robust beamforming via worst case sinr maximization minimum variance beamforming which uses a weight vector that maximizes the signal to interference plus noise ratio sinr is often sensitive to estimation error and uncertainty in the parameters steering vector and covariance matrix robust beamforming attempts to systematically alleviate this sensitivity by explicitly incorporating a data uncertainty model in the optimization problem in this paper we consider robust beamforming via worst case sinr maximization that is the problem of finding a weight vector that maximizes the worst case sinr over the uncertainty model we show that with a general convex uncertainty model the worst case sinr maximization problem can be solved by using convex optimization in particular when the uncertainty model can be represented by linear matrix inequalities the worst case sinr maximization problem can be solved via semidefinite programming the convex formulation result allows us to handle more general uncertainty models than prior work using a special form of uncertainty model we illustrate the method with a numerical example

a probabilistic analytic center cutting plane method for feasibility of uncertain lmis many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution x that satisfies a set of inequalities of the form f f x d 0 d d this set typically contains an infinite and uncountable number of inequalities and it has been proved that the related robust feasibility problem is numerically hard to solve in general in this paper we discuss a family of cutting plane methods that solve efficiently a probabilistically relaxed version of the problem specifically under suitable hypotheses we show that an analytic center cutting plane scheme based on a probabilistic oracle returns in a finite and pre specified number of iterations a solution x which is feasible for most of the members of f except possibly for a subset having arbitrarily small probability measure

equilibrium under uncertainty with sugeno payoff abstract this paper studies n player games where players beliefs about their opponents behavior are capacities fuzzy measures non additive probabilities the concept of an equilibrium under uncertainty was introduced by j dow and s werlang 5 for two players and was extended to n player games by j eichberger and d kelsey 8 expected utility was expressed by choquet integral we consider the concept of an equilibrium under uncertainty in this paper but with expected utility expressed by sugeno integral existence of such an equilibrium expressed by possibility or necessity capacities is demonstrated using some abstract non linear convexity on the space of capacities

floquet theory for integral and integro differential equations we study the extension of hill s method of infinite determinants to the case of integro differential equations with periodic coefficients and kernels we develop the analytical theory of such methods and we obtain certain qualitative properties of the equations that determine the boundaries between regions of dynamic stability and dynamic instability

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

numerical inverse scattering for the focusing and defocusing nonlinear schrodinger equations we solve the focusing and defocusing nonlinear schrodinger nls equations numerically by implementing the inverse scattering transform the computation of the scattering data and of the nls solution are both spectrally con vergent initial conditions in a suitable space are treated using the approach of biondini and bui 3 we numerically solve homogeneous robin boundary value problems on the half line finally using recent theoretical developments in the numerical approximation of riemann hilbert problems we prove that under mild assumptions our method of approximating solutions to the nls equations is uniformly accurate in their domain of definition

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

convergence of hill s method for nonselfadjoint operators by the introduction of a generalized evans function defined by an appropriate 2 modified fredholm determinant we give a simple proof of convergence in location and multiplicity of hill s method for numerical approximation of spectra of periodic coefficient ordinary differential operators our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite automatically satisfied in the scalar case notably this includes a large class of non self adjoint operators which previously had not been treated in a simple way the case of general coefficients depends on an interesting operator theoretic question regarding properties of toeplitz matrices

transfer method for characterizing the existence of maximal elements of binary relations on compact or noncompact sets this paper systematically studies the existence of maximal elements for unordered binary relations on compact or noncompact sets in a general topological space this is done by developing a method called transfer method to derive various necessary and sufficient conditions that characterize the existence of maximal elements for a binary relation in terms of 1 generalized transitivity conditions under certain topological assumptions 2 topological conditions under certain generalized transitivity assumptions and 3 generalized convexity conditions under certain topological assumptions there are two basic approaches in the literature to prove the existence by providing sufficient conditions one assumes certain convexity and continuity conditions for a topological vector space and the other assumes certain weakened transitivity and continuity conditions for a general topological space the results unify those two aproaches and generalize almost all of the existing results in the literature

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

optimization with reference based robust preference constraints we propose an optimization model using a novel robust preference relationship reference based almost stochastic dominance rsd the concept of rsd addresses the two problems in utility based dec

