an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

on robust hurwitz polynomials in this note kharitonov s theorem on robust hurwitz polynomials is simplified for low order polynomials specifically for n 3 4 and 5 the number of polynomials required to check robust stability is one two and three respectively instead of four furthermore it is shown that for n geq 6 the number of polynomials for robust stability checking is necessarily four thus further simplification is not possible the same simplifications arise in robust schur polynomials by using the bilinear transformation applications of these simplifications to two dimensional polynomials as well as to robustness for single parameters are indicated

designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

space time transmit code and receive filter design for colocated mimo radar this paper deals with the design of multiple input multiple output radar space time transmit code sttc and space time receive filter strf to enhance moving targets detection in the presence of signal dependent interferences an iterative procedure whose convergence is analytically proved is devised to maximize the signal to interference plus noise ratio sinr accounting for both a similarity constraint and a constant modulus requirement on the probing waveform each iteration of the algorithm involves the solution of hidden convex problems specifically both a convex problem whose solution is provided in closed form and a set of fractional programming problems that can be globally solved in polynomial time via the dinkelback s procedure are settled the computational complexity is linear with the number of iterations and polynomial with the sizes of the sttc and the strf in particular the proposed technique provides a monotonic sinr improvement without limitations on the size of the similarity constraint and ensures convergence to a stationary point filling these important gaps in the open literature besides the reported results highlight that the new devised procedure outperforms both in the optimized sinr value and the computational complexity than the available counterparts

fast unit modulus least squares with applications in beamforming unit modulus least squares uls problems arise in many applications including phase only beamforming sensor network localization synchronization phase retrieval and radar code design uls formulations can always be recast as unit modulus quadratic programs to which semidefinite relaxation sdr can be applied and is often the state of the art approach however sdr lifts the problem dimension i e the number of variables from n to n 2 which drastically increases the memory burden and computational cost when the problem size is already large e g when designing phase only beamformer weights for massive multiple input multiple output systems this paper focuses on scalable first order algorithms for the uls problem and some of its variants it advocates using simple gradient projection gp as a starting point for solving the uls problem establishes global convergence of gp to a karush kuhn tucker point for this np hard problem and bounds its iteration complexity then it proposes uls extensions tailored to reflect practical beamformer design objectives bringing in and exploiting new degrees of freedom to improve the beampattern designs simple variants of gp are proposed to deal with these extended uls problems simulations are used to showcase the effectiveness of the proposed algorithms in both the plain uls problem and in the context of phase only beamforming

testing normality in designs with many parameters goodness of fit tests are proposed for the assumption of normality of random errors in experimental designs where the variance of the response may vary with the levels of the covariates the exact distribution of standardized residuals is used to make the probability integral transform for use in tests based on the empirical distribution function a different mean and variance is estimated for each level of the covariate corresponding large sample theory is provided the proposed tests are robust to a possible misspecification of the model and permit data collected from several similar experiments to be pooled to improve the power of the test

hypothesis testing in linear regression when k n is large abstract this paper derives the asymptotic distribution of the f test for the significance of linear regression coefficients as both the number of regressors k and the number of observations n increase together so that their ratio remains positive in the limit the conventional critical values for this test statistic are too small and the standard version of the f test is invalid under this asymptotic theory this paper provides a correction to the f statistic that gives correctly sized tests both under this paper s limit theory and also under conventional asymptotic theory that keeps k finite this paper also presents simulations that indicate the new statistic can perform better in small samples than the conventional test the statistic is then used to reexamine olivei and tenreyro s results from olivei g tenreyro s 2007 the timing of monetary policy shocks the american economic review 97 636 663 and sala i martin s results from sala i martin x x 1997 i just ran two million regressions the american economic review 87 2 178 183

a unified framework for low autocorrelation sequence design via majorization minimization in this paper we consider the low autocorrelation sequence design problem we optimize a unified metric over a general constraint set the unified metric includes the integrated sidelobe level isl and the peak sidelobe level psl as special cases and the general constraint set contains the unimodular constraint peak to average ratio par constraint and similarity constraint to name a few the optimization technique we employ is the majorization minimization mm method which is iterative and enjoys guaranteed convergence to a stationary solution we carry out the mm method in two stages in the majorization stage we propose three majorizing functions two for the unified metric and one for the isl metric in the minimization stage we give closed form solutions for algorithmic updates under different constraints the update step can be implemented with a few fast fourier transformations ffts and or inverse ffts iffts we also show the connections between the mm and gradient projection method under our algorithmic scheme numerical simulations have shown that the proposed mm based algorithms can produce sequences with low autocorrelation and converge faster than the traditional gradient projection method and the state of the art algorithms

