capacity of mimo channels with antenna selection this correspondence studies the capacity of multiple input multiple output mimo channels in the presence of antenna selection antenna selection reduces the complexity of the radio devices and requires only a small amount of channel state feedback for high signal to noise ratio snr we define excess rate as the constant term in the expansion of the ergodic capacity in terms of snr it is shown that this value is representative of the channel state information csi at the transmitter the asymptotic behavior of the excess rate is then analyzed for three cases complete csi no csi and partial csi at transmitter antenna selection while water filling provides a excess rate that increases logarithmically in m the number of transmit antennas the excess rate of transmit antenna selection behaves only like log log m for the low snr case we use the concept of channel gain a measure introduced by verdu we show that channel gain for antenna selection increases only logarithmically in m as opposed to water filling channel gain which increases linearly in m the same techniques are also applied to the receive selection and corresponding results are noted in high and low snr regimes the methodology developed in this correspondence although motivated by antenna selection is fairly general and can be used for any system where partial csi is available at the mimo transmitter

the structure of max plus hyperplanes a max plus hyperplane briefly a hyperplane is the set of all points x x1 xn rmaxn satisfying an equation of the form a1x1 anxn an 1 b1x1 bnxn bn 1 that is max a1 x1 an xn an 1 max b1 x1 bn xn bn 1 with ai bi rmax i 1 n 1 where each side contains at least one term and where ai bi for at least one index i we show that the complements of max plus semispaces at finite points z rn are building blocks for the hyperplanes in rmaxn recall that a semispace at z is a maximal with respect to inclusion max plus convex subset of rmaxn z namely observing that up to a permutation of indices we may write the equation of any hyperplane h in one of the following two forms a1x1 apxp ap 1xp 1 aqxq a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m n 1 are finite or a1x1 apxp ap 1xp 1 aqxq an 1 a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m are finite and an 1 is either finite or we give a formula that expresses a nondegenerate strictly affine hyperplane i e with m n and an 1 as a union of complements of semispaces at a point z rn called the center of h with the boundary of a union of complements of other semispaces at z using this formula we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior we give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center and we characterize the cases in which the boundary bd h of a nondegenerate strictly affine hyperplane h is also a hyperplane next we give the relations between nondegenerate strictly affine hyperplanes h their centers z and their coefficients ai in the converse direction we show that any union of complements of semispaces at a point z rn with the boundary of any union of complements of some other semispaces at that point z is a nondegenerate strictly affine hyperplane we obtain a formula for the total number of strictly affine hyperplanes we give complete lists of all strictly affine hyperplanes for the cases n 1 and n 2 we show that each linear hyperplane h in rmaxn i e with an 1 can be decomposed as the union of four parts where each part is easy to describe in terms of complements of semispaces some of them in a lower dimensional space

a general stochastic outer approximations method optimization problems involving an infinite number of constraints are considered this paper presents a general stochastic outer approximations method which incorporates mechanisms for active search of relevant constraints and for dropping of irrelevant constraints the method extracts the characteristic features of several stochastic outer approximations algorithms suggested by wardi j optim theory appl 56 1988 pp 285 311 j optim theory appl 64 1990 pp 615 640 and furthermore develops the approach to get advantages of the eaves zangwill scheme similarly to gonzaga and polak siam j control optim 17 1979 pp 477 493 the method is based on the use of quasi optimality functions satisfying some general unrestricted assumptions these functions are usually employed in the stopping criteria of numerical techniques for solving simpler problems it is shown that the method s trajectories almost surely converge to the quasi optimal set following the proposed approach a stochastic algorithm for solving the approximation problem is constructed and studied the proposed general method can be considered as a developed eaves zangwill method applying the multistart technique at each iteration for the search of relevant constraints parameters

a new exchange method for convex semi infinite programming in this paper we propose a new exchange method for solving convex semi infinite programming csip problems we introduce a new dropping rule in the proposed exchange algorithm which only keeps those active constraints with positive lagrange multipliers moreover we exploit the idea of looking for eta infeasible indices of the lower level problem as the adding rule in our algorithm hence the algorithm does not require to solve a maximization problem over the index set at each iteration it only needs to find some points such that a certain computationally easy criterion is satisfied under some reasonable conditions the new adding dropping rule guarantees that our algorithm provides an approximate optimal solution for the csip problem in a finite number of iterations in the numerical experiments we apply the proposed algorithm to solve some test problems from the literature including some medium sized problems from complex approximation theory and fir filter design we compare our algorithm with an existing central cutting plane algorithm and with the semi infinite solver fseminf in matlab toolbox and we find that our algorithm solves the csip problem much faster for the fir filter design problem we show that our algorithm solves the problem better than some algorithms that were technically established for the problem

