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

regularizations of residue currents under assumptions about complete intersection we prove that coleff herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the coleff herrera product of the currents

compactness of the neumann operator on singular complex spaces let x be a hermitian complex space of pure dimension n we show that the neumann operator on p q forms is compact at isolated singularities of x if p q n 1 n and q 1 the main step is the construction of compact solution operators for the equation on such spaces which is based on a general characterization of compactness in function spaces on singular spaces and that leads also to a criterion for compactness of more general green operators on singular spaces

explicit serre duality on complex spaces abstract in this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization as well as a new analytic proof of serre duality on any reduced pure n dimensional paracompact complex space x at the core of the paper is the introduction of certain fine sheaves b x n q of currents on x of bidegree n q such that the dolbeault complex b x n becomes in a certain sense a dualizing complex in particular if x is cohen macaulay then b x n is an explicit fine resolution of the grothendieck dualizing sheaf

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

l 2 serre duality on domains in complex manifolds and applications an l 2 version of the serre duality on domains in complex manifolds involving duality of hilbert space realizations of the operator is established this duality is used to study the solution of the equation with prescribed support applications are given to closed extension of forms as well to bochner hartogs type extension of cr functions comp e with the quotient topology where we endow spaces of compactly supported forms with the natural inductive limit topology in fact condition that the two maps in 1 have closed range is also necessary for the duality theorem to hold see 9 also see 26 27 28 for further results of this type serre s original proof 33 is based on sheaf theory and the theory of topological vector spaces a different approach to this result in the case when is a compact complex manifold was given by kodaira using hodge theory see 23 or 7 in this note we extend kodaira s method to non compact hermitian manifolds to obtain an l 2 analog of the serre duality special cases of serre duality using l 2 methods have appeared before in many contexts see 25 or 11 theorem 5 1 7 and 19 20 for example our treatment aims to streamline and systematize these results with emphasis on non compact manifolds and point out its close relation with the choice of l 2 realizations of the cauchy riemann operator or alternatively choice of boundary conditions for the l 2 realizations of the formal complex laplacian e e e e the l 2 duality can be interpreted in many ways at one level it is a duality between the standard laplacian with neumann boundary conditions and thec laplacian with dual dirichlet boundary conditions using another approach results regarding solution of the equation in l 2 can be converted to statements regarding the solution of the c equation this leads to a solution of the cauchy problem i e solution of the equation with prescribed support at the heart of the matter lies the existence of

pure hodge structure on the l_2 cohomology of varieties with isolated singularities cheeger goresky and macpherson conjectured in cgm an l2 de rham theorem that the intersection cohomology of a projective variety v is naturally isomorphic to the l2 cohomology of the incomplete manifold v singv with metric induced by a projective embedding the early interest in this conjecture was motivated in large part by the hope that one could then put a pure hodge structure on the intersection cohomology of v and even extend the rest of the kahler package cgm to this context saito s1 s2 eventually established the kahler package for intersection cohomology without recourse to l2 cohomology techniques however interest in l2 cohomology did not disappear with this result since among other things l2 cohomology provides intrinsic geometric invariants of an arbitrary complex projective variety which are not apparent from the point of view of d modules for instance l2 coholomology groups depend on boundary conditions ps which as we show here must be treated carefully in order to give the correct hodge components for the l2 cohomology of a singular variety a related fact is that for incomplete manifolds the pure hodge structure and lefschetz decompositions are not direct consequences of the kahler condition as they are in the compact case indeed the primary obstruction to obtaining a hodge structure on the l2 cohomology is the following apparent technicality on an incomplete kahler manifold there are several potentially distinct definitions of a square integrable harmonic form for example a form h might be considered harmonic if dh 0 h or if h 0 h or simply if h 0 moreover there are further domain considerations one imposes boundary conditions which turn out to have no effect on cohomology in the case of d but are crucial for cohomology on a compact or even complete manifold all these definitions of harmonics coincide and one obtains the pure hodge structure by decomposing harmonic forms into their p q components the p q components are harmonic in the weakest sense they are in the kernel of the equality of the different notions of harmonic then allows one to realize these p q components as spaces of both and d cohomology classes the equivalence of the different definitions of harmonic is also required in order to obtain the lefschetz decomposition a local computation shows that interior product with the kahler form preserves the kernel of but one requires the equivalence to see that this also induces an endomorphism on the l2 cohomology

abelian differentials on singular varieties and variations on a theorem of lie griffiths we compare various notions of meromorphic and holomorphic differential forms on singular analytic varieties in particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety demanding that this current be closed we obtain a useful analytic description of the dualizing sheaf we then go on to generalize the lie griffiths theorem on one hand it is shown that a rational trace is enough to ensure algebraicity of the data on the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents in fact we prove a more general local version of the theorem

on the l 2 overline partial cohomology of certain complete kahler metrics let v be a compact and irreducible complex space of complex dimension v whose regular part is endowed with a complete hermitian metric h let pi m rightarrow v be a resolution of v under suitable assumptions on h we prove that begin aligned h v q 2 overline partial mathrm reg v h cong h v q overline partial m quad q 0 ldots v end aligned then we show that the previous isomorphism applies to the case of saper type kahler metrics as introduced by grant melles and milman and to the case of complete kahler metrics with finite volume and pinched negative sectional curvatures