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

robust solutions to a general class of approximation problems the solution of robust counterparts of optimization problems with uncertain data is currently attracting much interest in particular this has been considered in the context of approximation using total least squares here we consider the analogue of this in the errors in variables context where attention is focused on the particular errors which arise in the individual independent variable values in data fitting problems in addition to consideration of problems which are linear in the free parameters some suggestions are made for the treatment of nonlinear problems the emphasis throughout is on providing methods which are computationally tractable and the results parallel and extend earlier results on uncertain linear approximation problems

computing spectra of linear operators using the floquet fourier hill method in order to establish the stability of an equilibrium solution u of an infinite dimensional dynamical system u x u one is interested in the spectrum of the linear operator lu obtained by linearizing the dynamical system around u we use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator lu the method is particularly well suited to the case of periodic u although no periodic boundary conditions on the perturbations are imposed by incorporating the fundamentals of floquet theory an almost uniform approximation to the entire spectrum of the maximal extension is obtained as opposed to an approximation of a few selected elements the numerical component of the method is limited to i choosing the size of the matrices to be used and ii an eigenvalue solver such as the qr algorithm compared to often used finite difference approaches the method is an order of magnitude faster for comparable accuracy we illustrate that the method is efficiently extended to problems defined on the whole line

elliptic solutions of the defocusing nls equation are stable the stability of the stationary periodic solutions of the integrable one dimensional cubic defocusing nonlinear schrodinger nls equation is reasonably well understood especially for solutions of small amplitude in this paper we exploit the integrability of the nls equation to establish the spectral stability of all such stationary solutions this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem an additional argument using an appropriate krein signature allows us to conclude the nonlinear orbital stability of all stationary solutions of the defocusing nls equation with respect to so called subharmonic perturbations perturbations that have period equal to an integer multiple of the period of the amplitude of the solution all results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

multivariate nonnegative quadratic mappings in this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain with nonnegativity defined by a prespecified conic order in particular we consider the set cone of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone and study when it can be represented by linear matrix inequalities we also discuss the applications of the results in robust optimization especially the robust quadratic matrix inequalities and the robust linear programming models in the latter application the implementational errors of the solution are taken into account and the problem is formulated as a semidefinite program

linear semi infinite programming theory an updated survey abstract this paper presents an state of the art survey on linear semi infinite programming theory and its extensions in particular convex semi infinite programming this review updates a previous survey semi infinite programming non convex optim appl 25 1998 of the same authors on the same topic which was published in 1998

approximations of strongly continuous families of unbounded self adjoint operators the problem of approximating the discrete spectra of families of self adjoint operators that are merely strongly continuous is addressed it is well known that the spectrum need not vary continuously as a set under strong perturbations however it is shown that under an additional compactness assumption the spectrum does vary continuously and a family of symmetric finite dimensional approximations is constructed an important feature of these approximations is that they are valid for the entire family uniformly an application of this result to the study of plasma instabilities is illustrated

on the convergence of hill s method hill s method is a means to numerically approximate spectra of linear differential operators with periodic coefficients in this paper we address different issues related to the convergence of hill s method we show the method does not produce any spurious approximations and that for selfadjoint operators the method converges in a restricted sense furthermore assuming convergence of an eigenvalue we prove convergence of the associated eigenfunction approximation in the l 2 norm these results are not restricted to selfadjoint operators finally for certain selfadjoint operators we prove that the rate of convergence of hill s method to the least eigenvalue is faster than any polynomial power

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

evans functions jost functions and fredholm determinants the principal results of this paper consist of an intrinsic definition of the evans function in terms of newly introduced generalized matrix valued jost solutions for general first order matrix valued differential equations on the real line and a proof of the fact that the evans function a finite dimensional determinant by construction coincides with a modified fredholm determinant associated with a birman schwinger type integral operator up to an explicitly computable nonvanishing factor

compromise solutions for robust combinatorial optimization with variable sized uncertainty in classic robust optimization it is assumed that a set of possible parameter realizations the uncertainty set is modeled in a previous step and part of the input as recent work has shown finding the most suitable uncertainty set is in itself already a difficult task we consider robust problems where the uncertainty set is not completely defined only the shape is known but not its size such a setting is known as variable sized uncertainty in this work we present an approach how to find a single robust solution that performs well on average over all possible uncertainty set sizes we demonstrate that this approach can be solved efficiently for min max robust optimization but is more involved in the case of min max regret where positive and negative complexity results for the selection problem the minimum spanning tree problem and the shortest path problem are provided we introduce an iterative solution procedure and evaluate its performance in an experimental comparison

