an adaptive linear approximation algorithm for copositive programs we study linear optimization problems over the cone of copositive matrices these problems appear in nonconvex quadratic and binary optimization for instance the maximum clique problem and other combinatorial problems can be reformulated as such problems we present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit in contrast to previous approximation schemes our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not using these approximations we derive an adaptive linear approximation algorithm for copositive programs numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

irreducible elements of the copositive cone we call an element a of the n n copositive cone cn irreducible with respect to the nonnegative cone nn if it cannot be written as a nontrivial sum a c n of a copositive matrix c and an elementwise nonnegative matrix n note that our concept of irreducibility differs from the standard one normally studied in matrix theory this property was studied by baumert 1965 2 who gave a characterisation of irreducible matrices we demonstrate here that baumert s characterisation is incorrect and give a correct version of his theorem which establishes a necessary and sufficient condition for a copositive matrix to be irreducible for the case of 5 5 copositive matrices we give a complete characterisation of all irreducible matrices we show that those irreducible matrices in c5 which are not positive semidefinite can be parameterized in a semi trigonometric way finally we prove that every 5 5 copositive matrix which is not the sum of a nonnegative and a semidefinite matrix can be expressed as the sum of a nonnegative matrix with zero diagonal and a single irreducible matrix

renormalized equations for a weakly nonlinear duffing oscillator abstract we derive fluctuation renormalized transport equations for the averages position and velocity of a weakly nonlinear duffing oscillator we take the nonlinearity as the expansion parameter and use a modification of the van kampen expasion method also we study the spectral density of the equilibrium time correlation function of position comparison with other theories and computer data is carried out

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

copositive programming motivated bounds on the stability and the chromatic numbers the lovasz theta number of a graph g can be viewed as a semidefinite programming relaxation of the stability number of g it has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of g we introduce a related strengthening of the lovasz theta number toward the chromatic number of g which is shown to be equal to the fractional chromatic number of g solving copositive programs is np hard this motivates the study of tractable approximations of the copositive cone we investigate the parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs we provide some computational results indicating that the lovasz theta number can be strengthened significantly toward the fractional chromatic number of g on some hamming graphs

approximation algorithms for indefinite quadratic programming we consider approximation schemes for indefinite quadratic programming we argue that such an approximation can be found in polynomial time for fixed andt wheret denotes the number of negative eigenvalues of the quadratic term our algorithm is polynomial in 1 for fixedt and exponential int for fixed we next look at the special case of knapsack problems showing that a more efficient polynomial int approximation algorithm exists

finding a sparse vector in a subspace linear sparsity using alternating directions is it possible to find the sparsest vector direction in a generic subspace mathcal s subseteq mathbb r p with text dim left mathcal s right n this problem can be considered a homogeneous variant of the sparse recovery problem and finds connections to sparse dictionary learning sparse pca and many other problems in signal processing and machine learning in this paper we focus on a planted sparse model for the subspace the target sparse vector is embedded in an otherwise random subspace simple convex heuristics for this planted recovery problem provably break down when the fraction of nonzero entries in the target sparse vector substantially exceeds o 1 sqrt n in contrast we exhibit a relatively simple nonconvex approach based on alternating directions which provably succeeds even when the fraction of nonzero entries is omega 1 to the best of our knowledge this is the first practical algorithm to achieve linear scaling under the planted sparse model empirically our proposed algorithm also succeeds in more challenging data models e g sparse dictionary learning

a complementary pivoting approach to the maximum weight clique problem given an undirected graph with positive weights on the vertices the maximum weight clique problem mwcp is to find a subset of mutually adjacent vertices i e a clique having largest total weight the problem is known to be np hard even to approximate motivated by a recent quadratic programming formulation which generalizes an earlier remarkable result of motzkin and straus in this paper we propose a new framework for the mwcp based on the corresponding linear complementarity problem lcp we show that generically all stationary points of the mwcp quadratic program exhibit strict complementarity despite this regularity result however the lcp turns out to be inherently degenerate and we find that lemke s well known pivoting method equipped with standard degeneracy resolution strategies yields unsatisfactory experimental results we exploit the degeneracy inherent in the problem to develop a variant of lemke s algorithm which incorporates a new and effective look ahead pivot rule the resulting algorithm is tested extensively on various instances of random as well as dimacs benchmark graphs and the results obtained show the effectiveness of our method

