preconditioning and boundary conditions without h 2 estimates l 2 condition numbers and the distribution of the singular values this work deals with the behavior in the l sub 2 norm of the condition number and distribution of the l sub 2 singular values of the preconditioned operators b sub h sup minus 1 a sub h and a sub h b sub h sup minus 1 sub h where a sub h and b sub h are finite element discretizations of second order elliptic operators a and b in an earlier work manteuffel and parter proved that b sub h sup minus 1 a sub h a sub h b sub h sup minus 1 have a uniformly bounded l sub 2 condition number if and only if a sup and b sup a and b have the same boundary conditions this earlier work used the h sub 2 regularity of a and b as well as optimal l sub 2 error estimates and a quasi uniform grid for the finite element spaces in the present paper the authors first extend these condition number results to the case in which neither h sub 2 regularity and hence optimal l sub 2 error estimates nor the quasi uniformity assumption need to be satisfied instead it is assumed that the principle part of the preconditioning operator b is a scalar multiple 1 mu of the principle part of a

backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

finite element approximation of a phase field model for void electromigration we consider a fully practical finite element approximation of the nonlinear degenerate parabolic system begin align gamma textstyle frac partial u partial t nabla b u nabla w alpha phi 0 quad w gamma delta u gamma 1 psi u nonumber quad nabla c u nabla phi 0 nonumber end align subject to an initial condition u 0 cdot in 1 1 on u and flux boundary conditions on all three equations here gamma in mathbb r 0 alpha in mathbb r geq 0 psi is a nonsmooth double well potential and c u 1 u b u 1 u 2 are degenerate coefficients diffusional mobility the degeneracy in b restricts u cdot cdot in 1 1 the above in the limit gamma rightarrow 0 models the evolution of voids by surface diffusion in an electrically conducting solid in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenera

on the variational approximation of combined second and fourth order geometric evolution equations we present a variational formulation of combined motion by minus the laplacian of curvature and mean curvature flow as well as related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the equidistribution of mesh points and if applicable area conservation

cyclicity and invariant subspaces in dirichlet spaces let be a positive finite measure on the unit circle and d the associated dirichlet space the generalized brown shields conjecture asserts that an outer function f d is cyclic if and only if c z f 0 where c is the capacity associated with d and z f is the zero set of f in this paper we prove that this conjecture is true for measures with countable support we also give in this case a complete and explicit characterization of invariant subspaces

interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones we study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones while the analysis in l faybusovich positivity 1 1997 pp 331 357 depends on the optimization theory of convex log barrier functions our approach is based on the paper of monteiro and pang math oper res 23 1998 pp 39 60 where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices as an application of the results we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects consequently we show the existence of a path having the following properties a the path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions b any accumulation point of the path is a solution of the homogeneous model c if the original problem is solvable then every accumulation point of the path gives us a finite solution d if the original problem is strongly infeasible then under the assumption of lipschitz continuity any accumulation point of the path gives us a finite certificate proving infeasibility

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

lagrangian dual interior point methods for semidefinite programs this paper proposes a new predictor corrector interior point method for a class of semidefinite programs which numerically traces the central trajectory in a space of lagrange multipliers the distinguishing features of the method are full use of the bfgs quasi newton method in the corrector procedure and an application of the conjugate gradient method with an effective preconditioning matrix induced from the bfgs quasi newton method in the predictor procedure some preliminary numerical results are reported

kernel and capacity estimates in dirichlet spaces abstract we study the capacity in the sense of beurling deny associated with the dirichlet space d where is a finite positive borel measure on the unit circle first we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of d it allows us to give an estimate of the capacity of points and arcs of the unit circle we also provide new conditions on closed sets to be polar our method is based on sharp estimates of norms of some outer functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted sobolev space

on two interior point mappings for nonlinear semidefinite complementarity problems extending our previous work monteiro and pang 1996 this paper studies properties of two fundamental mappings associated with the family of interior point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices the first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case these trajectories completely fill up the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program unlike the approach based on the theory of maximal monotone maps taken by shida and shindoh 1996 and shida shindoh and kojima 1995 our approach is based on the theory of local homeomorphic maps in nonlinear analysis

fast numerical solution of the biharmonic dirichlet problem on rectangles a new method for the numerical solution of the first biharmonic dirichlet problem in a rectangular domain is presented for an n times n mesh the complexity of this algorithm is on the order of

