patch

basic solution to four three dimensional rectangular limited permeable cracks in transversely isotropic magneto electro elastic material the solution to four three dimensional rectangular cracks in magneto electro elastic material is proposed by using the generalized almansi s theorem and the schmidt method under limited permeable boundary conditions the problem is formulated through fourier transform as three pairs of dual integral equations in which the unknown variables are the displacement jumps across the crack surfaces for solving the dual integral equations the displacement jumps across the crack surfaces are directly expanded as a series of jacobi polynomials finally the effects of the electric permittivity the magnetic permeability of the air inside the crack the geometric shape of the rectangular crack and the distance among four rectangular cracks on the stress intensity factors the electric displacement intensity factors and the magnetic flux intensity factors in magneto electro elastic material are presented and the magneto electro elastic coupling effects are also discussed

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

mindlin plate theory best shear coefficient and higher spectra validity mindlin plate theory predicts three frequency spectra or equivalently three branches to a phase velocity dispersion diagram the lowest of which thew1mode provides rotatory inertia and shear deformation corrections to classical thin plate theory employing consistent truncation procedures to both the mindlin and the exact rayleigh lamb frequency equations valid for long wavelength and low phase velocity one finds thatw1mode agreement is achieved when the shear coefficient takes the valuek 5 6 v the mindlin prediction is then less than 0 5 in error when the wavelength is equal to the plate thickness and less than 1 in error as wavelength approaches zero the previously dismissed mindlinhmode is seen to be in exact frequency or phase velocity agreement with the second slowestshwave in the infinite plate as long as the shear coefficient for this mode takes the valuek 2 12 however thew2mode as with the second frequency spectrum of timoshenko beams should be regarded as the inevitable but meaningless consequence of an otherwise remarkable approximate engineering theory

analysis method of planar cracks of arbitrary shape in the isotropic plane of a three dimensional transversely isotropic magnetoelectroelastic medium abstract the hyper singular boundary integral equation method of crack analysis in three dimensional transversely isotropic magnetoelectroelastic media is proposed based on the fundamental solutions or green s functions of three dimensional transversely isotropic magnetoelectroelastic media and the corresponding somigliana identity the boundary integral equations for a planar crack of arbitrary shape in the plane of isotropy are obtained in terms of the extended displacement discontinuities across crack faces the extended displacement discontinuities include the displacement discontinuities the electric potential discontinuity and the magnetic potential discontinuity and correspondingly the extended tractions on crack face represent the conventional tractions the electric displacement and the magnetic induction boundary values the near crack tip fields and the intensity factors in terms of the extended displacement discontinuities are derived by boundary integral equation approach a solution method is proposed by use of the analogy between the boundary integral equations of the magnetoelectroelastic media and the purely elastic materials the influence of different electric and magnetic boundary conditions i e electrically and magnetically impermeable and permeable conditions electrically impermeable and magnetically permeable condition and electrically permeable and magnetically impermeable condition on the solutions is studied the crack opening model is proposed to consider the real crack opening and the electric and magnetic fields in the crack cavity under combined mechanical electric magnetic loadings an iteration approach is presented for the solution of the non linear model the exact solution is obtained for the case of uniformly applied loadings on the crack faces numerical results for a square crack under different electric and magnetic boundary conditions are displayed to demonstrate the proposed method

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

frequency maximization of laminated sandwich plates under general boundary conditions using layerwise optimization method with refined zigzag theory abstract the present paper extends the layerwise optimization lo procedure to the maximization problem of the fundamental frequencies of sandwich plates with fibrous composites and low stiffness core layers frequencies are calculated by the ritz method based on a refined zigzag theory rzt polynomial functions which satisfy at least geometrical boundary conditions with boundary indexes are employed as displacement functions and they enable satisfying arbitrary sets of boundary conditions for rectangular plates results of the experimental modal analysis validate the accuracy of the present calculations and a comparison with results of the classical laminated theory clpt and the first order share deformation theory fsdt supports the effectiveness of the present method optimized results are compared with other typical sets of lay up configurations and this shows the lo method as suitable means to the optimization problem for sandwich plates

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

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

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

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

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

optimal placement of constrained layer damping for reduction of interior noise in this paper an efficient constrained layer damping layout optimization method in structural acoustic systems is proposed to simulate vibro acoustic systems a hybrid model that uses finite elements for structures and boundary elements for the cavity is developed with a finite element formulation the intrinsic nonlinearities of viscoelastic damping materials with respect to frequency and temperature are included using a fractional derivative model the resulting complex eigenvalue problem is solved using an iterative scheme the optimal layout of the constrained viscoelastic layer damping on the structure is identified using a gradient based numerical search algorithm the sensitivity formulas are derived analytically for acoustic and structural responses the optimal damping layer layouts that minimize the added mass with minimum sound pressure level requirements are obtained for different temperatures

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 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

general solutions for coupled equations for piezoelectric media abstract three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium these solutions are expressed in terms of the two functions and f where satisfies a second degree partial differential equation and f a sixth degree partial differential equation respectively if the terms concerning the derivatives of time are removed the results become three general solutions for the corresponding equilibrium equations in which the function f can be represented by functions fi i 1 2 3 each of which satisfies a second degree partial differential equation by utilizing a generalized almansi theorem and the solution wang and zheng int j solids structures32 105 115 1995 obtained is proved to be consistent with one case of one of the three general solutions when the constants e15 e31 e33 0 the piezo electric coupling is absent then two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium one of which is the result hu acta scientia sin 2 2 145 151 1953 obtained the other one has not been published last the solution in the limiting explicit form for the problem for a half space with concentrated loads at the boundary is obtained by utilizing the general solutions

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

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

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

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

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

3d mode i crack analysis of piezoelectrics the finite part integral concept is first used to prove rigidly hypersingular integral equations for a mode i crack embedded an infinite transversely isotropic piezoelectric solid with three dimensional linear piezoelectricity theory investigations on the singular electroelastic fields in the vicinity of the crack are thereafter made by the dominant part analysis finally numerical solutions of several typical planar cracks with hypersingular integral equation method are performed to give the electroelastic intensity factor k fields and the energy release rate g accuracies of results are found to be very high