stability radii of positive linear difference equations under affine parameter perturbations in this paper we study stability radii of positive linear difference equations under arbitrary affine parameter perturbations it is shown that real and complex stability radii of positive equations coincide for block diagonal disturbances moreover for these stability radii estimates and computable formulae are derived which generalize to positive linear difference equations known results obtained earlier for positive linear discrete time systems of the form x k 1 ax k email protected n some illustrative examples are given

a primer of perron frobenius theory for matrix polynomials abstract we present an extension of perron frobenius theory to the spectra and numerical ranges of perron polynomials namely matrix polynomials of the form l i m a m 1 m 1 a 1 a 0 where the coefficient matrices are entrywise nonnegative our approach relies on the companion matrix linearization first we recount the generalization of the perron frobenius theorem to perron polynomials and report some of its consequences subsequently we examine the role of l in multistep difference equations and provide a multistep version of the fundamental theorem of demography finally we extend issos results on the numerical range of nonnegative matrices to perron polynomials

robust portfolio selection involving options under a marginal joint ellipsoidal uncertainty set in typical robust portfolio selection problems one mainly finds portfolios with the worst case return under a given uncertainty set in which asset returns can be realized a too large uncertainty set will lead to a too conservative robust portfolio however if the given uncertainty set is not large enough the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs the goal of this paper is to propose robust portfolio selection models under so called marginal joint ellipsoidal uncertainty set and to test the performance of the proposed models a robust portfolio selection model under a marginal joint ellipsoidal uncertainty set is proposed at first the model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set and relaxes the requirements on the uncertainty set then one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model convex programming approximations with second order cone and linear matrix inequalities constraints to both models are derived the proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs tests on real data of the chinese stock market and simulated options confirm the property of both the models test results show that 1 under the marginal joint uncertainty set the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model without options are better and greater than those generated from goldfarb and iyengar s model and 2 the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs

index tracking and enhanced indexing using mixed conditional value at risk abstract index tracking it and enhanced indexing ei are two forms of investment strategies which revolve around the movements of the benchmark index while it aims to match the performance of the benchmark index ei intents to outperform the same in this paper we seek to design portfolios for it and ei problems using mixed conditional value at risk mcvar we propose to use the two tail mcvar tmcvar measure to track the index optimizing tmcvar is a linear program which minimizes the upper deviation and the downside deviation from the benchmark index and hence meets the objective of it on the other hand we propose a two step procedure for ei problem in step one we design a discrete markov chain model to filter a few stocks on the basis of their high probability of gain over the benchmark index in step two we assign optimal weights to the filtered assets through maximizing any of the two variants of the starr ratio with mcvar or the starr ratio with deviation mcvar dmcvar maximizing the starr ratio either with mcvar or dmcvar is a linear program and hence tractable we analyze the empirical performance of the proposed models over 17 world wide indices using the rolling window approach we consider two it and four ei models from the literature for a comparative analysis it is exhibited that the proposed it model outperforms the other two it models over several performance measures including higher correlation value with the benchmark index and lower tracking error and the two proposed ei models outperform the other four ei models in achieving higher excess mean returns from the benchmark index

portfolio value at risk optimization for asymmetrically distributed asset returns we present a new approach to portfolio optimization by separating asset return distributions into positive and negative half spaces the approach minimizes a so called partitioned value at risk pvar measure by using the statistical information from the two half spaces respectively we show that the proposed pvar approach is a signiflcant improvement in several important aspects when compared to markowitz mean variance optimization approach first our approach which accommodates ambiguous asymmetric return distributions and captures portfolio risk in higher moments does not require asset distributions being elliptically symmetric or multivariate normal second using simulated and real data our approach generates better risk return tradeofis in the optimal portfolios the difierence between the two approaches increases in the degree of asymmetry in the underlying asset distributions third when given the support of asset returns our pvar measure becomes a coherent risk measure proposed by artzner et al 1999 whereas conventional risk measures such as variance and var fail to do so moreover our pvar measure is an asymmetric risk measure which is difierent from symmetric risk measures like variance and worst case mean covariance var wvar therefore our proposed pvar is a signiflcant addition to the existing portfolio risk measures we believe that the pvar approach can be very useful for better portfolio allocations than the mean variance or other symmetric risk metrics approach during market downturns when asset return distributions are often fat tailed or skewed