a set of stationary non maxwellian distributions abstract we propose a set of distribution functions stationary solutions of a fokker planck equation the frictional and diffusion coefficients having been expanded in a polynomial series with positive coefficients these distributions show tails depleted or increased with respect to the maxwellian distribution which can in any case be recovered if the coefficients of the expansion satisfy particular conditions the stationary non maxwellian distributions derived in this paper can be used in the calculations of rates in many different fields of physics where problems still have to be solved

numerical analysis of the smoluchowski equation using the split operator method abstract we apply the split operator method of feit fleck and steiger to the numerical solution of the smoluchowski equation we study the time evolution of the first few moments and of the equilibrium time correlation function for bistable potentials the time evolution of the probability density obtained by us is compared with the one predicted by the scaling theory of suzuki we also analyze the dependence of the total equilibrium relaxation time on the noise intensity d for small and large values of d the relaxation from an initial condition away from the unstable point is also discussed we find that the one well population is well described by a single exponential law with decay times which grow extremely large as the value of d is decreased

the copositive completion problem abstract an n n real symmetric matrix a is called strictly copositive if x t ax 0 0 whenever x r n satisfies x 0 x 0 and x 0 the strictly copositive matrix completion problem asks which partial strictly copositive matrices have a completion to a strictly copositive matrix we prove that every partial strictly copositive matrix has a strictly copositive matrix completion and give a lower bound on the values used in the completion we answer affirmatively an open question whether an n n copositive matrix a a ij with all diagonal entries a ii 1 stays copositive if each off diagonal entry of a is replaced by min a ij 1

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

on a new construction of geometric mean of n operators abstract for n positive definite operators a 1 a n ando li mathias defined geometric mean of n operators g a 1 a n by symmetric procedure it has many nice properties and is studied by many authors but the process is so complicated to compute in this paper we shall attempt to make a new construction of geometric mean of n operators which we can compute it easier than geometric mean by ando li mathias

a simplex algorithm for rational cp factorization in this paper we provide an algorithm similar to the simplex algorithm which determines a rational cp factorization of a given matrix whenever the matrix allows such a factorization this algorithm can be used to show that every integral completely positive 2 times 2 matrix has an integral cp factorization

weighted multivariable operator means of positive definite operators abstract in this paper we present a new weighted multivariable operator mean of positive definite operators over an arbitrary hilbert space which provides us the first generally applicable extension of the classical kubo ando theory of 2 variable operator means the construction is a weighted extension of the bini meini poloni symmetrization process originally given for the matrix geometric mean here to be able to consider such an iterative procedure we need a weighted version of every kubo ando mean in two variables therefore we also give a new construction for two arbitrary positive operators on a possibly infinite dimensional hilbert space that provides weighted counterparts to every not necessarily symmetric kubo ando mean and also agrees with the most well known weighted operator means

weighted matrix means and symmetrization procedures abstract here we prove the convergence of the ando li mathias and bini meini poloni procedures for matrix means actually it is proved here that for a two variable function which maps pairs of positive definite matrices to a positive definite matrix and is not greater than the square mean of two positive definite matrices the ando li mathias and bini meini poloni procedure converges in order to be able to set up the bini meini poloni procedure a weighted two variable matrix mean is also needed therefore a definition of a two variable weighted matrix mean corresponding to every symmetric matrix mean is also given it is also shown here that most of the properties considered by ando li and mathias for the n variable geometric mean hold for all of these n variable maps that we obtain by this two limiting process for all two variable matrix means as a consequence it also follows that the bini meini poloni procedure converges cubically for every matrix mean