a norm relaxed method of feasible directions for finely discretized problems from semi infinite programming abstract in this paper a class of finely discretized semi infinite programming sip problems is discussed combining the idea of the norm relaxed method of feasible directions mfd and the technique of updating discretization index set we present a new algorithm for solving the discretized semi infinite dsi problems from sip at each iteration the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem i e a quadratic program and some appropriate constraints are chosen to reduce the computational cost a high order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints under weak conditions such as mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses weak global convergence moreover the superlinear convergence is obtained under linearly independent constraint qualification licq and other assumptions in the end some elementary numerical experiments are reported

a nonsmooth levenberg marquardt method for solving semi infinite programming problems in this paper we first transform the semi infinite programming problem into the kkt system by the techniques in d h li l qi j tam s y wu a smoothing newton method for semi infinite programming j global optim 30 2004 169 194 l qi s y wu g l zhou semismooth newton methods for solving semi infinite programming problems j global optim 27 2003 215 232 then a nonsmooth and inexact levenberg marquardt method is proposed for solving this kkt system based on h dan n yamashita m fukushima convergence properties of the inexact levenberg marquardt method under local error bound conditions optimim methods softw 11 2002 605 626 this method is globally and superlinearly even quadratically convergent finally some numerical results are given

capacity limits of multiuser multiantenna cognitive networks unlike point to point cognitive radio where the constraint imposed by the primary rigidly curbs the secondary throughput multiple secondary users have the potential to efficiently harvest the spectrum and share it among themselves this paper analyzes the sum throughput of a multiuser cognitive radio system with multiantenna base stations either in the uplink or downlink mode the primary and secondary have n and n users respectively and their base stations have m and m antennas respectively we show that an uplink secondary throughput grows with m n 1 log n if the primary is a downlink system and grows with m m 1 log n if the primary is an uplink system these growth rates are shown to be optimal and can be obtained with a simple threshold based user selection rule in addition we show that the secondary throughput can grow proportional to while simultaneously the interference on the primary is forced down to zero asymptotically for a downlink secondary it is shown that the throughput grows with in the presence of either an uplink or downlink primary system in addition the interference on the primary can be made to go to zero asymptotically while the secondary throughput increases proportionally to the effect of unequal path loss and shadowing is also studied it is shown that under a broad class of path loss and shadowing models the secondary throughput growth rates remain unaffected

characterizing the statistical properties of mutual information in mimo channels we consider gaussian multiple input multiple output mimo frequency selective spatially correlated fading channels assuming that the channel is unknown at the transmitter and perfectly known at the receiver for gaussian codebooks using results from multivariate statistics we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal to noise ratio snr we show that our bound is tighter than previously reported analytical lower bounds and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity based on a closed form approximation of the variance of mutual information in correlated flat fading mimo channels we provide insights into the multiplexing diversity tradeoff for gaussian code books furthermore for a given total number of antennas we consider the problem of finding the optimal ergodic capacity maximizing number of transmit and receive antennas and we reveal the snr dependent nature of the maximization strategy finally we present numerical results and comparisons between our capacity bounds and previously reported bounds

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

linear precoding for finite alphabet inputs over mimo fading channels with statistical csi this paper investigates the linear precoder design that maximizes the average mutual information of multiple input multiple output fading channels with statistical channel state information known at the transmitter it formulates the design from the standpoint of finite alphabet inputs which leads to a problem that is very important in practice but extremely difficult in theory first the average mutual information lacks closed form expression and involves prohibitive computational burden second the optimization over the precoder is nonconcave and thus easily gets stuck in local maxima to address these issues this study first derives lower and upper bounds for the average mutual information in which the computational complexity is reduced by several orders of magnitude compared to calculating the average mutual information directly it proves that maximizing the bounds is asymptotically optimal and shows that with a constant shift the lower bound actually offers a very accurate approximation to the average mutual information for various fading channels this paper further proposes utilizing the lower bound as a low complexity and accurate alternative for developing a two step algorithm to find a near global optimal precoder numerical examples demonstrate the convergence and efficacy of the proposed algorithm compared to its conventional counterparts the proposed linear precoding method provides significant performance gain over existing precoding algorithms the gain becomes more substantial when the spatial correlation of mimo channels increases

the role of linear semi infinite programming in signal adapted qmf bank design we consider the problem of designing a perfect reconstruction fir quadrature mirror filter qmf bank h g adapted to input signal statistics with the coding gain as the adaptation criterion maximization of the coding gain has so far been viewed as a difficult nonlinear constrained optimization problem it is shown that the coding gain depends only on the product filter p z h z h z sup 1 and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints termed linear semi infinite programming sip problems the sought for original filter h z is obtained by deflation and spectral factorization of p z with the sip formulation every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms the natural regularity properties inherent in the sip formulation enhance the performance of these algorithms we present a comprehensive theoretical analysis of the sip problem and its dual characterize the optimal filters and analyze uniqueness and sensitivity issues all these properties are intimately related to those of the input signal and bring considerable insight into the nature of the adaptation process we present discretization and cutting plane algorithms and apply both methods to several examples