tight bounds for some risk measures with applications to robust portfolio selection in this paper we develop tight bounds on the expected values of several risk measures that are of interest to us this work is motivated by the robust optimization models arising from portfolio selection problems indeed the whole paper is centered around robust portfolio models and solutions the basic setting is to find a portfolio that maximizes respectively minimizes the expected utility respectively disutility values in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations first we show that the single stage portfolio selection problem within this framework whenever the disutility function is in the form of lower partial moments lpm or conditional value at risk cvar or value at risk var can be solved analytically the results lead to the solutions for single stage robust portfolio selection models furthermore the results also lead to a multistage adjustable robust optimization aro solution when the disutility function is the second order lpm exploring beyond the confines of convex optimization we also consider the so called s shaped value function which plays a key role in the prospect theory of kahneman and tversky the nonrobust version of the problem is shown to be np hard in general however we present an efficient procedure for solving the robust counterpart of the same portfolio selection problem in this particular case the consideration of the robustness actually helps to reduce the computational complexity finally we consider the situation whereby we have some additional information about the chance that a quadratic function of the random distribution reaches a certain threshold that information helps to further reduce the ambiguity in the robust model we show that the robust optimization problem in that case can be solved by means of semidefinite programming sdp if no more than two additional chance inequalities are to be incorporated

iterative processes with mixed errors for nonlinear equations with perturbed m accretive operators in banach spaces let x be a real uniformly smooth banach space of which the dual x has a frechet differentiable norm let a d a x 2 x be an m accretive operator with closed domain d a and bounded range r a and s x x a continuous and a strongly accretive operator with bounded range r i s it is proved that the ishikawa and mann iterative processes with mixed errors converge strongly to the unique solution of the equation z sx lax for given z x and l 0

some problems and results in the study of nonlinear analysis this paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis which were obtained by the author and some chinese mathematicians

varying confidence levels for cvar risk measures and minimax limits conditional value at risk cvar has been widely studied as a risk measure in this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with cvar appearing in the objective and also the constraints we start by considering a problem in which cvar is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one we make use of a consistent tail condition which ensures that the cvar of a random function will converge uniformly to its supremum as the confidence level increases and establish an error bound for the cvar optimal solution under second order growth conditions the results are extended to a minimization problem with a constraint on the cvar value which in the limit as the confidence level approaches one coincides with a problem having semi infinite constraints we study the sample average approximation scheme for the cvar constraints and establish an exponential rate of convergence for the sample averaged optimal solution we propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to cvar constrained problems out of sample our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

mann and ishikawa iterations with errors for asymptotically nonexpansive mappings abstract in a recent paper rhoades 1 presented some generalizations of schu 2 on the convergence of the mann and ishikawa iterations of asymptotically nonexpansive mappings in uniformly convex banach spaces we continue the study on the ishikawa and mann iteration process with errors and prove that if x is a uniformly convex banach space o e x closed bounded and convex and t e e is an asymptotically nonexpansive mapping then the ishikawa and mann iteration process with errors converges strongly to some fixed point of t

robust novelty detection via worst case cvar minimization novelty detection models aim to find the minimum volume set covering a given probability mass this paper proposes a robust single class support vector machine ssvm for novelty detection which is mainly based on the worst case conditional value at risk minimization by assuming that every input is subject to an uncertainty with a specified symmetric support this robust formulation results in a maximization term that is similar to the regularization term in the classical ssvm when the uncertainty set is ell 1 norm ell infty norm or box its training can be reformulated to a linear program while the uncertainty set is ell 2 norm or ellipsoidal its training is a tractable second order cone program the proposed method has a nice consistent statistical property as the training size goes to infinity the estimated normal region converges to the true provided that the magnitude of the uncertainty set decreases in a systematic way the experimental results on three data sets clearly demonstrate its superiority over three benchmark models

a computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set the separable uncertainty sets have been widely used in robust portfolio selection models e g see erdogan et al robust portfolio management manuscript department of industrial engineering and operations research columbia university new york 2004 goldfarb and iyengar math oper res 28 1 38 2003 tutuncu and koenig ann oper res 132 157 187 2004 for these uncertainty sets each type of uncertain parameters e g mean and covariance has its own uncertainty set as addressed in lu a new cone programming approach for robust portfolio selection technical report department of mathematics simon fraser university burnaby 2006 robust portfolio selection based on a joint ellipsoidal uncertainty set manuscript department of mathematics simon fraser university burnaby 2008 these separable uncertainty sets typically share two common properties i their actual confidence level namely the probability of uncertain parameters falling within the uncertainty set is unknown and it can be much higher than the desired one and ii they are fully or partially box type the associated consequences are that the resulting robust portfolios can be too conservative and moreover they are usually highly non diversified as observed in the computational experiments conducted in this paper and tutuncu and koenig ann oper res 132 157 187 2004 to combat these drawbacks the author of this paper introduced a joint ellipsoidal uncertainty set lu in a new cone programming approach for robust portfolio selection technical report department of mathematics simon fraser university burnaby 2006 robust portfolio selection based on a joint ellipsoidal uncertainty set manuscript department of mathematics simon fraser university burnaby 2008 and showed that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters for this uncertainty set we showed in lu a new cone programming approach for robust portfolio selection technical report department of mathematics simon fraser university burnaby 2006 robust portfolio selection based on a joint ellipsoidal uncertainty set manuscript department of mathematics simon fraser university burnaby 2008 that the corresponding robust maximum risk adjusted return rmrar model can be reformulated and solved as a cone programming problem in this paper we conduct computational experiments to compare the performance of the robust portfolios determined by the rmrar models with our joint uncertainty set lu in a new cone programming approach for robust portfolio selection technical report department of mathematics simon fraser university burnaby 2006 robust portfolio selection based on a joint ellipsoidal uncertainty set manuscript department of mathematics simon fraser university burnaby 2008 and goldfarb and iyengar s separable uncertainty set proposed in the seminal paper goldfarb and iyengar in math oper res 28 1 38 2003 our computational results demonstrate that our robust portfolio outperforms goldfarb and iyengar s in terms of wealth growth rate and transaction cost and moreover ours is fairly diversified but goldfarb and iyengar s is surprisingly highly non diversified