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

asymptotic behavior of eigenvalues of toeplitz operators on the weighted analytic spaces abstract in this paper we consider toeplitz operators on a class of analytic function spaces including standard bergman spaces on the unit disc and bargman fock spaces we give a complete characterization of those compact operators whose eigenvalues have a prescribed slow decay to zero

contact points and schatten composition operators we study composition operators on hardy and dirichlet spaces belonging to schatten classes we give some new examples and analyse the size of contact set of the symbol of such operators

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

general interior point maps and existence of weighted paths for nonlinear semidefinite complementarity problems extending the previous work of monteiro and pang 1998 this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com plementarity problems over the cone of symmetric positive semidefinite matrices instead of focusing our attention on a specific map as was done in the approach of monteiro and pang 1998 this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of monteiro and pang 1998 to this general map each fundamental map leads to a family of weighted continuous trajectories which include the central trajectory as a special case hence for complementarity problems over the cone of symmetric positive semidefinite matrices the notion of weighted central path depends on the fundamental map used to represent the central path

centers of monotone generalized complementarity problems let c be a full dimensional closed pointed and convex cone in a finite dimensional real vector space e with an inner product of x y e and m a maximal monotone subset of e e this paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with c and m find x y m c c such that 0 here c y e 0 for all x c denotes the dual cone of c the main result of the paper unifies and extends some results established for monotone complementarity problems in euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

generalized linear complementarity problems it has been shown by lemke that if a matrix is copositive plus on n then feasibility of the corresponding linear complementarity problem implies solvability in this article we show under suitable conditions that feasibility of ageneralized linear complementarity problem i e defined over a more general closed convex cone in a real hilbert space implies solvability whenever the operator is copositive plus on that cone we show that among all closed convex cones in a finite dimensional real hilbert space polyhedral cones are theonly ones with the property that every copositive plus feasible glcp is solvable we also prove a perturbation result for generalized linear complementarity problems

perturbation bounds for polynomials using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of ostrowski acta math 72 99 257 1940 elsner et al linear algebra appl 142 195 209 1990 a comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of beauzamy can math bull 42 1 3 12 1999 using the matrix theoretical approach we also improve the backward stability result of edelman and murakami proceedings of the fifth siam conference on applied linear algebra siam philapdelphia 1994 math comput 64 210 763 1995

barrier functions in interior point methods we show that the universal barrier function of a convex cone introduced by nesterov and nemirovskii is the logarithm of the characteristic function of the cone this interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone this provides a simple method for calculating the universal barrier for homogeneous cones we identify some known barriers as the universal barrier scaled by an appropriate constant we also calculate some new universal barrier functions our results connect the field of interior point methods to several branches of mathematics such as lie groups jordan algebras siegel domains differential geometry complex analysis of several variables etc

limiting behavior of trajectories generated by a continuation method for monotone complementarity problems defining the mapping f from the 2n dimensional euclidean space r2n into itself by fx y x1y1 xn yn y1 f1x yn fnx for every x y r2n we write the cp the complementarity problem with a mapping f rn rn as the system of equations fx y 0 and x y r 2n where r 2n denotes the nonnegative orthant of r2n under the assumption that f is a monotone function on r n we show that f maps the positive orthant r 2n of r2n homeomorphically this result then ensures the existence of a trajectory consisting of the solutions x y t of the system of equations fx y wt and x y t r 2n 0 1 where w is a continuous mapping from the unit interval 0 1 into fr 2n such that w0 0 if t 0 the system coincides with the cp we study limiting behavior of the trajectory as t 0 and give some sufficient conditions for the trajectory to lead to a solution of the cp this approach is an extension of the one used in the polynomially bounded algorithm recently given by kojima mizuno and yoshise for solving positive semidefinite linear complementarity problems