the dual parameterization approach to optimal least square fir filter design subject to maximum error constraints this paper is concerned with the design of linear phase finite impulse response fir digital filters for which the weighted least square error is minimized subject to maximum error constraints the design problem is formulated as a semi infinite quadratic optimization problem using a newly developed dual parameterization method in conjunction with the caratheodory s dimensional theorem an equivalent dual finite dimensional optimization problem is obtained the connection between the primal and the dual problems is established a computational procedure is devised for solving the dual finite dimensional optimization problem the optimal solution to the primal problem can then be readily obtained from the dual optimal solution for illustration examples are solved using the proposed computational procedure

zf detectors over correlated k fading mimo channels this paper provides a systematic characterization of zero forcing zf detectors over multiple input multiple output mimo channels that experience both small and large scale fading in particular we consider the generic k distribution rayleigh gamma distribution to model the composite fading fluctuations and also assume the general case of semi correlated small scale fading in the following novel exact analytical expressions for the achievable sum rate are derived followed by asymptotic expressions in the high and low signal to noise ratio snr regimes in these limiting cases two common and insightful affine expansions are studied followed by new closed form upper and lower bounds on the sum rate that remain tight for all snrs in the second part of the paper we present exact tractable expressions along with first order expansions for the symbol error rate ser and outage probability we also quantify the performance of zf detectors in terms of diversity order and array or coding gain the implications of the model parameters on the zf detector performance are investigated via monte carlo simulations which also validate the theoretical analysis

on the ergodic secret key agreement over spatially correlated multiple antenna channels with public discussion we consider secret key agreement with public discussion over multiple input multiple output mimo rayleigh fast fading channels under correlated environment we assume that transmit legitimate receiver and eavesdropper antennas are correlated the legitimate receiver and the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices first we derive the expression of the secret key capacity under the considered setup we prove that the optimal transmit strategy achieving the secret key capacity consists in transmitting independent gaussian signals along the eingenvectors of the transmit correlation matrix the powers allocated to each channel mode are determined as the solution to a numerical optimization problem a necessary and sufficient condition for beamforming i e transmitting along the strongest channel mode to be capacity achieving is derived moreover we analyze the impact of correlation matrices on the system performance finally we study the system s performance in the two extreme power regimes in the high power regime we provide closed form expressions of the gain loss due to correlation in the low signal to noise ratio snr regime we investigate the energy efficiency of the system by determining the minimum energy required for sharing a secret key bit and the wideband slope while highlighting the impact of correlation matrices

some outer approximation methods for semi infinite optimization problems abstract the paper starts with a simple model and convergence theorem for outer approximation methods this general framework is used to unifyingly derive and modify certain exchange methods cutting methods and discretization methods for semi infinite programming problems by that in particular a cutting plane method for convex semi infinite programs is developed for a practically reasonable specification the method is more generally stated the subproblems in the given algorithm are moderately sized quadratic problems and each step of the algorithm can be performed by means of finitely many operations

discretization in semi infinite programming the rate of convergence the discretization approach for solving semi infinite optimization problems is considered we are interested in the convergence rate of the error between the solution of the semi infinite problem and the solution of the discretized program depending on the discretization mesh size it will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way this is done for ordinary and for generalized semi infinite problems

a semi infinite programming algorithm for solving optimal power flow with transient stability constraints this paper proposes a new algorithm for solving a type of complicated optimal power flow opf problems in power systems i e opf problems with transient stability constraints ots the ots is converted into a semi infinite programming sip via some suitable function analysis then based on the kkt system of the reformulated sip a smoothing quasi newton algorithm is presented in which the numerical integration is used the convergence of the algorithm is established an ots problem in power system is tested which shows that the proposed algorithm is promising

a central cutting plane algorithm for convex semi infinite programming problems the central cutting plane algorithm for linear semi infinite programming sip is extended to nonlinear convex sip of the form min f x x in h g x t leq 0 text all t in s under differentiability assumptions that are weaker than those employed in superlinearly convergent algorithms a linear convergence rate is established that has additional important features these features are the ability to i generate a cut from any violated constraint ii invoke efficient constraint dropping rules for management of linear programming lp subproblem size iii provide an efficient grid management scheme to generate cuts and ultimately to test feasibility to a high degree of accuracy as well as to provide an automatic grid refinement for use in obtaining admissible starting solutions for the nonlinear system of first order conditions and iv provide primal and dual lagrangian sip feasible solutions in a finite number of iterations numerical tests are provided on a collection of prob