distributionally robust multi item newsvendor problems with multimodal demand distributions we present a risk averse multi dimensional newsvendor model for a class of products whose demands are strongly correlated and subject to fashion trends that are not fully understood at the time when orders are placed the demand distribution is known to be multimodal in the sense that there are spatially separated clusters of probability mass but otherwise lacks a complete description we assume that the newsvendor hedges against distributional ambiguity by minimizing the worst case risk of the order portfolio over all distributions that are compatible with the given modality information we demonstrate that the resulting distributionally robust optimization problem is mathrm np np hard but admits an efficient numerical solution in quadratic decision rules this approximation is conservative and computationally tractable moreover it achieves a high level of accuracy in numerical tests we further demonstrate that disregarding ambiguity or multimodality can lead to unstable solutions that perform poorly in stress test experiments

robust risk management estimating the probabilities by which different events might occur is usually a delicate task subject to many sources of inaccuracies moreover these probabilities can change over time leading to a very difficult evaluation of the risk induced by any particular decision given a set of probability measures and a set of nominal risk measures we define in this paper the concept of robust risk measure as the worst possible of our risks when each of our probability measures is likely to occur we study how some properties of this new object can be related with those of our nominal risk measures such as convexity or coherence we introduce a robust version of the conditional value at risk cvar and of entropy based risk measures we show how to compute and optimize the robust cvar using convex duality methods and illustrate its behavior using data from the new york stock exchange and from the nasdaq between 2005 and 2010

twenty years of linear programming based portfolio optimization markowitz formulated the portfolio optimization problem through two criteria the expected return and the risk as a measure of the variability of the return the classical markowitz model uses the variance as the risk measure and is a quadratic programming problem many attempts have been made to linearize the portfolio optimization problem several different risk measures have been proposed which are computationally attractive as for discrete random variables they give rise to linear programming lp problems about twenty years ago the mean absolute deviation mad model drew a lot of attention resulting in much research and speeding up development of other lp models further the lp models based on the conditional value at risk cvar have a great impact on new developments in portfolio optimization during the first decade of the 21st century the lp solvability may become relevant for real life decisions when portfolios have to meet side constraints and take into account transaction costs or when large size instances have to be solved in this paper we review the variety of lp solvable portfolio optimization models presented in the literature the real features that have been modeled and the solution approaches to the resulting models in most of the cases mixed integer linear programming milp models we also discuss the impact of the inclusion of the real features

the block numerical range of matrix polynomials block numerical ranges of matrix polynomials especially the quadratic numerical range are considered the main results concern spectral inclusion boundedness of the block numerical range an estimate of the resolvent in terms of the quadratic numerical range geometrical properties of the quadratic numerical range and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions as an application we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials

on spectra of expansion graphs and matrix polynomials abstract an expansion graph of a directed weighted graph g0 is obtained by replacing certain edges of g0 by disjoint chains the adjacency matrix of the expansion graph is a partial linearization of a monic matrix polynomial we prove results on common properties of a monic operator polynomial and its partial linearization the graph g0 is connected if and only if each expansion graph of g0 is connected in this case we compute the index of imprimitivity of the adjacency matrix of some special expansion graphs of g0

on the shape of numerical range of matrix polynomials abstract the numerical range of an n n matrix polynomial p a m m a 1 a 0 is defined by w p c x p x 0 x c n x 0 in this paper we investigate the shape of w p by using the notion of local dimension the numerical range of first order matrix polynomials is always simply connected the special cases of diagonal matrix polynomials and 2 2 matrix polynomials are also considered

on a mann type implicit iteration process for continuous pseudo contractive mappings abstract let k be a nonempty closed convex subset of a banach space e t k k a continuous pseudo contractive mapping suppose that n is a real sequence in 0 1 satisfying appropriate conditions then for arbitrary x 0 k the mann type implicit iteration process x n given by x n n x n 1 1 n t x n n 0 strongly and weakly converges to a fixed point of t respectively