design of par constrained sequences for mimo channel estimation via majorization minimization communication systems and radars widely employ sequences of low peak to average power ratio par or unimodulus to meet the hardware constraints and maximize the power efficiency numerous works have probed the unimodular sequence design especially attempting to obtain good correlation properties regarding channel estimation however sequences of such properties do not necessarily qualify for the mission and tailored unimodular sequences for the specific criterion concerned are more desirable when we have access to the prior knowledge of the channel impulse response in this paper we formulate the problem of unimodular sequence design by optimizing minimum mean square error and conditional mutual information respectively the problems turn out nonconvex and we develop efficient algorithms based on the majorization minimization framework with convergence guaranteed more general we also examine optimal sequence design with low par constraints numerical examples demonstrate the improved results of mean square error signal to noise ratio and conditional mutual information by using our proposed training sequences with the efficiency of the derived algorithms illustrated

quadratic maximization and semidefinite relaxation in this paper we study a class of quadratic maximization problems and their semidefinite programming sdp relaxation for a special subclass of the problems we show that the sdp relaxation provides an exact optimal solution another subclass which is hard guarantees that the sdp relaxation yields an approximate solution with a worst case performance ratio of 0 87856 this is a generalization of the well known result of goemans and williamson for the maximum cut problem finally we discuss extensions of these results in the presence of a certain type of sign restrictions

hermitian completely positive matrices abstract in this paper we introduce the hermitian completely positive hcp matrix which has a hermitian completely positive hcp decomposition with all real and imaginary parts of the decomposition vectors being nonnegative some properties of the hermitian completely positive matrix are given a semidefinite algorithm is also proposed for checking whether a hermitian matrix is hcp or not if a matrix is not hcp a certificate for it can be obtained if it is an hcp decomposition can be obtained

robust strict positive realness new results for interval plant controller families a frequency domain approach is used to derive several robust strict positive realness results for interval plants and interval plant plus controller families of transfer functions based on simple frequency domain properties of transfer functions the approach provides a framework for obtaining new results and constructing easy proofs of several important existing results on robust strict positive realness the main new result states that the minimum of the real part of the transfer functions belonging to an interval plant controller family is achieved on one of the 32 kharitonov segments of the interval plant the argument used in the proof is of wider interest and suggests easy ways of proving that robustness of other different frequency properties of interval plant plus controller families of transfer functions such as robust stability of h sub infinity norm computation can be deduced from a fixed number of segments of transfer functions of the family

a coordinate descent framework to design low psl isl sequences this paper is focused on the design of phase sequences with good aperiodic autocorrelation properties in terms of peak sidelobe level and integrated sidelobe level the problem is formulated as a biobjective pareto optimization forcing either a continuous or a discrete phase constraint at the design stage an iterative procedure based on the coordinate descent method is introduced to deal with the resulting optimization problems that are nonconvex and np hard in general each iteration of the devised method requires the solution of a nonconvex min max problem it is handled either through a novel bisection or an fft based method respectively for the continuous and the discrete phase constraint additionally a heuristic approach to initialize the procedures employing the l p norm minimization technique is proposed simulation results illustrate that the proposed methodologies can outperform some counterparts providing sequences with good autocorrelation features especially in the discrete phase binary case

designing unimodular codes via quadratic optimization the np hard problem of optimizing a quadratic form over the unimodular vector set arises in radar code design scenarios as well as other active sensing and communication applications to tackle this problem which we call unimodular quadratic program uqp several computational approaches are devised and studied power method like iterations are introduced for local optimization of uqp furthermore a monotonically error bound improving technique merit is proposed to obtain the global optimum or a local optimum of uqp with good sub optimality guarantees the provided sub optimality guarantees are case dependent and may outperform the 4 approximation guarantee of semi definite relaxation several numerical examples are presented to illustrate the performance of the proposed method the examples show that for several cases including rank deficient matrices the proposed methods can solve uqps efficiently in the sense of sub optimality guarantee and computational time

goodness of fit tests for mixed model diagnostics a simple goodness of fit test is proposed for checking distributional assumptions involved in a mixed linear model an estimated critical value of the test statistic is derived and is shown to be asymptotically correct under mild conditions as a special case the test may be applied to linear regression models to formally check distribution of the errors finite sample performance of the proposed test is examined and compared with that of a previously proposed test by simulations