a note on unifying absolute and relative perturbation bounds abstract perturbation bounds for invariant subspaces and eigenvalues of complex matrices are presented that lead to absolute as well as a large class of relative bounds in particular it is shown that absolute bounds such as those by davis and kahan bauer and fike and hoffman and wielandt and some relative bounds are special cases of universal bounds as a consequence we obtain a new relative bound for subspaces of normal matrices which contains a deviation of the matrix from positive definiteness we also investigate how row scaling affects eigenvalues and their sensitivity to perturbations and we illustrate how the departure from normality can affect the condition number with respect to inversion of the scaled eigenvectors

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

a note on the variation of the spectrum of an arbitrary matrix abstract let a and b be two n n matrices with spectra a 1 n and b 1 n suppose that the nonsingular matrix q satisfies q 1 aq diag j 1 j p where each submatrix j i i 1 p is a jordan block then there exists a permutation of 1 n such that j 1 n j j 2 n 1 n p max q 1 b a q f q 1 b a q f m and for j 1 n j j n 1 n p max n q 1 b a q 2 n q 1 b a q 2 m where m is the order of the largest jordan block of a and f and 2 denote respectively the frobenius norm and the spectral norm

composition operators acting on weighted dirichlet spaces abstract we study composition operators acting on weighted dirichlet spaces we obtain estimates for the essential norm describe the membership in schatten von neumann ideals and characterize the composition operators with closed range

a condensed representation of almost normal matrices abstract in this paper we study the structure of almost normal matrices that is the matrices for which there exists a rank one matrix c such that a h a aa h ca ac necessary and sufficient conditions for a matrix to belong to the class are given and a canonical representation as a block tridiagonal matrix is shown the approach is constructive and in the paper it is explained how starting from a 1 1 or 2 2 matrix we can generate almost normal matrices moreover given an n n almost normal matrix we can compute the block tridiagonal representation with a finite procedure

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

trace ideal criteria for toeplitz operators for a complex measure on the open unit disk u define an operator t on a hilbert space h of analytic functions with reproducing kernel k z w by u for a certain scale of hilbert spaces h 1 which includes the hardy space h2 and weighted bergman spaces a2 conditions are obtained which imply t belongs to a schatten ideal sp if is a positive measure then these conditions are necessary and sufficient application to composition operators and restriction operators are indicated

new perturbation bounds for the spectrum of a normal matrix abstract let a c n n and a c n n be two normal matrices with spectra i i 1 n and i i 1 n respectively the celebrated hoffman wielandt theorem states that there exists a permutation of 1 n such that i 1 n i i 2 1 2 is no larger than the frobenius norm of a a however if either a or a is non normal this result does not hold in general in this paper we present several novel upper bounds for i 1 n i i 2 1 2 provided that a is normal and a is arbitrary some of these estimates involving the departure from normality of a have generalized the hoffman wielandt theorem furthermore we give new perturbation bounds for the spectrum of a hermitian matrix

timely communication efficient algorithms for solving a fourth order equation with the spectral galerkin method we show that one can derive an o n 3 spectral galerkin method for fourth order biharmonic type elliptic equations based on the use of chebyshev polynomials the use of chebyshev polynomials provides a fast transform between physical and spectral space which is advantageous when a sequence of problems must be solved e g as part of a nonlinear iteration this improves the result of shen siam j sci comput 16 1995 pp 74 87 which reported an o n 4 algorithm inferior to the o n 3 method developed earlier shen siam j sci comput 15 1994 pp 1440 1451 based on legendre polynomials but less practical in the case of multiple problems we further compare our method with an improved implementation of the legendre galerkin method based on the same approach