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

approximation in normal linear spaces abstract a historical account is given of the development of methods for solving approximation problems set in normed linear spaces approximation of both real functions and real data is considered with particular reference to lp or lp and chebyshev norms as well as coverage of methods for the usual linear problems an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters and special attention is paid to some particular nonlinear approximating families

on tropical fractional linear programming very recently tropical counterparts of fractional linear programs have been studied some algorithms were proposed for solving them with techniques ranging from bisection methods to homeomorphisms to formal power series in this paper some algorithms are also proposed they mainly rely in the ability of finding the greatest and smallest solutions of tropical equations a subject that was discussed in a previous work of the authors 13

capacity scaling in mimo wireless systems under correlated fading previous studies have shown that single user systems employing n element antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to n assuming independent rayleigh fading between antenna pairs we explore the capacity of dual antenna array systems under correlated fading via theoretical analysis and ray tracing simulations we derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases the latter is derived under some assumptions about the scaling of the fading correlation structure in both cases the theoretic capacity growth is linear in n but the growth rate is 10 20 smaller in the presence of correlated fading we analyze our assumption of separable transmit receive correlations via simulations based on a ray tracing propagation model results show that empirical capacities converge to the limit capacity predicted from our asymptotic theory even at moderate n 16 we present results for both the cases when the transmitter does and does not know the channel realization

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

a new proof of the strong duality theorem for semidefinite programming semidefinite programs are convex optimization problems arising in a wide variety of applications and are the extension of linear programming most methods for linear programming have been generalized to semidefinite programs just as in linear programming duality theorem plays a basic and an important role in theory as well as in algorithmics based on the discretization method and convergence property this paper proposes a new proof of the strong duality theorem for semidefinite programming which is different from other common proofs and is more simple

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

densities of idempotent measures and large deviations considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by maslov then idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure in this paper we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied these conditions depend on the lattice structure of the semiring and on the boolean algebra in which the measure is defined as an application we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle as defined by varadhan

lipschitz modulus of linear and convex inequality systems with the hausdorff metric this paper analyzes the lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in mathbb r n to start with we deal with the parameter space of linear finite semi infinite systems identified with the corresponding sets of coefficient vectors which are assumed to be closed subsets of mathbb r n 1 in this framework the size of perturbations is measured by means of the extended hausdorff distance a direct antecedent extensively studied in the literature comes from considering the parameter space of all linear systems with a fixed index set t where the chebyshev extended distance is used to measure perturbations in the present work we propose an appropriate indexation strategy which allows us to establish the equality of the lipschitz moduli of the feasible set mappings in both parametric contexts as well as to benefit from existing results in the chebyshev setting for transferring them to the hausdorff one in a second stage the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the lipschitz behavior of convex systems via linearization techniques

modifications of the first remez algorithm this paper proves the convergence of several modifications of the first remez algorithm for the solution of linear and nonlinear chebyshev approximation problems on compact b subset mathbb r s while the first remez algorithm in its original form requires the determination of the global maximum of the error function on all of b in each iteration the algorithms given here are based on its being sufficient to compute the maximum of the kth error function on a grid b k 1 where b k k geqq 0 is a prescribed sequence of finite point sets in b with density tending to zero interpreted differently some results on the discretization of chebyshev approximation problems which do not use full grids b k in b but only small subsets of them are provided the paper concludes with some numerical examples for the solution of linear multivariate problems

on the numerical treatment of linearly constrained semi infinite optimization problems abstract we consider the application of two primal algorithms to solve linear semi infinite programming problems depending on a real parameter combining a simplex type strategy with a feasible direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently the second algorithm runs a feasible direction method first and then switches to the purification procedure the linear programming subproblems that yield the search direction involve only a small subset of the constraints these subsets are updated at each iteration using a multi local optimization algorithm numerical test examples taken from the literature in order to compare the numerical effort with other methods show the efficiency of the proposed algorithms

discretization method for semi definite programming semi definite programs are convex optimization problems arising in a wide variety of applications and the extension of linear programming most methods for linear programming have been generalized to semi definite programs this paper discusses the discretization method in semi definite programming the convergence and the convergent rate of error between the optimal value of the semi definite programming problems and the optimal value of the discretized problems are obtained an approximately optimal division is given under certain assumptions with the significance of the convergence property the duality result in semi definite programs is proved in a simple way which is different from the other common proofs