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

conditional value at risk and average value at risk estimation and asymptotics we discuss linear regression approaches to the estimation of law invariant conditional risk measures two estimation procedures are considered and compared one is based on residual analysis of the standard least squares method and the other is in the spirit of the m estimation approach used in robust statistics in particular value at risk and average value at risk measures are discussed in detail large sample statistical inference of the estimators is derived furthermore finite sample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensive monte carlo study empirical results on the real data different financial asset classes are also provided to illustrate the performance of the estimators

the numerical range of a nonnegative matrix abstract we offer an almost self contained development of perron frobenius type results for the numerical range of an irreducible nonnegative matrix rederiving and completing the previous work of issos nylen and tam and tam and yang on this topic we solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin some related results are obtained and some open problems are also posed

robustness of optimal portfolios under risk and stochastic dominance constraints abstract solutions of portfolio optimization problems are often influenced by a model misspecification or by errors due to approximation estimation and incomplete information the obtained results recommendations for the risk and portfolio manager should be then carefully analyzed we shall deal with output analysis and stress testing with respect to uncertainty or perturbations of input data for static risk constrained portfolio optimization problems by means of the contamination technique dependence of the set of feasible solutions on the probability distribution rules out the straightforward construction of convexity based global contamination bounds results obtained in our paper dupacova j kopa m 2012 robustness in stochastic programs with risk constraints annals of operations research 200 55 74 were derived for the risk and second order stochastic dominance constraints under suitable smoothness and or convexity assumptions that are fulfilled e g for the markowitz mean variance model in this paper we relax these assumptions having in mind the first order stochastic dominance and probabilistic risk constraints local bounds for problems of a special structure are obtained under suitable conditions on the structure of the problem and for discrete distributions we shall exploit the contamination technique to derive a new robust first order stochastic dominance portfolio efficiency test

worst case robust omega ratio abstract the omega ratio is a recent performance measure proposed to overcome the known shortcomings of the sharpe ratio until recently the omega ratio was thought to be computationally intractable and research was focused on heuristic optimization procedures we have shown elsewhere that the omega ratio optimization is equivalent to a linear program and hence can be solved exactly in polynomial time this permits the investigation of more complex and realistic variants of the problem the standard formulation of the omega ratio requires perfect information for the probability distribution of the asset returns in this paper we investigate the problem arising from the probability distribution of the asset returns being only partially known we introduce the robust variant of the conventional omega ratio that hedges against uncertainty in the probability distribution we examine the worst case omega ratio optimization problem under three types of uncertainty mixture distribution box and ellipsoidal uncertainty and show that the problem remains tractable

diagonal lyapunov krasovskii functionals for discrete time positive systems with delay we consider the existence of diagonal lyapunov krasovskii l k functionals for positive discrete time systems subject to time delay in particular we show that the existence of a diagonal functional is necessary and sufficient for delay independent stability of a positive linear time delay system we extend this result and provide conditions for the existence of diagonal l k functionals for classes of nonlinear positive time delay systems which are not necessarily order preserving we also describe sufficient conditions for the existence of common diagonal l k functionals for switched positive systems subject to time delay

self inversive hilbert space operator polynomials with spectrum on the unit circle abstract hilbert space operator polynomials with self inversive structure are studied if the inner numerical radius of an associated polynomial is greater than or equal to one then the spectrum lies on the unit circle and consists of normal approximate characteristic values

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

a probabilistic model for minmax regret in combinatorial optimization in this paper we propose a new probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients the interval uncertainty representation of data is supplemented with information on the marginal distributions as a decision criterion we minimize the worst case conditional value at risk of regret the proposed model includes the interval data minmax regret model as a special case for the class of combinatorial optimization problems with a compact convex hull representation a polynomial sized mixed integer linear program is formulated when a the range and mean are known and b the range mean and mean absolute deviation are known and a mixed integer second order cone program is formulated when c the range mean and standard deviation are known for the subset selection problem of choosing k elements of maximum total weight out of a set of n elements the probabilistic regret model is shown to be solvable in polynomial time in the instances a and b above this extends the current known polynomial complexity result for minmax regret subset selection with range information only

spectral properties of matrix polynomials in the max algebra we consider the spectral properties of matrix polynomials over the max algebra in particular we show how the perron frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra we also present a number of inequalities for the largest max eigenvalue of a matrix polynomial