design of phase codes for radar performance optimization with a similarity constraint this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity constraint with a prefixed radar code this is tantamount to forcing a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence we consider the cases of both continuous and finite phase alphabet and formulate the code design in terms of a nonconvex np hard quadratic optimization problem in order to approximate the optimal solutions we propose techniques with polynomial computational complexity based on the method of semidefinite program sdp relaxation and randomization moreover we also derive approximation bounds yielding a ldquomeasure of goodnessrdquo of the devised algorithms at the analysis stage we assess the performance of the new encoding techniques both in terms of detection performance and ambiguity function under different choices for the similarity parameter we also show that the new algorithms achieve an accurate approximation of the optimal solution with a modest number of randomizations

mimo radar waveform design with constant modulus and similarity constraints we consider the problem of waveform design for multiple input multiple output mimo radar in the presence of signal dependent interference embedded in white gaussian disturbance we present two sequential optimization procedures to maximize the signal to interference plus noise ratio sinr accounting for a constant modulus constraint as well as a similarity constraint involving a known radar waveform with some desired properties e g in terms of pulse compression and ambiguity the presented sequential optimization algorithms based on a relaxation method yield solutions with good accuracy their computational complexity is linear in the number of iterations and trials in the randomized procedure and polynomial in the receive filter length finally we evaluate the proposed techniques by considering their sinr performance beam pattern as well as pulse compression property via numerical simulations

constructing binary sequences with good correlation properties an efficient analytical computational interplay binary sequence sets with asymptotically optimal auto cross correlation peak sidelobe level psl growth have been known in the literature for a long time and their construction has been studied both analytically and numerically in contrast it has been a long standing problem whether we can construct a family of binary sequences whose auto correlation psl grows in an optimal manner in this paper we devise a construction method of binary sequences with asymptotically optimal psl growth from the sequence sets with good correlation properties a key component of the design follows from the observation that if the psl of the sequence set grows optimally or nearly optimally then the psl of the constructed binary sequence will experience a similar growth as a consequence the proposed construction is simple to implement and is shown to be accomplished in polynomial time with such a construction we not only bridge the gap between analytical construction and computational search but also pave the way to settle the long standing design problem of binary sequences with an optimal growth of the auto correlation psl

successive qcqp refinement for mimo radar waveform design under practical constraints the authors address the problem of designing a waveform for multiple input multiple output mimo radar under the important practical constraints of constant modulus and waveform similarity incorporating these constraints in an analytically tractable manner is a longstanding open challenge this is due to the fact that the optimization problem that results from signal to interference plus noise ratio sinr maximization subject to these constraints is a hard non convex problem the authors develop a new analytical approach that involves solving a sequence of convex quadratically constrained quadratic programing qcqp problems which they prove converges to a sub optimal solution because an improvement in sinr results via solving each problem in the sequence they call the method successive qcqp refinement sqr furthermore the proposed sqr method can be easily extended to incorporate emerging requirements of spectral coexistence as shown briefly in this paper the authors evaluate sqr against other candidate techniques with respect to sinr performance beam pattern and pulse compression properties in a variety of scenarios results show that sqr outperforms state of the art methods that also employ constant modulus and or similarity constraints while being computationally less burdensome

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

residual bootstrap tests in linear models with many regressors abstract this paper is concerned with bootstrap hypothesis testing in linear regression models with many regressors i show that bootstrap f l r and l m tests are asymptotically valid even when the numbers of estimated parameters and tested restrictions are not asymptotically negligible fractions of the sample size one of the conditions for these results is that the regressors come from an asymptotically balanced design depending on the number of restrictions tested and on the errors distribution violation of that condition might render the bootstrap tests asymptotically invalid in that case i propose bootstrapping calhoun s 2011 g statistic or modified versions of the l r and l m statistics and show that these procedures remain asymptotically valid monte carlo simulations indicate that the bootstrap tests often outperform asymptotic ones however analyzing the approximate third cumulant of the f statistic reveals that the bootstrap test does not generally provide the usual higher order asymptotic refinements nevertheless it is found that the bootstrap third cumulant partially matches the population third cumulant which might explain the bootstrap s good finite sample performances especially when the errors come from a symmetric distribution