optimal signaling and selection verification for single transmit antenna selection in marked contrast with the ideal error free feedback assumption that is common in the literature practical systems are likely to have severely bandwidth limited error prone feedback channels we consider the scenario where feedback from the receiver is used by the transmitter to select the best antenna out of many available antennas for data transmission feedback errors cause the transmitter to select an antenna different from the one signaled by the receiver we show that optimizing the signaling assignment which maps the antenna indices to the feedback codewords improves performance without introducing any additional redundancy for a system that uses error prone feedback to transmit quadrature phase shift keying modulated data from a single antenna selected from many available spatially correlated antennas we derive closed form approximations for the data symbol error probability for an arbitrary number of receive antennas we use these to systematically find the optimal signaling assignments using a low complexity algorithm the optimal signaling is intimately coupled to how the receiver performs selection verification i e how it decodes the data signal when due to feedback errors it does not always know which antenna was used for data transmission we show that ignoring feedback errors at the receiver can lead to an unacceptable performance degradation and develop optimal and suboptimal blind and nonblind selection verification methods with a small side information overhead nonblind verification approaches the ideal perfect selection verification performance

stacked ostbc error performance and rate analysis it is well known that the alamouti scheme is the only space time code from orthogonal designs achieving the capacity of a multiple input multiple output mimo wireless communication system with nt 2 transmit antennas and nr 1 receive antenna in this paper we propose the n times stacked alamouti scheme for nt 2n transmit antennas and show that this scheme achieves the capacity in the case of nr 1 receive antenna this result may regarded as an extension of the alamouti case for the more general case of more than one receive antenna we show that if the number of transmit antennas is higher than the number of receive antennas we achieve a high portion of the capacity with this scheme further we show that the mimo capacity is at most twice the rate achieved with the proposed scheme for all signal to noise ratio snr we derive lower and upper bounds for the rate achieved with this scheme and compare it with upper and lower bounds for the capacity in addition to the capacity analysis based on the assumption of a coherent channel we analyze the error rate performance of the stacked orthogonal space time block code ostbc with the optimal maximum likelihood ml detector and with the suboptimal lattice reduction lr aided zero forcing detector we compare the error rate performance of the stacked ostbc with spatial multiplexing sm and full diversity achieving schemes finally we illustrate the theoretical results by numerical simulations

a new exact penalty method for semi infinite programming problems in this paper we consider a class of nonlinear semi infinite optimization problems these problems involve continuous inequality constraints that need to be satisfied at every point in an infinite index set as well as conventional equality and inequality constraints by introducing a novel penalty function to penalize constraint violations we form an approximate optimization problem in which the penalty function is minimized subject to only bound constraints we then show that this penalty function is exact that is when the penalty parameter is sufficiently large any local solution of the approximate problem can be used to generate a corresponding local solution of the original problem on this basis the original problem can be solved as a sequence of approximate nonlinear programming problems we conclude the paper with some numerical results demonstrating the applicability of our approach to pid control and filter design

a new method for chebyshev approximation of complex valued functions problems of this type appear in many connections and have been treated by various authors cf 2 12 13 considering the computation of best rational chebyshev approximations to complex valued functions by descent algorithms ellacott and williams 6 report that the main portion of computer time is spent in the solution of the linear subproblems they have applied the very slowly convergent lawson algorithm to these subproblems and they remark that a fast linear algorithm would bring about a significant improvement in efficiency in our paper we describe a new algorithm which seems to be the first one which works for the continuous case where t is not a finite set too in our method we transform the complex approximation problem into a real linear optimization problem with infinitely many constraints a so called semi infinite program sip cf 5 8 10 this is solved by a two step procedure in the first step we apply the stabilized simplex algorithm to a discrete approximation of the sip and in the second we make use of the newton raphson method in order to obtain very rapidly the solution of the sip and thus of our original approximation problem

on the use of consistent approximations in the solution of semi infinite optimization and optimal control problems we consider a pair consisting of an optimization problem and its optimality function p and define consistency of approximating problem optimality function pairs p n n to p in terms of the epigraphical convergence of the p n to p and the hypographical convergence of the optimality functions n to we then show that standard discretization techniques decompose semi infinite optimization and optimal control problems into families of finite dimensional problems which together with associated optimality functions are consistent discretizations to the original problems we then present two types of techniques for using consistent approximations in obtaining an approximate solution of the original problems the first is a filter type technique similar to that used in conjunction with penalty functions the second one is an adaptive discretization technique that can be viewed as an implementation of a conceptual algorithm for solving the original problems

spatio temporal coding for wireless communication multipath signal propagation has long been viewed as an impairment to reliable communication in wireless channels this paper shows that the presence of multipath greatly improves achievable data rate if the appropriate communication structure is employed a compact model is developed for the multiple input multiple output mimo dispersive spatially selective wireless communication channel the multivariate information capacity is analyzed for high signal to noise ratio snr conditions the mimo channel can exhibit a capacity slope in bits per decibel of power increase that is proportional to the minimum of the number multipath components the number of input antennas or the number of output antennas this desirable result is contrasted with the lower capacity slope of the well studied case with multiple antennas at only one side of the radio link a spatio temporal vector coding stvc communication structure is suggested as a means for achieving mimo channel capacity the complexity of stvc motivates a more practical reduced complexity discrete matrix multitone dmmt space frequency coding approach both of these structures are shown to be asymptotically optimum an adaptive lattice trellis coding technique is suggested as a method for coding across the space and frequency dimensions that exist in the dmmt channel experimental examples that support the theoretical results are presented