toward unified analysis and controller synthesis for a class of hybrid systems abstract this work defines a new class of hybrid systems called state based switched sbs systems that have numerous important engineering applications the characterizing feature of these systems is that the discrete event dynamics are associated with the continuous time state making a specific function be equal to zero the choice of this function is application specific and for the closed loop sbs systems defined in this paper it is related to the execution of a desired set of tasks from a pre specified mission plan for this broad class of sbs systems the paper presents a unified analysis and controller synthesis methodology based on lyapunov theory depending on the details of the mission plan the closed loop hybrid system will be divided into two subclasses sequential and non sequential the controller design procedure for both subclasses consists of the same two steps finding a control law and finding a stabilizing switching rule for static state and output feedback of sequential hybrid systems the paper proposes a new hybrid sequential sliding mode controller it is proven that the control mission can be accomplished for sequential hybrid systems under static state and output feedback using this new controller a similar framework is investigated for the more complex class of nonsequential hybrid systems and a systematic procedure for designing the switching rule is presented for some specific instances of these systems

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

factorization and cutting planes for completely positive matrices by copositive projection we consider the problem of projecting a matrix onto the cones of copositive and completely positive matrices as this can not be done directly we use polyhedral approximations of the cones with the help of these projections we obtain a technique to compute factorizations of completely positive matrices we also describe a method to determine a cutting plane which cuts off an arbitrary matrix from the completely positive or copositive cone

an efficient memetic algorithm for themax bisection problem the max bisection problem consists in partitioning the vertices of a weighted undirected graph into two equally sized subsets so as to maximize the sum of the weights of crossing edges it is an np hard combinatorial optimization problem that arises in many applications in this paper we present a memetic algorithm for the max bisection problem which integrates a new fast local search procedure a crossover operator and a pool updating strategy these strategies achieve a balance between intensification and diversification extensive experiments were performed on a number of benchmark instances with 800 to 10 000 vertices from the literature the proposed memetic algorithm improved the best known solutions for all benchmark instances tested in this paper the improvement in terms of cut value over the circut by burer et al ranging from 0 02 to 4 15 percent and the average time of our proposed memetic algorithm is much lower than that of circut it shows that the proposed memetic algorithm can find high quality solutions in an acceptable running time

stochastic evolution of systems of particles obeying an exclusion principle abstract we present numerical solutions of a nonlinear fokker planck equation to describe a fermi gas the model equation g kaniadakis p quarati phys rev e 49 1994 5103 includes a nonlinear term due to the pauli exclusion principle which has been disregarded up to now a new numerical method to study the time evolution of the distribution function is presented the method can be applied to any potential and allows us to describe accurately and efficiently a fermi gas in this work we focus on the differences between the results of the linear case in which the effects of the exclusion principle are not considered and the nonlinear case in which these effects are taken into account

new deterministic approaches to the least square mean abstract in this paper we present new deterministic approximations to the least square mean also called geometric mean or barycenter of a finite collection of positive definite matrices let a 1 a 2 a m be any elements of m d c where the set m d c is the open cone in the real vector space of selfadjoint matrices h n we consider a sequence of blocks of m matrices that is a 1 a m a 1 a m a 1 a m we take a permutation on every block and then take the usual inductive mean of that new sequence the main result of this work is that the inductive mean of this block permutation sequence approximate the least square mean on m d c this generalizes a theorem obtain by holbrook even more we have an estimate for the rate of convergence

multivariate nonnegative quadratic mappings in this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain with nonnegativity defined by a prespecified conic order in particular we consider the set cone of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone and study when it can be represented by linear matrix inequalities we also discuss the applications of the results in robust optimization especially the robust quadratic matrix inequalities and the robust linear programming models in the latter application the implementational errors of the solution are taken into account and the problem is formulated as a semidefinite program

on the solution of concave knapsack problems we consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint we characterize strict local miniimizers of concave minimization problems subject to linear constraints and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is np hard it is possible to determine a local minimizer of this problem with at most o n logn operations and 1 logn evaluations of the function if the function is quadratic this algorithm requires at most o n logn operations

a cone programming approach to the bilinear matrix inequality problem and its geometry we discuss an approach for solving the bilinear matrix inequality bmi based on its connections with certain problems defined over matrix cones these problems are among others the cone generalization of the linear programming lp and the linear complementarity problem lcp referred to as the cone lp and the cone lcp respectively specifically we show that solving a given bmi is equivalent to examining the solution set of a suitably constructed cone lp or cone lcp this approach facilitates our understanding of the geometry of the bmi and opens up new avenues for the development of the computational procedures for its solution