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

the boundary of the numerical range of matrix polynomials abstract some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper we also consider isolated points of the numerical range and the location of the numerical range in a circular annulus

credit risk optimization with conditional value at risk criterion this paper examines a new approach for credit risk optimization the model is based on the conditional value at risk cvar risk measure the expected loss exceeding value at risk cvar is also known as mean excess mean shortfall or tail var this model can simultaneously adjust all positions in a portfolio of financial instruments in order to minimize cvar subject to trading and return constraints the credit risk distribution is generated by monte carlo simulations and the optimization problem is solved effectively by linear programming the algorithm is very efficient it can handle hundreds of instruments and thousands of scenarios in reasonable computer time the approach is demonstrated with a portfolio of emerging market bonds

on computing the generalized crawford number of a matrix abstract we use methods of geometric computing combined with hermitean matrix eigenvalue eigenvector evaluations to find the generalized crawford number of a pair of hermitean matrices s 1 and s 2 quickly and to high precision the classical crawford number is defined as the minimal distance from zero in c to the field of values of the associated complex matrix a s 1 is 2 if zero lies outside the field of values of a we describe test and compare a geometry based matlab code for finding for the generalized crawford number that measures the smallest distance of zero from the boundary of the field of values of a even if zero lies inside

the numerical range of linear pencils of 2 by 2 matrices abstract let a b m n the numerical range of a linear pencil a b is defined by w a b t c tw aw w bw 0 for some non zero w c n the numerical range of a linear pencil of n by n matrices coincides with the union of the numerical ranges of linear pencils of 2 by 2 matrices we emphasize on computation of the numerical range of linear pencils of 2 by 2 matrices let a b m 2 with constant diagonals we show that the boundary of w a b lies on a rational curve of degree at most 4 by using lagrange s multiplier and implicit function we describe the boundary of w a b for general 2 by 2 matices

robustness to dependency in portfolio optimization using overlapping marginals in this paper we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses for a fr echet class of discrete distributions with overlapping marginals we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming to guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information we make use of a graph theoretic property known as the running intersection property building on this property we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst case conditional value at risk measure lastly we use a data driven approach with financial return data to identify the fr echet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample based approach

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

birkhoff james e orthogonality sets of vectors and vector valued polynomials abstract consider a complex normed linear space x and let x with 0 motivated by recent works on rectangular matrices and on normed linear spaces we study the birkhoff james e orthogonality set of with respect to give an alternative definition for this set and explore its rich structure we also introduce the birkhoff james e orthogonality set of polynomials in one complex variable whose coefficients are members of x and survey and record extensions of results on matrix polynomials to these vector valued polynomials

convergence theorem of common fixed points for lipschitzian pseudocontraction semigroups in banach spaces the purpose of this paper is to study the convergence problems of the implicity iteration process for an asymptotically nonexpansive semigroups in general banach spaces the results presented in this paper extend and improve the corresponding results announced by many authors

geometrical properties of numerical range of matrix polynomials abstract through the linearization of a matrix polynomial p the symmetry and the sharp points of the numerical range p are studied

on the connectedness of numerical range of matrix polynomials abstract an investigation on nonconnectedness of numerical range for monic matrix polynomials l is undertaking here the distribution of eigenvalues of l to the components of numerical range and some other algebraic properties are also presented

solving matrix polynomial equations arising in queueing problems abstract the matrix equation i 0 n a i x i 0 where the a i s are m m matrices is encountered in the numerical solution of markov chains which model queueing problems we provide here a unifying framework in terms of mobius mapping to relate different resolution algorithms having a quadratic convergence this allows us to compare algorithms like logarithmic reduction lr and cyclic reduction cr which extend graeffe s iteration to matrix polynomials and the invariant subspace is approach which extends cardinal s algorithm we devise new iterative techniques having quadratic convergence and present numerical experiments

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces in this paper we consider an ishikawa type iteration process with errors which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces furthermore we obtain the corresponding results in banach spaces our results extend and generalize many known results

a new concept for block operator matrices the quadratic numerical range abstract in this paper a new concept for 2 2 block operator matrices the quadratic numerical range is studied the main results are a spectral inclusion theorem an estimate of the resolvent in terms of the quadratic numerical range factorization theorems for the schur complements and a theorem about angular operator representations of spectral invariant subspaces which implies e g the existence of solutions of the corresponding riccati equations and a block diagonalization all results are new in the operator as well as in the matrix case

a classification of matrices of class z abstract we generalize the classes n 0 and f 0 studied by k fan g johnson and r smith schur complements and lattices are examined for matrices in these classes

robust optimization of mixed cvar starr ratio using copulas abstract we introduce the robust optimization models for two variants of stable tail adjusted return ratio starr one with mixed conditional value at risk mcvar and the other with deviation mcvar dmcvar under joint ambiguity in the distribution modeled using copulas the two models are shown to be computationally tractable linear programs we apply a two step procedure to capture the joint dependence structure among the assets we first extract the filtered residuals from the return series of each asset using autoregressive moving average glosten jagannathan runkle generalized autoregressive conditional heteroscedastic arma gjr garch model subsequently we exploit the regular vine copulas to model the joint dependence among the transformed residuals the tree structure in the regular vines is accomplished using kendall s tau we compare the performance of the proposed two robust models with their conventional counterparts when the joint distribution in the latter is described using gaussian copula only we also examine the performance of the obtained portfolios against those from the markowitz model and multivariate garch models using the rolling window analysis we illustrate the superior performance of the proposed robust models than their conventional counterpart models on excess mean returns sortino ratio rachev ratio var ratio and treynor ratio on three data sets comprising of indices across the globe