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

nash points ky fan inequality and equilibria of abstract economies in max plus and b convexity abstract b convexity was introduced in w briec c horvath b convexity optimization 53 2004 103 127 separation and hahn banach like theorems can be found in g adilov a m rubinov b convex sets and functions numer funct anal optim 27 2006 237 257 and w briec c d horvath a rubinov separation in b convexity pacific j optim 1 2005 13 30 we show here that all the basic results related to fixed point theorems are available in b convexity ky fan inequality existence of nash equilibria and existence of equilibria for abstract economies are established in the framework of b convexity monotone analysis or analysis on maslov semimodules v n kolokoltsov v p maslov idempotent analysis and its applications math appl vol 401 kluwer academic 1997 v p litvinov v p maslov g b shpitz idempotent functional analysis an algebraic approach math notes 69 2001 696 729 v p maslov s n samborski eds idempotent analysis advances in soviet mathematics amer math soc providence ri 1992 is the natural framework for these results from this point of view max plus convexity and b convexity are isomorphic maslov semimodules structures over isomorphic semirings therefore all the results of this paper hold in the context of max plus convexity

randomized methods for design of uncertain systems in this paper we study randomized methods for feedback design of uncertain systems the first contribution is to derive the sample complexity of various constrained control problems in particular we show the key role played by the binomial distribution and related tail inequalities and compute the sample complexity this contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm these results are then applied to the analysis of worst case performance and design with robust optimization the second contribution of the paper is to introduce a general class of sequential algorithms denoted as sequential probabilistic validation spv in these sequential algorithms at each iteration a candidate solution is probabilistically validated and corrected if necessary to meet the required specifications the results we derive provide the sample complexity which guarantees that the solutions obtained with spv algorithms meet some pre specified probabilistic accuracy and confidence the performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification

on the spectral stability of solitary wave solutions of the vector nonlinear schrodinger equation we consider a system of coupled cubic nonlinear schrodinger nls equations where the interaction coefficients jk are real the spectral stability of solitary wave solutions both bright and dark is examined both analytically and numerically our results build on preceding work by nguyen et al and others specifically we present closed form solitary wave solutions with trivial and nontrivial phase profiles their spectral stability is examined analytically by determining the locus of their essential spectrum their full stability spectrum is computed numerically using a large period limit of hill s method we find that all nontrivial phase solutions are unstable while some trivial phase solutions are spectrally stable to our knowledge this paper presents the first investigation of the stability of the solitary waves of the coupled cubic nls equation without the restriction that all components j are proportional to sech

extended matrix cube theorems with applications to µ theory in control in this paper we study semi infinite systems of linear matrix inequalities which are generically np hard for these systems we introduce computationally tractable approximations and derive quantitative guarantees of their quality as applications we discuss the problem of maximizing a hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value with the help of our complex matrix cube theorem we demonstrate that the standard scaling upper bound on m is a tight upper bound on the largest level of structured perturbations of the matrixm for which all perturbed matrices share a common lyapunov certificate for the discrete time stability

investment models based on clustered scenario trees abstract stochastic programming is widely applied in financial decision problems in particular when we need to carry out the actual calculations for portfolio selection problems we have to assign a value for each expected return and the associated conditional probability in advance these estimated random parameters often rely on a scenario tree representing the distribution of the underlying asset returns one of the drawbacks is that the estimated parameters may be deviated from the actual ones therefore robustness is considered so as to cope with the issue of parameter inaccuracy in view of this we propose a clustered scenario tree approach which accommodates the parameter inaccuracy problem in the context of a scenario tree

parameter estimation with multiple sources and levels of uncertainties least squares designs are sensitive to errors in the data which can be due to several factors including the approximation of complex models by simpler ones the presence of unavoidable experimental errors when collecting data or even due to unknown or unmodeled effects we formulate a new design criterion that treats multiple sources of uncertainties in the data with possibly varied degrees of intensity we show that the solution has a regularized form with one regularization parameter for each source of uncertainty the parameters turn out to be model dependent and can be determined optimally as the nonnegative roots of certain coupled equations applications in array signal processing and image processing are considered

kkm and nash equilibria type theorems in topological ordered spaces abstract in this paper we obatin a generalized kkm theorem a generalized fan browder fixed theorem and an existence theorem of nash equilibria in topological ordered spaces