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

on the capacity of generalized k fading mimo channels this correspondence explores the ergodic capacity of multiple input multiple output mimo systems operating in generalized k fading conditions using some recent results on majorization theory we derive an analytical capacity bound which is applicable for arbitrary values of the signal to noise ratio snr and number of antenna elements in addition we deduce simple bound approximations in the high snr regime and demonstrate that the effects of small and large scale fading are decoupled a similar statistical analysis is carried out for mimo channels under k fading which represents a special case of generalized k fading that can be tackled via the wishart matrix theory the implications of the model parameters on the bound performance are also investigated via monte carlo simulations

max plus a b invariant spaces and control of timed discrete event systems the concept of a b invariant subspace or controlled invariant of a linear dynamical system is extended to linear systems over the max plus semiring although this extension presents several difficulties which are similar to those encountered in the same kind of extension to linear dynamical systems over rings it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings sufficient conditions are given for computing the maximal a b invariant subspace contained in a given space and the existence of linear state feedbacks is discussed an application to the study of transportation networks which evolve according to a timetable is considered

a note on an implementation of a method for quadratic semi infinite programming for convex quadratic semi infinite programming problems afortran package is described a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements powell s 1983 numerically stable convex quadratic programming implementation is used to solve the qp subproblems

outage mutual information of space time mimo channels we derive analytical expressions for the probability density function pdf of the random mutual information between transmitted and received vector signals of a random space time independent and identically distributed i i d multiple input multiple output mimo channel assuming that the transmitted signals from the multiple antennas are gaussian i i d we show that this pdf can be well approximated by a gaussian distribution and such a gaussian approximation is based on expressions for the given pdfs mean and variance that we derive we prove that at high signal to noise ratio snr every factor of 2 increase in snr leads to an increase in outage rate in the amount of min m n bits where m and n denote the number of transmit and receive antennas respectively a simple expression for the moment generating function mgf of the mutual information pdf is also provided based on which we establish normality of the pdf when both m and n are large and the snr is large

a globally convergent algorithm with adaptively refined discretization for semi infinite optimization problems arising in engineering design although most of the algorithms that have been proposed for the solution of semi infinite optimization problems make use at each iteration of a set of local maximizers over the range of the independent parameter the question of suitably approximating such maximizers is generally left aside it has been suggested that this issue can be addressed by means of an adaptively refined discretization of the interval of variation of the independent parameter the algorithm proposed in the paper makes use of such a technique and by means of a certain memory mechanism avoids the potential lack of convergence suffered by an existing algorithm while requiring a relatively small number of gradient evaluations

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

a projected lagrangian algorithm for semi infinite programming a globally convergent algorithm is presented for the solution of a wide class of semi infinite programming problems the method is based on the solution of a sequence of equality constrained quadratic programming problems and usually has a second order convergence rate numerical results illustrating the method are given

an interior point sequential quadratic programming type method for log determinant semi infinite programs abstract in this paper we consider a nonlinear semi infinite program that minimizes a function including a log determinant logdet function over positive definite matrix constraints and infinitely many convex inequality constraints we call this problem siplog where sip stands for semi infinite program and log comes from log det function the main purpose of the paper is to develop an algorithm for efficiently computing a karush kuhn tucker kkt point for the siplog more specifically we propose an interior point sequential quadratic programming type method that solves inexactly a sequence of semi infinite quadratic programs approximating the siplog furthermore to generate a search direction in the dual matrix space associated with the semi definite constraint we solve scaled newton equations that yield the family of monteiro zhang directions we prove that the proposed method weakly converges to a kkt point under some mild assumptions finally we conduct some numerical experiments to demonstrate the efficiency of the proposed method

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

on the use of e most active constraints in an exact penalty function method for nonlinear optimization this note presents an algorithm for nonlinear programming problems which utilizes the e most active constraint strategy in an exact penalty function method with trust region the algorithm is particularly suitable for problems containing a large number of constraints the global convergence of the proposed algorithm is proved the results of limited computational experiments for discretized semi infinite programming problems are also reported to demonstrate the effectiveness of the present approach

data throughputs of noise limited mimo cellular systems we present the closed form approximated expressions of the cumulative distribution function of the spectral efficiency for noise limited multiple input multiple output mimo cellular systems with minimum mean square error based ordered successive interference cancellation mmse osic or zero forcing zf receivers in a composite channel model which includes both rayleigh and log normal fading when the users have independent and identically distributed i i d channels we analyze the spectral efficiency of the mimo time division multiple access tdma systems with maximum data rate scheduler numerical experiments show that these results can consistently produce highly accurate results