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

semidefinite approximations for global unconstrained polynomial optimization we consider the problem of minimizing a polynomial function on mathbb r n known to be hard even for degree 4 polynomials therefore approximation algorithms are of interest lasserre siam j optim 11 2001 pp 796 817 and parrilo math program 96 2003 pp 293 320 have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations we propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming the method is applied to several examples

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method abstract we establish in this paper a group of closed form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic hermitian matrix function x q x p x with respect to the variable matrix x by using a linearization method and some known formulas for extremum ranks and inertias of linear hermitian matrix functions where both p and q are complex hermitian matrices and x is the conjugate transpose of x we then derive the global maximum and minimum ranks and inertias of the two quadratic hermitian matrix functions 1 x q 1 x p 1 x and 2 x q 2 x p 2 x subject to a consistent matrix equation a x b respectively by using some pure algebraic operations of matrices and their generalized inverses as consequences we establish necessary and sufficient conditions for the solutions of the matrix equation a x b to satisfy the quadratic hermitian matrix equalities x p 1 x q 1 and x p 2 x q 2 respectively and for the quadratic matrix inequalities x p 1 x q 1 and x p 2 x q 2 in the lowner partial ordering to hold respectively in addition we give complete solutions to four lowner partial ordering optimization problems on the matrix functions 1 x and 2 x subject to a x b examples are also presented to illustrative applications of the equality constrained quadratic optimizations in some matrix completion problems

diffusion in a potential field path integral approach abstract within the framework of the path integral formalism we develop a simple method to solve the fokker planck diffusion problems in a potential field including the decay of an unstable state unlike previous approaches it yields a unified treatment of all time regimes and is legitimate for any value of the diffusion coefficient an attractive feature of our method is that it provides numerical results in regions where there are no analytical solutions in particular being valid for an arbitrary shape of the potential it is very useful for studying a diffusion problem near and in the critical point where the standard treatments break down moreover our method after a simple transformation permits to obtain also the first passage times of the process

diffusion in a bistable potential the functional integral approach we study with the help of the onsager machlup functional integral approach the distributionp of a single stochastic variable the evolution of which is described by a fokker planck equation with a first moment deriving from a bistable potential we set up the approximation scheme appropriate in this approach to the limit of constant and small diffusion coefficient two regimes are to be distinguished very long times kramers regime are treated within the frame of a free instanton molecule gas approximation and at intermediate times suzuki regime a standard semiclassical calculation is legitimate we thus rederive exactly the results obtained from the mode expansion and wkb method

operator means of probability measures and generalized karcher equations abstract in this article we consider means of positive bounded linear operators on a hilbert space we present a complete theory that provides a framework which extends the theory of the karcher mean its approximating matrix power means and a large part of kubo ando theory to arbitrary many variables in fact to the case of probability measures with bounded support on the cone of positive definite operators this framework characterizes each operator mean extrinsically as unique solutions of generalized karcher equations which are obtained by exchanging the matrix logarithm function in the karcher equation to arbitrary operator monotone functions over the positive real half line if the underlying hilbert space is finite dimensional then these generalized karcher equations are riemannian gradients of convex combinations of strictly geodesically convex log determinant divergence functions hence these new means are their global minimizers in analogy to the case of the karcher mean as pointed out our framework is based on fundamental contraction results with respect to the thompson metric which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above

the application of statistical linearization to the problem of decay from an unstable state abstract it is shown that the method of statistical linearization for the approximate treatment of stochastic nonlinear systems driven by gaussian white noise may be applied successfully to systems which evolve from a state of unstable equilibrium thereby developing non gaussian statistics the essential idea is the replacement of the full ensemble of realizations by subensembles according to the values taken by a small number of linear functionals of the noise the method is illustrated by applying it to two problems i strongly damped linear motion of a particle in a symmetric bistable potential ii the single mode laser in both cases it is found that low order moments can be calculated quite easily to an accuracy of a few per cent and that corrections with error estimates can be found by using the approximation to provide a control variate for numerical simulation