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

l 2 properties of the overline partial and the overline partial neumann operator on spaces with isolated singularities let x be a hermitian complex space of pure dimension with only isolated singularities and pi m rightarrow x a resolution of singularities let omega subset subset x be a domain with no singularities in the boundary omega omega setminus mathrm sing x and omega pi 1 omega we relate l 2 properties of the overline partial and the overline partial neumann operator on omega to properties of the corresponding operators on omega where the situation is classically well understood outside some middle degrees there are compact solution operators for the overline partial equation on omega exactly if there are such operators on the resolution omega and the overline partial neumann operator is compact on omega exactly if it is compact on omega

the bar partial equation on a non reduced analytic space let x be a possibly non reduced analytic space of pure dimension we introduce a notion of overline partial equation on x and prove a dolbeault grothendieck lemma we obtain fine sheaves mathcal a x q of 0 q currents so that the associated dolbeault complex yields a resolution of the structure sheaf mathscr o x our construction is based on intrinsic semi global koppelman formulas on x

various approaches to products of residue currents abstract we describe various approaches to coleff herrera products of residue currents r j of cauchy fantappie leray type associated to holomorphic mappings f j more precisely we study to which extent exterior products of natural regularizations of the individual currents r j yield regularizations of the corresponding coleff herrera products our results hold globally on an arbitrary pure dimensional complex space

l2 overline partial cohomology groups of some singular complex spaces let x be a pure n dimensional where n 2 complex analytic subset in n with an isolated singularity at 0 in this paper we express the l 2 0 q overline partial cohomology groups for all q with 1 q n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data we also obtain identifications of the l 2 0 q overline partial cohomology groups of the smooth points of x in terms of resolution data when x is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities

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

decomposition of residue currents given a submodule j o r 0 and a free resolution of j one can de ne a certain vector valued residue current whose annihilator is j we make a decomposition of the current with respect to assj that corresponds to a primary decomposition of j as a tool we introduce a class of currents that includes usual residue and principal value currents in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets

two undecidability results for chain code picture languages it is undecidable whether or not two 1 retreat bounded regular languages describe exactly the same set of pictures or they describe a picture in common

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

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

weighted koppelman formulas and the equation on an analytic space let x be an analytic space of pure dimension we introduce a formalism to generate intrinsic weighted koppelman formulas on x that provide solutions to the partial derivative over bar equation we obtain new existence results for the partial derivative over bar equation as well as new proofs of various known results c 2011 elsevier inc all rights reserved

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

degenerating hermitian metrics and spectral geometry of the canonical bundle abstract let x h be a compact and irreducible hermitian complex space of complex dimension m in this paper we are interested in the dolbeault operator acting on the space of l 2 sections of the canonical bundle of reg x the regular part of x more precisely let d m 0 l 2 m 0 reg x h l 2 m 1 reg x h be an arbitrarily fixed closed extension of m 0 l 2 m 0 reg x h l 2 m 1 reg x h where the domain of the latter operator is c m 0 reg x we establish various properties such as closed range of d m 0 compactness of the inclusion d d m 0 l 2 m 0 reg x h where d d m 0 the domain of d m 0 is endowed with the corresponding graph norm and discreteness of the spectrum of the associated hodge kodaira laplacian d m 0 d m 0 with an estimate for the growth of its eigenvalues several corollaries such as trace class property for the heat operator associated to d m 0 d m 0 with an estimate for its trace are derived finally in the last part we provide several applications to the hodge kodaira laplacian in the setting of both compact irreducible hermitian complex spaces with isolated singularities and complex projective surfaces

some l2 results for overline partial on projective varieties with general singularities let x be an irreducible n dimensional projective variety in bbb c p n with arbitrary singular locus we prove that the l 2 overline partial p 1 cohomology groups with respect to the fubini study metric of the regular part of x are finite dimensional

on the duality theorem on an analytic variety the duality theorem for coleff herrera products on a complex manifold says that if f f 1 f p defines a complete intersection then the annihilator of the coleff herrera product f equals locally the ideal generated by f this does not hold unrestrictedly on an analytic variety z we give necessary and in many cases sufficient conditions for when the duality theorem holds these conditions are related to how the zero set of f intersects certain singularity subvarieties of the sheaf mathcal o z

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

perfectly quilted rectangular snake tilings we introduce a particular form of snake tilings to define picture languages and relate the obtained family to the recognizable picture languages as defined by wang tiles the correspondence for substitution tilings is even closer and hence is applicable to the hilbert curve

koppelman formulas on grassmannians we construct koppelman formulas on grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual as an application we obtain new explicit proofs of some vanishing theorems of the bott borel weil type by solving the corresponding equation we also relate the projection part of our formulas to the bergman kernels associated to the line bundles

the bar partial operator on algebraic curves for a singular algebraic curve we show that all closed extensions of bar partial are fredholm and we give a general index formula in particular we prove a modified version of a conjecture due to macpherson