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

parametric approximation of willmore flow and related geometric evolution equations we present various variational approximations of willmore flow in mathbb r d d 2 3 as well as the classic willmore flow we also consider variants that are a volume preserving and b volume and area preserving the latter evolution law is the so called helfrich flow in addition we consider motion by gaus curvature the presented fully discrete schemes are easy to solve as they are linear at each time level and they have good properties with respect to the distribution of mesh points finally we present numerous numerical experiments including simulations for energies appearing in the modeling of biological cell membranes

optimality of multilevel preconditioners for local mesh refinement in three dimensions in this article we establish optimality of the bramble pasciak xu bpx norm equivalence and optimality of the wavelet modified or stabilized hierarchical basis whb preconditioner in the setting of local 3d mesh refinement in the analysis of whb methods a critical first step is to establish the optimality of bpx norm equivalence for the refinement procedures under consideration while the available optimality results for the bpx norm have been constructed primarily in the setting of uniformly refined meshes a notable exception is the local 2d red green result due to dahmen and kunoth the purpose of this article is to extend this original 2d optimality result to the local 3d red green refinement procedure introduced by bornemann erdmann and kornhuber and then to use this result to extend the whb optimality results from the quasi uniform setting to local 2d and 3d red green refinement scenarios the bpx extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally riesz stable this construction turns out to rest not only on the shape regularity of the refined elements but also critically on a number of geometrical properties we establish between neighboring simplices produced by the bornemann erdmann kornhuber bek refinement procedure it is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement indicating that a practical implementable version of the resulting bpx preconditioner for the bek refinement setting has provably optimal linear computational complexity per iteration an interesting implication of the optimality of the whb preconditioner is the a priori h1 stability of the l2 projection the existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied the theoretical framework employed supports arbitrary spatial dimension d geq 1 and requires no coefficient smoothness assumptions beyond those required for well posedness in h1

perturbation analysis for matrix joint block diagonalization abstract the matrix joint block diagonalization problem jbdp of a given matrix set a a i i 1 m is about finding a nonsingular matrix w such that all w t a i w are block diagonal it includes the matrix joint diagonalization problem jdp as a special case for which all w t a i w are required diagonal generically such a matrix w may not exist but there are practical applications such as multidimensional independent component analysis mica for which it does exist under the ideal situation i e no noise is presented however in practice noises do get in and as a consequence the matrix set is only approximately block diagonalizable i e one can only make all w t a i w nearly block diagonal at best where w is an approximation to w obtained usually by computation the main goal of this paper is to develop a perturbation theory for jbdp to address among others the question how accurate this w is previously such a theory for jdp has been discussed but no effort had been attempted for jbdp because in large part there is no quantitative way to describe solution uniqueness of jbdp until 2017 when cai and liu 2017 9 successfully obtained a necessary and sufficient uniqueness condition based on the condition in this article we will establish a perturbation theory for jbdp our main contributions include an error bound for the approximate block diagonalizer w and a backward error analysis for jbdp numerical tests are presented to validate the theoretical results

toeplitz operators on harmonically weighted bergman spaces and applications to composition operators on dirichlet spaces abstract in this paper we consider toeplitz operators acting on bergman spaces associated with harmonic weights we give a complete description of those operators which belong to p schatten classes we also obtain an upper estimate for the reproducing kernel of these spaces applications to composition operators on dirichlet spaces are given

surface diffusion of graphs variational formulation error analysis and simulation surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for graphs and derive a priori error estimates for a time continuous finite element discretization we also introduce a semi implicit time discretization and a schur complement approach to solve the resulting fully discrete linear systems after computational verification of the orders of convergence for polynomial degrees 1 and 2 we show several simulations in one dimension and two dimensions with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time such as infinite slopes and cracks

nearest stable system using successive convex approximations stability is a crucial property in the study of dynamical systems we focus on the problem of enforcing the stability of a system a posteriori the system can be a matrix or a polynomial either in continuous time or in discrete time we present an algorithm that constructs a sequence of successive stable iterates that tend to a nearby stable approximation x of a given system a the stable iterates are obtained by projecting a onto the convex approximations of the set of stable systems some possible applications for this method are correcting the error arising from some noise in system identification and a possible solver for bilinear matrix inequalities based on convex approximations in the case of polynomials a fair complexity is achieved by finding a closed form solution to first order optimality conditions