robust adaptive beamforming based on steering vector estimation with as little as possible prior information a general notion of robustness for robust adaptive beamforming rab problem and a unified principle for minimum variance distortionless response mvdr rab techniques design are formulated this principle is to use standard mvdr beamformer in tandem with an estimate of the desired signal steering vector found based on some imprecise prior information differences between various mvdr rab techniques occur only because of the differences in the assumed prior information and the corresponding signal steering vector estimation techniques a new mvdr rab technique which uses as little as possible and easy to obtain imprecise prior information is developed the objective for estimating the steering vector is the maximization of the beamformer output power while the constraints are the normalization condition and the requirement that the estimate does not converge to any of the interference steering vectors and their linear combinations the prior information used is only the imprecise knowledge of the antenna array geometry and angular sector in which the actual steering vector lies mathematically the proposed mvdr rab is expressed as the well known non convex quadratically constrained quadratic programming problem with two constraints which can be efficiently and exactly solved some new results for the corresponding optimization problem such as a new algebraic way of finding the rank one solution from the general rank solution of the relaxed problem and the condition under which the solution of the relaxed problem is guaranteed to be rank one are derived our simulation results demonstrate the superiority of the proposed method over other previously developed rab techniques

inhomogeneous polynomial optimization over a convex set an approximation approach in this paper we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set the focus is on the design and analysis of polynomial time approximation algorithms the methods are able to deal with optimization models with polynomial objective functions in any fixed degrees in particular we first study the problem of maximizing an inhomogeneous polynomial over the euclidean ball a polynomial time approximation algorithm is proposed for this problem with an assured relative worst case performance ratio which is dependent only on the dimensions of the model the method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co centered ellipsoids furthermore the constraint set is extended to a general convex compact set specifically we propose a polynomial time approximation algorithm with a relative worst case performance ratio for polynomial optimization over some convex compact sets e g a polytope finally numerical results are reported revealing good practical performance of the proposed algorithms for solving some randomly generated instances

parameter space methods for robust control design a guided tour the results obtained in studies of robust stability and stabilizability of control systems with parametric structured uncertainties are reviewed both the algebraic methods based upon characteristic equations and the methods using lyapunov functions and riccati equations are discussed and compared in the context of algebraic methods most promising are the kharitonov type approach and the optimization procedure of embedding a geometric figure of some kind inside the stability regions of the parameter space maximizing its size using minimax or some other mathematical programming technique in the framework of lyapunov s direct method the dominant approach has been a quadratic function estimation of stability regions in the parameter space in large sale systems the concept of vector lyapunov functions has been used with the possibility of choosing quadratic forms norm like functions and their combinations

stability of families of polynomials geometric considerations in coefficient space the stability robustness of both continuous and discrete time dynamical systems is tantamount to the stability of a whole family of characteristic polynomials the authors treat such families as sets in coefficient space and examine some of their geometric properties specifically they examine cases where the stability of the extreme points of a set implies the stability of the complete family they are also concerned with the construction and convexity of the stability region in coefficient space

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

estimating the innovation distribution in nonparametric autoregression we prove a bahadur representation for a residual based estimator of the innovation distribution function in a nonparametric autoregressive model the residuals are based on a local linear smoother for the autoregression function our result implies a functional central limit theorem for the residual based estimator

on binary probing signals and instrumental variables receivers for radar the so called merit factor approach mfa to radar binary sequence design has led to several theoretical contributions in fairly diverse research areas including information theory computer science combinatorial optimization and analytical number theory however the mfa which basically aims at minimizing the clutter effect on radar performance implicitly assumes the use of a least squares ls receiver that is optimal only when there is no clutter this problem can be eliminated by using a more general optimal instrumental variables iv receiver in lieu of the ls receiver the iv receiver can reject clutter more efficiently than the ls receiver additionally the binary sequence design problem associated with the iv approach has an interesting form

on robust stability of interval control systems the authors develop results on the robust stability of a nonlinear control system containing both parametric as well as unstructured uncertainty the basic system considered is that of the classical lur e problem of nonlinear control theory a robust version of the lur e problem consisting of a family of linear time invariant systems subjected simultaneously to bounded parameter variations and feedback perturbations from a family of sector bounded nonlinear gains is presently treated by using the kharitonov theorem to develop some extremal results on positive realness of interval transfer functions i e a family of rational transfer functions with bounded independent coefficient perturbations the authors determine the size of a sector of nonlinear feedback gains for which absolute stability can be guaranteed these calculations amount to the determination of the stability margin of the system under joint parametric and nonlinear feedback perturbations