multi objective mean variance skewness model for nonconvex and stochastic optimal power flow considering wind power and load uncertainties abstract this paper presents a multi objective mean variance skewness mvs model for reliably assessing the problem of nonconvex and stochastic optimal power flow nsopf considering valve point effect stochastic load and uncertain integrated wind power the mvs model aims to maximize both the expected return and skewness while simultaneously minimizing the risk which is formulated as a competing and conflicting three objective optimization problem based on such a model we propose a multi objective optimization algorithm multiple preys based evolutionary predator and prey strategy mpepps to provide pareto solutions which show the trade off relationship among the expected return the skewness and the risk of the dispatching objective subsequently a multi criteria decision making method the technique for order preference by similarity to an ideal solution topsis is applied for determining the final dispatch solution the objective of this paper is to develop a reliable model to assess the nsopf from the perspective of economics and reliability of power system operation and propose an efficient algorithm to obtain a solution that considers all of the possible load and wind power simultaneously simulation results based on a modified ieee 30 bus power system demonstrate the reliability and effectiveness of the mvs and mpepps in solving nsopf

numerical range of linear pencils abstract consider a linear pencil a b where a and b are n n complex matrices the numerical range of a b is defined as w a b c x a b x 0 for some nonzero x c n in this paper we study the geometrical properties of w a b with emphasis to its boundary an answer to the problem of the numerical approximation of w a b when one of the coefficients a and b is hermitian is presented the numerical range of a matrix on an indefinite inner product space is also considered

distributionally robust reward risk ratio optimization with moment constraints reward risk ratio optimization is an important mathematical approach in finance we revisit the model by considering a situation where an investor does not have complete information on the distribution of the underlying uncertainty and consequently a robust action is taken to mitigate the risk arising from ambiguity of the true distribution we consider a distributionally robust reward risk ratio optimization model varied from the ex ante sharpe ratio where the ambiguity set is constructed through prior moment information and the return function is not necessarily linear we transform the robust optimization problem into a nonlinear semi infinite programming problem through standard lagrange dualization and then use the well known entropic risk measure to construct an approximation of the semi infinite constraints we solve the latter by an implicit dinkelbach method finally we apply the proposed robust model and numerical scheme to a portfolio optimization problem and report some preliminary numerical

bounds for eigenvalues of matrix polynomials upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial or matrix the bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices they generalize various existing bounds for scalar polynomials and single matrices a variety of tools are used in the derivations including block companion matrices gershgorin s theorem the numerical radius and associated scalar polynomials numerical experiments show that the bounds can be surprisingly sharp on practical problems

weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations abstract let h be a real hilbert space k a nonempty closed convex subset of h and t k k a lipschitz pseudocontraction with a nonempty fixed point set weak and strong convergence theorems for the iterative approximation of fixed points of t are proved furthermore if t h h is a lipschitz monotone operator and f r t where r t denotes the range of t weak and strong convergence theorems for the iterative approximation of solutions of the operator equation tx f are proved

birkhoff james approximate orthogonality sets and numerical ranges abstract in this paper the notion of birkhoff james approximate orthogonality sets is introduced for rectangular matrices and matrix polynomials the proposed definition yields a natural generalization of standard numerical range and q numerical range and also of recent extensions sharing with them several geometric properties

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

the influence of the marked reduced graph of a nonnegative matrix on the jordan form and on related properties a survey abstract for a nonnegative matrix p we discuss the relation of its marked reduced graph to that part of the jordan form that is associated with the perron frobenius root to the nonnegativity of the eigenvectors and generalized eigenvectors to the nonnegativity of solutions of linear equations and to the asymptotic growth of powers of the matrix results are often stated in terms of m matrices and standard results on irreducible matrices are assumed we give examples to illustrate the theorems surveyed

general a monotone operators and perturbed iterations for nonlinear set valued relaxed cocoercive operator inclusion problems in this paper we introduce the notion of general a monotone operators in banach spaces study some properties of general a monotone operator and the new proximal mapping associated with the general a monotone operators by using alber s inequalities nalder s results and the new proximal mapping technique we also construct two new class of perturbed iterative algorithms with mixed errors for solving a new class of nonlinear set valued relaxed cocoercive operator inclusions and study applications of general a monotone operators to the approximation solvability of the nonlinear set valued relaxed cocoercive operator inclusion problems in banach spaces furthermore the general a monotone operators are illustrated by some examples the results presented in this paper improve and generalize the corresponding results on strongly monotone quasi variational inclusions and nonlinear implicit quasi variational inclusions