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

an approximation technique for the numerical solution of a stefan problem a nonlinear approximation technique for the numerical solution of certain free boundary problems is proposed the method is shown for a degenerate one dimensional stefan problem for this problem an error estimate which is independent of the used algorithm is derived numerical examples are discussed

an accelerated central cutting plane algorithm for linear semi infinite programming an algorithm for linear semi infinite programming is presented which accelerates the convergence of the central cutting plane algorithm first proposed in 4 compared with other algorithms the algorithm in 4 has the advantage of being applicable under mild conditions and of providing feasible solutions however its convergence has been shown to be rather slow in practical instances the algorithm proposed in this paper introduces a simple acceleration scheme which gives faster convergence as confirmed by several examples as well as an interval of prefixed length containing the optimum value it is also shown that the algorithm provides a solution of the dual problem and that it can be used for convex semi infinite programming too

a method for detecting damage induced nonlinearities in structures using information theory abstract in this work a new approach is presented for detecting the presence of damage induced nonlinearities in structures from measurements of structural dynamics two different information theoretic it measures the time delayed mutual information and the time delayed transfer entropy are used to provide a probabilistic measure of the coupling between structural components these measures may be used to capture both linear and nonlinear relationships among time series data the formula for both quantities is derived for a linear five degree of freedom system subject to gaussian excitation an algorithm is then described for computing the it metrics from time series data and results are shown to agree with theory we then show that as the coupling between the structure s components changes from linear to nonlinear the information flow can be used to indicate the degree of nonlinearity deviations from a linear model are quantified statistically by generating surrogate data sets that by construction possess only linear second order correlations we then apply the proposed algorithms to both the original data and the surrogates differences in the results are shown to be proportional to the degree of nonlinearity this result is shown to be independent of global changes in stiffness and is therefore unaffected by certain models of environmental variability furthermore the method provides an absolute measure of nonlinearity and therefore does not require a baseline data set for making comparisons this approach is discussed in the context of structural health monitoring where damage is often associated with structural nonlinearity

a one phase algorithm for semi infinite linear programming we present an algorithm for solving a large class of semi infinite linear programming problems this algorithm has several advantages it handles feasibility and optimality together it has very weak restrictions on the constraints it allows cuts that are not near the most violated cut and it solves the primal and the dual problems simultaneously we prove the convergence of this algorithm in two steps first we show that the algorithm can find ane optimal solution after finitely many iterations then we use this result to show that it can find an optimal solution in the limit we also estimate how good ane optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding ane optimal solution under some assumptions this algorithm is generalized to solve a class of nonlinear semi infinite programming problems applications to convex programming are discussed

the maximum entropy method on the mean applications to linear programming and superresolution in this paper we give two different results we propose new methods to solve classical optimization problems in linear programming we also obtain precise quantitative results for the superresolution phenomenon as observed earlier by practical searchers on specific algorithms the common background of our work is the generalized moment problem which is known to be connected with linear programming and superresolution we describe the maximum entropy method on the mean that provides solution to the problem and leads to computational criteria to decide the existence of solutions or not

solving semi infinite optimization problems with interior point techniques we introduce a new numerical solution method for semi infinite optimization problems with convex lower level problems the method is based on a reformulation of the semi infinite problem as a stackelberg game and the use of regularized nonlinear complementarity problem functions this approach leads to central path conditions for the lower level problems where for a given path parameter a smooth nonlinear finite optimization problem has to be solved the solution of the semi infinite optimization problem then amounts to driving the path parameter to zero we show convergence properties of the method and give a number of numerical examples from design centering and from robust optimization where actually so called generalized semi infinite optimization problems are solved the presented method is easy to implement and in our examples it works well for dimensions of the semi infinite index set at least up to 150

impact of scattering on the capacity diversity and propagation range of multiple antenna channels the impact of scattering condition and array configuration on performances are inseparable in early analyses of multiple antenna systems an array independent scattering model is introduced where three basic scattering mechanisms are modeled performance results become more intrinsic property of the scattering channel itself for linear arrays of length l in an environment of total angle spread omega the ergodic capacity is shown to increase linearly with l omega for large arrays when antenna arrays reduce to practical sizes the capacity scaling depends on the signal to noise ratio snr as well this implies that the number of antennas used should also depend on the snr in terms of outage capacity the tradeoff between spatial multiplexing gain and diversity gain is shown to be very sensitive to the underlying scattering mechanisms finally as omega varies with the propagation range the tradeoff among multiplexing gain diversity gain and propagation range is studied

an implementation of a discretization method for semi infinite programming for linear semi infinite programming problems a discretization method is presented a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements a numerically stable simplex algorithm is used to solve the lp subproblems numerical examples from bivariate chebyshev approximation are presented