minimal zeros of copositive matrices abstract let a be an element of the copositive cone c n a zero u of a is a nonzero nonnegative vector such that u t a u 0 the support of u is the index set supp u 1 n corresponding to the positive entries of u a zero u of a is called minimal if there does not exist another zero v of a such that its support supp v is a strict subset of supp u we investigate the properties of minimal zeros of copositive matrices and their supports special attention is devoted to copositive matrices which are irreducible with respect to the cone s n of positive semi definite matrices i e matrices which cannot be written as a sum of a copositive and a nonzero positive semi definite matrix we give a necessary and sufficient condition for irreducibility of a matrix a with respect to s n in terms of its minimal zeros a similar condition is given for the irreducibility with respect to the cone n n of entry wise nonnegative matrices for n 5 matrices which are irreducible with respect to both s 5 and n 5 are extremal for n 6 a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided

means in metric spaces and the center of mass in this article k convex metric spaces are considered where a several variable mapping is provided as a limit point of an iteration scheme based on the midpoint map in the metric space itself this mapping considered as a mean of its variables has some properties which relates it to the center of mass of these variables in the metric space sufficient conditions are given here for the two points to be identical as well as upper bounds on their distances from one another the asymptotic rate of convergence of the iterative process defining the mean is also determined here the case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied here as examples of k convex metric spaces

solution of fokker planck equation using trotter s formula solution of fokker planck equation using trotter s formula is discussed the method is illustrated on the linear fokker planck equation and the ornstein uhlenbeck solution is obtained for the case of a general nonlinear fokker planck equation the method yields an integral representation amenable to approximations in the lowest order approximation suzuki s scaling result emerges physical interpretation and limitations of the approximations are also discussed

a new bound and reduce approach of nonconvex quadratic programming problems abstract for the nonconvex quadratic programming problem a new linear programming relaxation bound and reduce algorithm is proposed and its convergence is proved in this algorithm a new hyper rectangle partition technique and a new linear programming relaxation tactics are used at the same time the hyper rectangular reduction method is used to raise its convergent speed the numerical results demonstrate the effectiveness and feasibility of the proposed algorithm

a complete characterization of the gap between convexity and sos convexity our first contribution in this paper is to prove that three natural sum of squares sos based sufficient conditions for convexity of polynomials via the definition of convexity its first order characterization and its second order characterization are equivalent these three equivalent algebraic conditions henceforth referred to as sos convexity can be checked by semidefinite programming whereas deciding convexity is np hard if we denote the set of convex and sosconvex polynomials in n variables of degree d with cn d and cn d respectively then our main contribution is to prove that cn d cn d if and only if n 1 or d 2 or n d 2 4 we also present a complete characterization for forms homogeneous polynomials except for the case n d 3 4 which is joint work with g blekherman and is to be published elsewhere our result states that the set cn d of convex forms in n variables of degree d equals the set cn d of sos convex forms if and only if n 2 or d 2 or n d 3 4 to prove these results we present in particular explicit examples of polynomials in c2 6 c2 6 and c3 4 c3 4 and forms in c3 6 c3 6 and c4 4 c4 4 and a general procedure for constructing forms in cn d 2 cn d 2 from nonnegative but not sos forms in n variables and degree d although for disparate reasons the remarkable outcome is that convex polynomials resp forms are sos convex exactly in cases where nonnegative polynomials resp forms are sums of squares as characterized by hilbert

a triangulation approach to asymptotically exact conditions for fuzzy summations many takagi sugeno t s fuzzy control synthesis problems in the literature are expressed as the problem of finding decision variables in a double convex sum fuzzy summation of positive definite matrices matricespsila coefficients in the summation take values in the standard simplex this paper presents a triangulation approach to the problem of generating simplicial partitions of the standard simplex in order to set up a family of sufficient conditions and in parallel another family of necessary ones for fuzzy summations the conditions proposed in this paper are asymptotically exact as the size of the involved simplices decreases its conservativeness vanishes for a sufficiently fine partition sufficiently dense mesh of vertex points the set of conditions is in the form of linear matrix inequalities lmis for which efficient software is available