the nearest definite pair for the hermitian generalized eigenvalue problem the generalized eigenvalue problem ax lambda bx has special properties when a b is a hermitian and definite pair given a general hermitian pair a b it is of interest to find the nearest definite pair having a specified crawford number delta 0 we solve the problem in terms of the inner numerical radius associated with the field of values of a ib we show that once the problem has been solved it is trivial to rotate the perturbed pair a da b db to a pair widetilde a widetilde b for which lambda min widetilde b achieves its maximum value delta which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by inverting b numerical examples are given to illustrate the analysis

an estimation of the spectral radius of a product of block matrices let c r c ij r 1 2 r be block m x m matrices where c ij r are nonnegative n i x n j matrices for i j 1 2 m let be a consistent matrix norm denote for each r by b r c ij r an m m matrix the relation of the spectral radii r r 1 c r and r r 1 b r is studied in this paper it is shown with two proofs that r r 1 c r r r 1 b r as shown in one of the proofs r r 1 b r can be reduced so that it gives a better estimation of r r 1 c r

definite triples of hermitian matrices and matrix polynomials let a b and c be three n n nonzero hermitian matrices the triple a b c is called defnite if the convex hull of the joint numerical range f a b c x ax x bx x cx r3 x cn x x 1 does not contain 0 0 0 if the triple a b c is nondefinite then the numerical ranges of the matrix polynomials q a v3 b v2 c v1 v3 v2 v1 0 and l a 2 b ic 1 2 1 0 coincide with the whole complex plane providing no information as a consequence it is of particular interest to characterize a definite triple a b c and find the distance between 0 0 0 and the boundary f a b c the distance between a nondefinite triple a b c and the nearest definite triples with specified properties is also investigated moreover applications of definite triples on matrix polynomials of special interest are presented

tail variance of portfolio under generalized laplace distribution many popular downside risk measures such as tail conditional expectation and conditional value at risk only characterize the tail expectation but pay no attention to the tail variance beyond the value at risk this is a severe deficiency of risk management in finance and insurance industry especially in measuring extreme risk with large losses we derive the explicit formulae of the tail variance of portfolio under the assumption of generalized laplace distribution and mixture generalized laplace distribution as well some numerical results of parameters related to the tail variance of portfolio are also provided finally we present an example of application to optimization tail mean variance portfolio the empirical results show that the performance of optimal portfolio can be efficiently improved by controlling the tail variability of returns distribution

worst case conditional value at risk with application to robust portfolio management this paper considers the worst case conditional value at risk cvar in the situation where only partial information on the underlying probability distribution is available the minimization of the worst case cvar under mixture distribution uncertainty box uncertainty and ellipsoidal uncertainty are investigated the application of the worst case cvar to robust portfolio optimization is proposed and the corresponding problems are cast as linear programs and second order cone programs that can be solved efficiently market data simulation and monte carlo simulation examples are presented to illustrate the proposed approach

generalisation of the perron frobenius theory to matrix pencils abstract we present a new extension of the well known perron frobenius theorem to regular matrix pairs e a the new extension is based on projector chains and is motivated from the solution of positive differential algebraic systems or descriptor systems we present several examples where the new condition holds whereas conditions in previous literature are not satisfied

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

generating scenario trees for multistage decision problems in models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models if the uncertainties are expressed in terms of multivariate continuous distributions or a discrete distribution with far too many outcomes we normally face two possibilities either creating a decision model with internal sampling or trying to find a simple discrete approximation of the given distribution that serves as input to the model this paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties users are free to specify any statistical properties they find relevant and the method can handle inconsistencies in the specifications the basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties we illustrate the method by single and multiple period problems the results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications we discuss how to verify that the relevant statistical properties are captured in these specifications and argue that what are the relevant properties will be problem dependent

point equation of the boundary of the numerical range of a matrix polynomial abstract the numerical range of an n n matrix polynomial p a m m a m 1 m 1 a 1 a 0 is defined by w p c x p x 0 x c n x 0 for the linear pencil p i a the range w p coincides with the numerical range of matrix a f a x ax x c n x x 1 in this paper we obtain necessary conditions for the origin to be a boundary point of f a as a consequence an algebraic curve of degree at most 2 n n 1 m which contains the boundary of w p is constructed

real eigenvalues of certain tridiagonal matrix polynomials with queueing applications many queueing problems lead to tridiagonal lambda matrices containing polynomials that have except for the diagonal non negative coefficients this paper deals with the question addressed in the literature only for special cases whether the eigenvalues corresponding to such lambda matrices are real in most cases they are as the theorems of this paper show but sometimes complex eigenvalues occur our results are derived by using sturm sequences in addition to simplifying the proofs of our theorems sturm sequences are also valuable to verify whether or not a given interval contains eigenvalues