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

empirical process of residuals for high dimensional linear models we give a stochastic expansion for the empirical distribution function f n of residuals in a p dimensional linear model this expansion holds for p increasing with n it shows that for high dimensional linear models f n strongly depends on the chosen estimator of the parameter of the linear model in particular if one uses an ml estimator ml which is motivated by a wrongly specified error distribution function g then f n is biased toward g for p 2 n this bias effect is of larger order than the stochastic fluctuations of the empirical process hence the statistical analysis may just reproduce the assumptions imposed

new algorithms for designing unimodular sequences with good correlation properties unimodular i e constant modulus sequences with good autocorrelation properties are useful in several areas including communications and radar the integrated sidelobe level isl of the correlation function is often used to express the goodness of the correlation properties of a given sequence in this paper we present several cyclic algorithms for the local minimization of isl related metrics these cyclic algorithms can be initialized with a good existing sequence such as a golomb sequence a frank sequence or even a pseudo random sequence to illustrate the performance of the proposed algorithms we present a number of examples including the design of sequences that have virtually zero autocorrelation sidelobes in a specified lag interval and of long sequences that could hardly be handled by means of other algorithms previously suggested in the literature

relative entropy based waveform design for mimo radar detection in the presence of clutter and interference this paper considers the waveform design problem for multiple input multiple output mimo radars in order to improve the detection performance of the systems we assume that target echoes are embedded in signal dependent clutter as well as colored interference considering the optimal neyman pearson detector obtaining waveforms which maximize the detection probability for a fixed value of the probability of false alarm is intractable therefore we employ relative entropy associated with the detection problem as the figure of merit for the waveform design we devise an iterative method based on minorization maximization mm technique to tackle the nonconvex design problem this method also includes a novel trick for replacing a nonconvex constraint set associated with an equivalent form of the design problem with a convex one iteratively the proposed method increases the design metric monotonically and is guaranteed to converge we extend the devised method for using in design with peak to average power ratio par and similarity constraints the method can be applied to both statistical and colocated mimo radars several numerical examples are included to demonstrate the effectiveness of the proposed method

convexity properties of polynomials with assigned root location this paper presents some geometrical results on the domain of generalized stability for a family of nth order polynomials regions of root location such that the convex hull of the corresponding domain of stability in the coefficient space is a polyhedron are investigated and specific regions for which the convex hull is an n 1 vertex polyhedron are derived the discrete time stability domain falls in the latter class of regions implications of the results for the design of filters solving the robust strict positive realness for families of rational transfer functions with uncertainty in the numerator are also developed

design of optimized radar codes with a peak to average power ratio constraint this paper considers the problem of radar waveform design in the presence of colored gaussian disturbance under a peak to average power ratio par and an energy constraint first of all we focus on the selection of the radar signal optimizing the signal to noise ratio snr in correspondence of a given expected target doppler frequency algorithm 1 then through a max min approach we make robust the technique with respect to the received doppler algorithm 2 namely we optimize the worst case snr under the same constraints as in the previous problem since algorithms 1 and 2 do not impose any condition on the waveform phase we also devise their phase quantized versions algorithms 3 and 4 respectively which force the waveform phase to lie within a finite alphabet all the problems are formulated in terms of nonconvex quadratic optimization programs with either a finite or an infinite number of quadratic constraints we prove that these problems are np hard and hence introduce design techniques relying on semidefinite programming sdp relaxation and randomization as well as on the theory of trigonometric polynomials providing high quality suboptimal solutions with a polynomial time computational complexity finally we analyze the performance of the new waveform design algorithms in terms of detection performance and robustness with respect to doppler shifts

blind ml detection of orthogonal space time block codes efficient high performance implementations orthogonal space time block codes ostbcs have attracted much attention owing to their simple code construction maximal diversity gain and low maximum likelihood ml detection complexity when channel state information csi is available at the receiver this paper addresses the problem of ml ostbc detection with unknown csi focusing on the binary and quaternary psk constellations we show that blind ml ostbc detection can be simplified to a boolean quadratic program bqp from an optimization viewpoint the bqp is still a computationally hard problem and we propose two alternatives for dealing with this inherent complexity first we consider the semidefinite relaxation sdr approach which leads to a suboptimal but accurate blind ml detection algorithm with an affordable worst case computational cost we also consider the sphere decoding approach which leads to an exact blind ml detection algorithm that remains computationally expensive in the worst case but generally incurs a reasonable average computational cost for the two algorithms we study implementation methods that can significantly reduce the computational complexity simulation results indicate that the two blind ml detection algorithms are competitive in that the bit error performance of the two algorithms is almost the same and is noticeably better than that of some other existing blind detectors moreover numerical studies show that the sdr algorithm provides better complexity performance than the sphere decoder in the worst case sense and vice versa in the average sense

code design to optimize radar detection performance under accuracy and similarity constraints this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of linearly coded pulse trains and determine the optimum radar code according to the following criterion maximization of the detection performance under a control on the region of achievable doppler estimation accuracies and imposing a similarity constraint with a prefixed radar code this last constraint is tantamount to requiring a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence the resulting optimization problem is nonconvex and quadratic in order to solve it we propose a technique with polynomial computational complexity based on the relaxation of the original problem into a semidefinite program thus the best code is determined through a rank one decomposition of an optimal solution of the relaxed problem at the analysis stage we assess the performance of the new encoding technique in terms of detection performance region of achievable doppler estimation accuracies and ambiguity function