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

a standard branch and bound approach for nonlinear semi infinite problems this paper considers nonlinear semi infinite problems which contain at least one semi infinite constraint sic the standard branch and bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to sics this is achieved by defining the interval evaluation of parametrized functions and their generalized gradients by also adapting numerical constraint programming techniques to quantified inequalities and by introducing linear relaxations and restrictions for sics the overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature in comparison with the best state of the art alternative

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

generalized semi infinite programming a tutorial this tutorial presents an introduction to generalized semi infinite programming gsip which in recent years became a vivid field of active research in mathematical programming a gsip problem is characterized by an infinite number of inequality constraints and the corresponding index set depends additionally on the decision variables there exist a wide range of applications which give rise to gsip models some of them are discussed in the present paper furthermore geometric and topological properties of the feasible set and in particular the difference to the standard semi infinite case are analyzed by using first order approximations of the feasible set corresponding constraint qualifications are developed then necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so called lower level problem are used finally an overview of numerical methods is given

a cutting surface algorithm for semi infinite convex programming with an application to moment robust optimization we present and analyze a central cutting surface algorithm for general semi infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments moments of arbitrary order as well as nonpolynomial moments can be included in the formulation we show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk aversion with classic robust optimization at one end of the spectrum and stochastic programming at the other although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty the cutting surface method for general semi infinite convex programs is also of independent interest the proposed method is applicable to problems with nondifferentiable semi infinite constraints indexed by an infinite dimensional index set examples comparing the cutting surface algorit

penalty and smoothing methods for convex semi infinite programming in this paper we consider min max convex semi infinite programming to solve these problems we introduce a unified framework concerning remez type algorithms and integral methods coupled with penalty and smoothing methods this framework subsumes well known classical algorithms but also provides some new methods with interesting properties convergence of the primal and dual sequences are proved under minimal assumptions

a truncated projected newton type algorithm for large scale semi infinite programming in this paper a truncated projected newton type algorithm is presented for solving large scale semi infinite programming problems this is a hybrid method of a truncated projected newton direction and a modified projected gradient direction the truncated projected newton method is used to solve the constrained nonlinear system in order to guarantee global convergence a robust loss function is chosen as the merit function and the projected gradient method inserted is used to decrease the merit function this algorithm is suitable for handling large scale problems and possesses superlinear convergence rate the global convergence of this algorithm is proved and the convergence rate is analyzed the detailed implementation is discussed and some numerical tests for solving large scale semi infinite programming problems with examples up to 2000 decision variables are reported

ergodic capacity upper bound for dual mimo ricean systems simplified derivation and asymptotic tightness an analytical upper bound on the ergodic capacity of multiple input multiple output mimo systems is deduced with the aid of a simplified approach that relies on a fundamental power normalization given their high practical usability we are particularly interested in dual configurations where both ends deploy two antenna elements contrary to the majority of related studies where only the common case of rayleigh fading is considered our analysis is extended to account for ricean fading where a deterministic line of sight los component exists in the communication link and both ends are affected by spatial correlation in the following it is shown that the proposed bound is applicable for any arbitrary signal to noise ratio snr and rank of the mean channel matrix in fact we consider both conventional and optimized mimo configurations with equal los eigenvalues moreover the tightness of the bound is explored where it is demonstrated that as the snr tends to zero the bound becomes asymptotically tight at high snrs the offset between empirical capacity and the bound is analytically computed which implies that an explicit asymptotic capacity expression can ultimately be obtained

relaxation based bounds for semi infinite programs finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi infinite programs sips involving smooth potentially nonconvex objective function and constraints the lower bounding problem is obtained by a formulation that combines the first and second order kkt necessary conditions of the lower level problem with a discretization of the parameter set the resulting mathematical program with equilibrium constraints mpec is a relaxation of the original sip and furnishes valid lower bounds if the parameter set is subdivided the optimal solution value of the lower bounding problem converges to the optimal solution value of the sip the upper bounding problem is based on convex and linear relaxations of the lower level problem resulting in a restriction of the sip if the parameter set is subdivided the constructed relaxations converge to the original lower level program the existence of sip slater points ensures convergence of the upper bounding problems to the optimal solution value of the sip several alternatives for the upper bounding problem are presented and discussed numerical results are presented for a number of test problems from the literature

a nonmonotonic trust region algorithm for a class of semi infinite minimax programming based on the discretization methods for solving semi infinite programming problems this paper presents a new nonmonotonic trust region algorithm for a class of semi infinite minimax programming problem under some mild assumptions the global convergence of the proposed algorithm is given numerical tests are reported that show the efficiency of the proposed method