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

the dolbeault complex with weights according to normal crossings in the present paper we define a dolbeault complex with weights according to normal crossings which is a useful tool for studying the bar partial equation on singular complex spaces by resolution of singularities where normal crossings appear naturally the major difficulty is to prove that this complex is locally exact we do that by constructing a local bar partial solution operator which involves only cauchy s integral formula in one complex variable and behaves well for lp forms with weights according to normal crossings

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

undecidability of existential properties in picture languages abstract we are interested in the description of a set of pictures by string languages by using several semantics segments 12 segments with blank moves 8 and pixels 10 we give a method to code the post correspondence problem in rational picture languages in order to show the undecidability of existence of words satisfying a given property in a rational language in particular we study the properties like is a self avoiding word is a minimal picture word or describes a connected picture

koppelman formulas on the a1 singularity in the present paper we study the regularity of the andersson samuelsson koppelman integral operator on the a 1 singularity particularly we prove l p and c 0 estimates as applications we obtain l p homotopy formulas for the dbar equation on the a 1 singularity and we prove that the a forms introduced by andersson samuelsson are continuous on the a 1 singularity

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

global effective versions of the briancon skoda huneke theorem we prove global effective versions of the briancon skoda huneke theorem our results extend to singular varieties a result of hickel on the membership problem in polynomial ideals in cn and a related theorem of ein and lazarsfeld for smooth projective varieties the proofs rely on known geometric estimates and new results on multivariable residue calculus

complexity and decidability for chain code picture languages abstract a picture description is a word over the alphabet u d r l where u means go one unit line up from the current point and d r and l are interpreted analogously with down right and left instead of up by this a picture description describes a walk in the plane its trace is the picture it describes a set of picture descriptions describes a chain code picture language this paper investigates complexity and decidability questions for these picture languages thus it is shown that the membership problem is np complete for regular picture languages i e picture languages described by regular languages of picture descriptions and that it is undecidable whether two regular picture description languages describe a picture in common after this we investigate so called stripe picture languages all pictures are within a stripe defined by two parallel lines providing better complexity and decidability results membership is decidable in linear time for regular stripe picture languages emptiness of intersection and equivalence is decidable for regular stripe picture languages

integral representation with weights ii division and interpolation let f be a r m matrix of holomorphic functions that is generically surjective we provide explicit integral representation of holomorphic such that f provided that is holomorphic and annihilates a certain residue current with support on the set where f is not surjective we also consider formulas for interpolation as applications we obtain generalizations of various results previously known for the case r 1

the equation on a positive current we study the induced equation on a positive current in a complex manifold we extend the l 2 estimates for the equation to harmonic currents of bidimension 1 1 satisfying a frobenius type condition we also show that the l 2 estimates are satisfied for the equation on a positive closed current of bidegree 1 1 on a pseudoconvex domain in n

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

integral representation with weights i we describe a new approach to representation formulas for holomorphic functions including the cauchy fantappie leray formula and provide a general method to generate weighted formulas

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

a dolbeault grothendieck lemma on complex spaces via koppelman formulas let x be a complex space of pure dimension we introduce fine sheaves a x q of 0 q currents which coincides with the sheaves of smooth forms on the regular part of x so that the associated dolbeault complex yields a resolution of the structure sheaf hol x our construction is based on intrinsic and quite explicit semi global koppelman formulas

the membership and equivalence problems for picture languages abstract the paper describes two principal results 1 there is a nondeterministic polynomial time algorithm to determine whether an arbitrary picture is a member of a context free picture language and 2 the equivalence problem for regular picture languages is not decidable not even partially decidable this solves open questions described in earlier publications furthermore these results are best possible that is the membership problem for context free picture languages was earlier shown to be np hard and the complement of the equivalence problem for regular picture languages is easily seen to be partially decidable

the equation on homogeneous varieties with an isolated singularity let x be a regular irreducible variety in mathbb cp n 1 y the associated homogeneous variety in mathbb c n and n the restriction of the universal bundle of mathbb cp n 1 to x in the present paper we compute the obstructions to solving the overline partial equation in the lp sense on y for 1 p in terms of cohomology groups h q x mathcal o n mu that allows to identify obstructions explicitly if x is specified more precisely for example if it is equivalent to mathbb cp 1 or an elliptic curve

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

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

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

using string languages to describe picture languages a picture is a set of unit lines from the cartesian plane considered as a square grid a word over the alphabet l r u d is a picture description in the sense that it represents a traversal of a picture where the interpretation of the symbols l r u d is l go one unit line to the left of the current point r go one unit line to the right of the current point u go one unit line up from the current point and d go one unit line down from the current point a set of picture descriptions forms a picture description language this paper investigates the basic properties of pictures and picture description languages from the formal language theory point of view

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