sequence design to minimize the weighted integrated and peak sidelobe levels sequences with low aperiodic autocorrelation sidelobes are well known to have extensive applications in active sensing and communication systems in this paper we first consider the problem of minimizing the weighted integrated sidelobe level wisl which can be used to design sequences with impulse like autocorrelation and a zero or low correlation zone two algorithms based on the general majorization minimization method are developed to tackle the wisl minimization problem with guaranteed convergence to a stationary point the proposed methods are then extended to optimize the ell p norm of the autocorrelation sidelobes which leads to a way to minimize the peak sidelobe level psl criterion all the proposed algorithms can be implemented via the fast fourier transform fft and thus are computationally efficient an acceleration scheme is considered to further accelerate the algorithms numerical experiments show that the proposed algorithms can efficiently generate sequences with virtually zero autocorrelation sidelobes in a specified lag interval and can also produce very long sequences with much smaller psl compared with some well known analytical sequences

an information theoretic approach to robust constrained code design for mimo radars the problem of transmit code design in multiple input multiple output mimo radar systems is addressed in this paper the problem is considered in both colocated and widely separated antenna radars in the presence of signal dependent interference and target mobility due to complexity of exact expressions of the optimal detector in the mimo radars information theoretic criteria are employed as design metrics which results in a general form of nonconvex optimization problems in order to tackle the problem a novel technique based on the minorization maximization algorithm is proposed and solutions are presented under practical constraints on the transmit code namely energy constraint peak to average power ratio constraint and similarity constraint furthermore the devised method is extended to be robust against uncertainties of the clutter and interference statistics finally numerical examples are used to show the performance of the proposed technique in different situations

robust absolute stability of lur e control systems in parameter space abstract this paper deals with the problem of robust absolute stability analysis for nonlinear lur e control systems in the presence of system parameter variations the well known popov criterion for absolute stability is used in order to characterize the boundary of the region of absolute stability in the parameter plane when the coefficients of the transfer function of the linear plant are polynomial functions of the uncertain parameters for a scalar parameter a method is given to determine the maximal interval of variation around a fixed nominal value preserving absolute stability this result is also used to derive a technique for checking absolute stability of lur e systems with parameters in given planar uncertainty sets numerical examples showing the application of the method are reported

exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure this work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems including quadratic and polynomial optimization problems the main objective of this paper is to investigate how the hidden structure of a given real complex valued optimization problem makes it easy to solve to this end three conic relaxations are proposed necessary and sufficient conditions are derived for the exactness of each of these relaxations and it is shown that these conditions are satisfied if the optimization problem is highly structured more precisely the structure of the optimization problem is mapped into a generalized weighted graph where each edge is associated with a weight set extracted from the coefficients of the optimization problem in the real valued case it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph it is also proved that if some of these conditi

weak convergence of the empirical process of residuals in linear models with many parameters when fitting by least squares a linear model with an intercept term with p parameters to n data points the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided p 3 log 2 p n 0 for any error density having finite variance and a bounded first derivative no further conditions are imposed on the sequence of design matrices the result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

synthetic aperture radar autofocus via semidefinite relaxation the autofocus problem in synthetic aperture radar imaging amounts to estimating unknown phase errors caused by unknown platform or target motion at the heart of three state of the art autofocus algorithms namely phase gradient autofocus multichannel autofocus mca and fourier domain multichannel autofocus fmca is the solution of a constant modulus quadratic program cmqp currently these algorithms solve a cmqp by using an eigenvalue relaxation approach we propose an alternative relaxation approach based on semidefinite programming which has recently attracted considerable attention in other signal processing problems experimental results show that our proposed methods provide promising performance improvements for mca and fmca through an increase in computational complexity

efficient computation of the binary vector that maximizes a rank deficient quadratic form the maximization of a full rank quadratic form over the binary alphabet can be performed through exponential complexity exhaustive search however if the rank of the form is not a function of the problem size then it can be maximized in polynomial time by introducing auxiliary spherical coordinates we show that the rank deficient quadratic form maximization problem is converted into a double maximization of a linear form over a multidimensional continuous set the multidimensional set is partitioned into a polynomial size set of regions which are associated with distinct candidate binary vectors and the optimal binary vector belongs to the polynomial size set of candidate vectors thus the size of the candidate set is reduced from exponential to polynomial we also develop an algorithm that constructs the polynomial size candidate set in polynomial time and show that it is fully parallelizable and rank scalable finally we demonstrate the efficiency of the proposed algorithm in the context of adaptive spreading code design