perron frobenius type results on the numerical range we present results connecting the shape of the numerical range to intrinsic properties of a matrix a w hena is a nonnegative matrix these results are to a large extent analogous to the perron frobenius theory especially as it pertains to irreducibility and cyclicity in the combinatorial sense special attention is given to polygonal circular and elliptic numerical ranges the main vehicles for obtaining these results are the hermitian and skew hermitian parts of a as well as levinger s transformation aa 1 a a 2002 elsevier science

perron frobenius theorems for the numerical range of semi monic matrix polynomials abstract we present an extension of the perron frobenius theory to the numerical ranges of semi monic perron frobenius polynomials namely matrix polynomials of the form q m i l a l a 0 m i a where the coefficients are entrywise nonnegative matrices our approach relies on the function numerical radius a and the infinite graph g m a 0 a l our main result describes the cyclic distribution of the elements of the numerical range of q on the circles with radius satisfying m numerical radius a

global existence and mann iterative algorithms of positive solutions for first order nonlinear neutral delay differential equations abstract this paper deals with the first order nonlinear neutral delay differential equation d dt x t p t x t f t x 1 t x 2 t x n t 0 t t 0 where 0 p c t 0 r f c t 0 r n r and l c t 0 r with limt l t for l 1 2 n by using the banach fixed point theorem we prove the global existence of uncountably many bounded positive solutions for the above equation relative to all ranges of the function p construct some mann type iterative algorithms with errors to approximate these positive solutions and discuss several error estimates between the sequences generated by the iterative algorithms and these positive solutions seven examples are presented to illuminate the results obtained in this paper

matrix polynomials with spectral radius equal to the numerical radius in this paper we study a class of matrix polynomials with the property that spectral radius and numerical radius coincide special attention is paid to the spectrum on the boundary of the numerical range

strong convergence theorems of total quasi φ asymptotically nonexpansive semi groups in banach spaces the purpose of this article is first to introduce the concept of total quasi f asymptotically nonexpansive semigroups which contains many kinds of semi groups as its special cases and then to modify the halpern mann type iteration algorithm for this kind of semi groups to have the strong convergence under a limit condition only in the framework of banach spaces the results presented in the paper improve and extend the corresponding recent results announced by many authors

an iterative method for total quasi φ asymptotically nonexpansive semi groups and generalized mixed equilibrium problems in banach spaces in this paper we introduce an iterative scheme by the modified halpern mann type for total quasi asymptotically nonexpansive semi groups and prove the strong convergence theorem using our new iterative process which converges strongly to a common element of the set of common fixed points for total quasi asymptotically nonexpansive semi groups and the set of solutions of generalized mixed equilibrium problem in a uniformly smooth and strictly convex banach space with the kadec klee property using the properties of the generalized f projection operator our results extend and improve the corresponding recent results in the literature

distributionally robust control of constrained stochastic systems we investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process i e when only the first two moments of the disturbance distribution are known we consider two types of distributionally robust constraints in the first case we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments these constraints are commonly referred to as distributionally robust chance constraints in the second case we impose conditional value at risk cvar constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information such constraints are referred to as distributionally robust cvar constraints with second order moment specifications we propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems we prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust cvar and chance constrained problems the proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives

constructing risk measures from uncertainty sets we illustrate the correspondence between uncertainty sets in robust optimization and some popular risk measures in finance and show how robust optimization can be used to generalize the concepts of these risk measures we also show that by using properly defined uncertainty sets in robust optimization models one can construct coherent risk measures and address the issue of the computational tractability of the resulting formulations our results have implications for efficient portfolio optimization under different measures of risk

robustifying convex risk measures for linear portfolios a nonparametric approach this paper introduces a framework for robustifying convex law invariant risk measures the robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure which represent the investors beliefs about the distribution of future asset losses it is shown that under mild conditions the infinite dimensional optimization problem of finding the worst case risk can be solved analytically and closed form expressions for the robust risk measures are obtained using these results robust versions of several risk measures including the standard deviation the conditional value at risk and the general class of distortion functionals are derived the resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems and a numerical study shows that in most cases they perform significantly better out of sample than their nonrobust variants in terms of risk expected losses and turnover

convergence analysis for distributionally robust optimization and equilibrium problems in this paper we study distributionally robust optimization approaches for a one stage stochastic minimization problem where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of the worst possible distribution from that set we consider the case when the distributional set which is also known as the ambiguity set varies and its impact on the optimal value and the optimal solutions a typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size the analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming the discussion is extended briefly to a stochastic nash equilibrium problem wh