on meeting the peak correlation bounds in this paper we study the problem of meeting peak periodic or aperiodic correlation bounds for complex valued sets of sequences to this end the welch levenstein and exponential bounds on the peak inner product of sequence sets are considered and used to provide compound peak correlation bounds in both periodic and aperiodic cases the peak aperiodic correlation bound is further improved by using the intrinsic dimension deficiencies associated with its formulation in comparison to the compound bound the new aperiodic bound contributes an improvement of more than 35 for some specific values of the sequence length n and set cardinality m we study the tightness of the provided bounds by using both analytical and computational tools in particular novel algorithms based on alternating projections are devised to approach a given peak periodic or aperiodic correlation bound several numerical examples are presented to assess the tightness of the provided correlation bounds as well as to illustrate the effectiveness of the proposed methods for meeting these bounds

cognitive radar based sequence design via sinr maximization the ambiguity function plays an important role in radar systems in fact many radar design problems can be interpreted from the perspective of persuing desired ambiguity functions to adapt to various application scenes in this paper we consider designing a radar sequence subject to a peak to average power ratio par constraint to maximize the signal to interference plus noise ratio which can also be interpreted as designing a sequence with a desired ambiguity function from an optimization point of view this is equivalent to optimizing a complex quartic function with the par constraint an efficient algorithm based on the general majorization minimization mm method is developed to solve this problem with guaranteed convergence to a stationary point under some mild conditions in addition the unit modulus constraint as a special case is considered and another algorithm is proposed which is the combination of the general mm and the coordinate descent method numerical experiments show that the proposed algorithms can shape a desired ambiguity function based on the prior information and the performance is much better compared with the existing methods

constant modulus mimo radar waveform design with minimum peak sidelobe transmit beampattern this paper addresses the problem of minimum peak sidelobe transmit beampattern design for colocated multiple input multiple output radar systems two new methods are proposed to achieve it via designing constant modulus cm waveforms directly one is to maximize the ratio of the minimum mainlobe level to the peak sidelobe level psl without specifying pattern masks and another is to minimize psl under the mainlobe ripple and unspecified waveform modulus constraints the resultant optimization problems are difficult to solve due to the coupled numerator and denominator of the quadratic fractional programming problem formulation in the former and double sided quadratic constraints and unknown waveform modulus in the latter for the former we decouple the numerator and denominator by equivalent transformation and introduction of boundary type auxiliary variables in order that feasible methods are derived to tackle them independently and efficiently for the latter we simplify the coupled double sided quadratic constraints on the same waveform vector into those on different single auxiliary variables to tackle the corresponding nonconvex optimization problem numerical examples show that the proposed algorithms can attain sufficient low peak sidelobe beampattern levels under the cm constraints

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

semidefinite relaxation bounds for indefinite homogeneous quadratic optimization this paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming sdp relaxation we consider two quadratic optimization models 1 min x c x mid x a k x ge 1 k 0 1 ldots m x in mathbb f n and 2 max x c x mid x a k x le 1 k 0 1 ldots m x in mathbb f n where mathbb f is either the real field mathbb r or the complex field mathbb c and a k c are symmetric matrices for the minimization model 1 we prove that if the matrix c and all but one of the a k s are positive semidefinite then the ratio between the optimal value of 1 and its sdp relaxation is upper bounded by o m 2 when mathbb f mathbb r and by o m when mathbb f mathbb c moreover when two or more of the a k s are indefinite this ratio can be arbitrarily large for the maximization model 2 we show that if c and at most one of the a k s are indefinite while other a k s are positive semidefinite then the ratio between the optimal value of 2 and its sdp relaxation is bounded from below by o 1 log m for both the real and the complex case this result improves the bound based on the so called approximate s lemma of ben tal nemirovski and roos siam j optim 13 2002 pp 535 560 when two or more of the a k s in 2 are indefinite we derive a general bound in terms of the problem data and the sdp solution for both optimization models we present examples to show that the derived approximation bounds are essentially tight

ambiguity function shaping for cognitive radar via complex quartic optimization in this paper we propose a cognitive approach to design phase only modulated waveforms sharing a desired range doppler response the idea is to minimize the average value of the ambiguity function of the transmitted signal over some range doppler bins which are identified exploiting a plurality of knowledge sources from a technical point of view this is tantamount to optimizing a real and homogeneous complex quartic order polynomial with a constant modulus constraint on each optimization variable after proving some interesting properties of the considered problem we devise a polynomial time waveform optimization procedure based on the maximum block improvement mbi method and the theory of conjugate partial symmetric conjugate super symmetric fourth order tensors at the analysis stage we assess the performance of the proposed technique showing its capability to properly shape the range doppler response of the transmitted waveform

quadratic optimization with similarity constraint for unimodular sequence synthesis this paper considers unimodular sequence synthesis under similarity constraint for both the continuous and discrete phase cases a computationally efficient iterative algorithm for the continuous phase case ia cpc is proposed to sequentially optimize the quadratic objective function the quadratic optimization problem is turned into multiple one dimensional optimization problems with closed form solutions for the discrete phase case we present an iterative block optimization algorithm specifically we partition the design variables into k blocks and then we sequentially optimize each block via exhaustive search while fixing the remaining k 1 blocks finally we evaluate the computational costs and performance gains of the proposed algorithms in comparison with power method like and semidefinite relaxation related techniques

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

maximal rank of m x n x mn k tensors it is shown that the maximal rank of m times n times mn k tensors with k leqq min m 1 2 2 n 1 2 2 is greater than mn 4 sqrt 2k o 1

computational design of sequences with good correlation properties in this paper we introduce a computational framework based on an iterative twisted approximation itrox and a set of associated algorithms for various sequence design problems the proposed computational framework can be used to obtain sequences or complementary sets of sequences possessing good periodic or aperiodic correlation properties and in an extended form to construct zero or low correlation zone sequences furthermore as constrained e g finite alphabets are of interest in many applications we introduce a modified version of our general framework that can be useful in these cases several applications of itrox are studied and numerical examples focusing on the construction of real valued and binary sequences are provided to illustrate the performance of itrox for each application

a doppler robust max min approach to radar code design this correspondence considers the problem of robust waveform design in the presence of colored gaussian disturbance under a similarity and an energy constraint we resort to a max min approach where the worst case detection performance over the possible doppler shifts is optimized with respect to the radar waveform under the previously mentioned constraints the resulting optimization problem is a non convex quadratically constrained quadratic program qcqp with an infinite number of constraints which is np hard in general and typically difficult to solve hence we propose an algorithm with a polynomial computational complexity to generate a good sub optimal solution for the aforementioned qcqp the analysis conducted in comparison with some known radar waveforms shows that the sub optimal solutions by the algorithm lead to high quality radar signals

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

nonconvex phase synchronization we estimate n phases angles from noisy pairwise relative phase measurements the task is modeled as a nonconvex least squares optimization problem it was recently shown that this problem can be solved in polynomial time via convex relaxation under some conditions on the noise in this paper under similar but more restrictive conditions we show that a modified version of the power method converges to the global optimum this is simpler and empirically faster than convex approaches empirically they both succeed in the same regime further analysis shows that in the same noise regime as previously studied second order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient despite nonconvexity

approximation bounds for quadratic optimization with homogeneous quadratic constraints we consider the np hard problem of finding a minimum norm vector in n dimensional real or complex euclidean space subject to m concave homogeneous quadratic constraints we show that a semidefinite programming sdp relaxation for this nonconvex quadratically constrained quadratic program qp provides an o m 2 approximation in the real case and an o m approximation in the complex case moreover we show that these bounds are tight up to a constant factor when the hessian of each constraint function is of rank 1 namely outer products of some given so called steering vectors and the phase spread of the entries of these steering vectors are bounded away from pi 2 we establish a certain constant factor approximation depending on the phase spread but independent of m and n for both the sdp relaxation and a convex qp restriction of the original np hard problem finally we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints we show that an sdp relaxation for this nonconvex qp provides an o 1 ln m approximation which is analogous to a result of nemirovski e math program 86 1999 pp 463 473 for the real case

on the complexity of purely complex spl mu computation and related problems in multidimensional systems in this paper the following robust control problems are shown to be np hard given a purely complex uncertainty structure spl delta determine if 1 spl mu sub spl delta m 1 for a given rational matrix m 2 spl par m spl middot spl par sub spl mu 1 for a given rational transfer matrix m s and 3 inf q spl isin h sup spl infin spl par f t q spl par sub spl mu 1 for a given linear fractional transformation f t q with rational coefficients in other words purely complex spl mu computation analysis and synthesis problems are np hard it is also shown that checking stability and computing the h sup spl infin norm of a multidimensional system are np hard problems therefore it is rather unlikely to find nonconservative polynomial time algorithms for solving the problems in complete generality

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied