h control of uncertain fuzzy continuous time systems a new kind of dynamical fuzzy model is proposed to represent the continuous time complex system which includes both linguistic information and system uncertainties a new stability analysis and control system design approach is developed for this kind of dynamical fuzzy model since the fuzzy model considers the norm bounded disturbance the controller design problem becomes a h control problem a constructive algorithm is developed to obtain the h feedback control law finally an example is given to illustrate the application of the method

stability analysis and control of rigid body systems with impacts and friction many critical tasks in robotics such as locomotion or manipulation involve collisions between a rigid body and the environment or between multiple bodies methods based on sums of squares sos for numerical computation of lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems and can additionally be used to automatically synthesize stabilizing feedback controllers here we present a method for applying sums of squares verification to rigid bodies with coulomb friction undergoing discontinuous inelastic impact events the proposed algorithm explicitly generates lyapunov certificates for stability positive invariance and safety over admissible non penetrating states and contact forces we leverage the complementarity formulation of contact which naturally generates the semialgebraic constraints that define this admissible region the approach is demonstrated on multiple robotics examples including simple models of a walking robot a perching aircraft and control design of a balancing robot

an extension of sums of squares relaxations to polynomial optimization problems over symmetric cones this paper is based on a recent work by kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs let varepsilon and varepsilon be a finite dimensional real vector space and a symmetric cone embedded in varepsilon examples of varepsilon and varepsilon include a pair of the n dimensional euclidean space and its nonnegative orthant a pair of the n dimensional euclidean space and n dimensional second order cones and a pair of the space of m m real symmetric or complex hermitian matrices and the cone of their positive semidefinite matrices sums of squares relaxations are further extended to a polynomial optimization problem over varepsilon i e a minimization of a real valued polynomial a x in the n dimensional real variable vector x over a compact feasible region bf x b bf x in varepsilon where b x denotes an varepsilon valued polynomial in x it is shown under a certain moderate assumption on the varepsilon valued polynomial b x that optimal values of a sequence of sums of squares relaxations of the problem which are converted into a sequence of semidefinite programs when they are numerically solved converge to the optimal value of the problem

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

robust global optimization with polynomials we consider the optimization problems maxz minx k p z x and minx k maxz p z x where the criterion p is a polynomial linear in the variables z the set can be described by lmis and k is a basic closed semi algebraic set the first problem is a robust analogue of the generic sdp problem maxz p z whereas the second problem is a robust analogue of the generic problem minx k p x of minimizing a polynomial over a semi algebraic set we show that the optimal values of both robust optimization problems can be approximated as closely as desired by solving a hierarchy of sdp relaxations we also relate and compare the sdp relaxations associated with the max min and the min max robust optimization problems

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

entropy numbers s numbers and eigenvalue problems abstract we establish inequalities between entropy numbers and approximation numbers for operators acting between banach spaces furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a banach space the results are compared with the classical inequalities of bernstein and jackson

global stability analysis of fluid flows using sum of squares this paper introduces a new method for proving global stability of uid ows through the construction of lyapunov functionals for nite dimensional approximations of uid systems we show how one can exploit recently developed optimization methods based on sum of squares decomposition to construct a polynomial lyapunov function we then show how these methods can be extended to innite dimensional navier stokes systems using robust optimization techniques crucially this extension requires only the solution of innite dimension al linear eigenvalue problems and nite dimensional sum of squares optimization problems we further show that subject to minor technical constraints a general polynomial lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower bound on the maximum reynolds number for which a ow is globally stable if the ow does remain globally stable for reynolds numbers at least slightly beyond the energy stability limit such polynomial functions can be searched for eciently using the sos technique we propose

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

new upper bounds for kissing numbers from semidefinite programming recently a schrijver derived new upper bounds for binary codes using semidefinite programming in this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions in particular our computations give the known values for the cases

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

discriminant of symmetric matrices as a sum of squares and the orthogonal group it is proved that the discriminant of n n real symmetric matrices can be written as a sum of squares where the number of summands equals the dimension of the space of n variable spherical harmonics of degree n the representation theory of the orthogonal group is applied to express the discriminant of 3 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares it is proved that the discriminant of 4 4 real symmetric matrices can be written as a sum of seven squares these improve results of kummer from 1843 and borchardt from 1846 2010 wiley periodicals inc

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

convergent sdp relaxations in polynomial optimization with sparsity we consider a polynomial programming problem p on a compact basic semialgebraic set k subset r n described by m polynomial inequalities g j x geq0 and with criterion f in r x we propose a hierarchy of semidefinite relaxations in the spirit of those of waki e siam j optim 17 2006 pp 218 242 in particular the sdp relaxation of order r has the following two features a the number of variables is o kappa 2r where kappa max kappa 1 kappa 2 with kappa 1 resp kappa 2 being the maximum number of variables appearing in the monomials of f resp appearing in a single constraint g j x geq0 b the largest size of the linear matrix inequalities lmis is o kappa r this is to compare with the respective number of variables o n 2r and lmi size o n r in the original sdp relaxations defined in j b lasserre siam j optim 11 2001 pp 796 817 therefore great computational savings are expected in case of sparsity in the data g j f i e when k is small a frequent case in practical applications of interest the novelty with respect to h waki s kim m kojima and m maramatsu siam j optim 17 2006 pp 218 242 is that we prove convergence to the global optimum of p when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications and known as the running intersection property in graph theory in such cases and as a by product we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set a sparse version of putinar s positivstellensatz m putinar indiana univ math j 42 1993 pp 969 984

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

design of continuous twisting algorithm abstract for the double integrator with matched lipschitz disturbances we propose a continuous homogeneous controller providing finite time stability of the origin the disturbance is compensated exactly in finite time using a discontinuous function through an integral action since the controller is dynamic the closed loop is a third order system that achieves a third order sliding mode in the steady state the stability and robustness properties of the controller are proven using a smooth and homogeneous strict lyapunov function lf in a first stage the gains of the controller and the lf are designed using a method based on polya s theorem in a second stage the controller s gains are adjusted through a sum of squares representation of the lf

generalised absolute stability and sum of squares this paper introduces a general framework for analysing nonlinear systems using absolute stability theory and sum of squares programming the technique decomposes a vector field into a system with a polynomial vector field in feedback with a nonlinear memoryless term which is contained in a generalised sector inequality with polynomial bounds this decomposition can be used to model uncertainty in the nonlinearity or to bound difficult to analyse terms by simpler polynomial functions such as time varying non polynomial or higher order nonlinearities conditions for stability and regions of attraction are found using polynomial and lur e type lyapunov functions which generalise those used for the derivation of the multivariable circle and popov criteria in classical absolute stability the technique extends both absolute stability theory and the applicability of sum of squares programming the usefulness of the technique is demonstrated with illustrative examples

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

an asymptotically exact approach to robust semidefinite programming problems with function variables this technical note provides an approximate approach to a semidefinite programming problem with a parameter dependent constraint and a function variable this problem covers a variety of control problems including a robust stability performance analysis with a parameter dependent lyapunov function in the proposed approach the original problem is approximated by a standard semidefinite programming problem through two steps first the function variable is approximated by a finite dimensional variable second the parameter dependent constraint is approximated by a finite number of parameter independent constraints both steps produce approximation error on the sum of these approximation errors this technical note provides an upper bound this bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error moreover this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem

a convex optimization model for finding non negative polynomials this paper presents a convex optimization model for the problem of finding some polynomials for which certain linear combinations are non negative polynomials this model is then applied to solve several filter design problems we first reformulate some low pass filter design problems with finite or infinite impulse response as optimization problems over non negative real or complex polynomials whose feasibility problems can be solved by applying our model the whole optimization problems are then solved by using a combination of a bisection search procedure on an appropriate parameter and our convex optimization model to solve the feasibility problems some numerical examples illustrate the method

sparse sos relaxations for minimizing functions that are summations of small polynomials this paper discusses how to find the global minimum of functions that are summations of small polynomials small means involving a small number of variables some sparse sum of squares sos techniques are proposed we compare their computational complexity and lower bounds with prior sos relaxations under certain conditions we also discuss how to extract the global minimizers from these sparse relaxations the proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems numerical experiments are presented which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems lastly we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

on weak approximation of stochastic differential equations through hard bounds by mathematical programming upper and lower hard bounds of the expected value on stochastic differential equations can be obtained with the help of mathematical programming and the dynkin formula without recourse to monte carlo sample paths simulation in this paper we show that feasible solutions of those optimization approaches further provide useful additional information namely feasible solutions provide upper and lower bounds for arbitrary intermediate times and or different initial states we also show that the optimization approach can be applied to stochastic differential equations with a random initial state our theoretical analyses are illustrated by numerical results on the survival probability for a square root diffusion on the moment estimation of a doleans dade stochastic exponential with jumps and on pricing of a barrier basket option under the multiasset black scholes model

positive semidefinite diagonal minus tail forms are sums of squares by a diagonal minus tail form of even degree we understand a real homogeneous polynomial f x1 xn f x d x t x where the diagonal part d x is a sum of terms of the form b i x i 2d with all bi 0 and the tail t x a sum of terms a i 1i 2 cdots i n x 1 i 1 cdots x n i n with a i 1i 2 cdots i n 0 and at least two i 1 we show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials we also give an easily tested sufficient condition for a polynomial to be a sum of squares of polynomials sos and show that the class of polynomials passing this test is wider than the class passing lasserre s recent conditions another sufficient condition for a polynomial to be sos like lasserre s piecewise linear in its coefficients is also given

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

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

semi infinite programming using high degree polynomial interpolants and semidefinite programming in a common formulation of semi infinite programs the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval if this function is sufficiently closely approximable by a polynomial or a rational function then the semi infinite program can be reformulated as an equivalent semidefinite program which in turn can be solved with interior point methods very efficiently to high accuracy on the other hand solving this semidefinite program is challenging if the polynomials involved are of high degree due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum of squares representation of nonnegative polynomials are badly scaled we combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties using sum of squares interpolants specifically it is shown that the c

a region dividing technique for constructing the sum of squares approximations to robust semidefinite programs in this technical note we present a novel approach to robust semidefinite programs of which coefficient matrices depend polynomially on uncertain parameters the approach is based on approximation with the sum of squares polynomials but in contrast to the conventional sum of squares approach the quality of approximation is improved by dividing the parameter region into several subregions the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer an advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division a numerical example on polynomial optimization is presented to show usefulness of the present approach

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

stability and performance verification of optimization based controllers this paper presents a method to verify closed loop properties of optimization based controllers for deterministic and stochastic constrained polynomial discrete time dynamical systems the closed loop properties amenable to the proposed technique include global and local stability performance with respect to a given cost function both in a deterministic and stochastic setting and the mathcal l 2 gain the method applies to a wide range of practical control problems for instance a dynamical controller e g a pid plus input saturation model predictive control with state estimation inexact model and soft constraints or a general optimization based controller where the underlying problem is solved with a fixed number of iterations of a first order method are all amenable to the proposed approach the approach is based on the observation that the control input generated by an optimization based controller satisfies the associated karush kuhn tucker kkt conditions which provided all data is polynomial are a system of polynomial equalities and inequalities the closed loop properties can then be analyzed using sum of squares sos programming

regularization methods for sdp relaxations in large scale polynomial optimization we study how to solve semidefinite programming sdp relaxations for large scale polynomial optimization when interior point methods are used typically only small or moderately large problems could be solved this paper studies regularization methods for solving polynomial optimization problems we describe these methods for semidefinite optimization with block structures and then apply them to solve large scale polynomial optimization problems the performance is tested on various numerical examples with regularization methods significantly bigger problems could be solved on a regular computer which is almost impossible with interior point methods

optimizing n variate n k nomials for small k we give a high precision polynomial time approximation scheme for the supremum of any honest n variate n 2 nomial with a constant term allowing real exponents as well as real coefficients our complexity bounds count field operations and inequality checks and are quadratic in n and the logarithm of a certain condition number for the special case of n variate n 2 nomials with integer exponents the log of our condition number is sub quadratic in the sparse size the best previous complexity bounds were exponential in the sparse size even for n fixed along the way we partially extend the theory of viro diagrams and a discriminants to real exponents we also show that for any fixed 0 deciding whether the supremum of an n variate n n nomial exceeds a given number is npr complete

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

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

the tracial moment problem and trace optimization of polynomials the main topic addressed in this paper is trace optimization of polynomials in noncommuting nc variables given an nc polynomial f what is the smallest trace f underline a can attain for a tuple of matrices underline a a relaxation using semidefinite programming sdp based on sums of hermitian squares and commutators is proposed while this relaxation is not always exact it gives effectively computable bounds on the optima to test for exactness the solution of the dual sdp is investigated if it satisfies a certain condition called flatness then the relaxation is exact in this case it is shown how to extract global trace optimizers with a procedure based on two ingredients the first is the solution to the truncated tracial moment problem and the other crucial component is the numerical implementation of the artin wedderburn theorem for matrix algebras due to murota kanno kojima kojima and maehara trace optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side two topics with many applications the most prominent being to linear systems engineering and quantum physics the optimization problems discussed here facilitate new possibilities for applications e g in operator algebras and statistical physics

complexity estimates depending on condition and round off error this paper has two agendas one is to develop the foundations of round off in computation the other is to describe an algorithm for deciding feasibility for polynomial systems of equations and inequalities together with its complexity analysis and its round off properties each role reinforces the other

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

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

polyhedra related to integer convex polynomial systems this paper deals with the reformulation of a polynomial integer program for deducing a linear integer relaxation of such a program a class of polyhedra that are associated with nonlinear functions is introduced a characterization of the family of polynomials for which our approach leads to an equivalent linear integer program is given finally the family of so called integer convex polynomials is defined and polyhedra related to such a polynomial are investigated

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

fuzzy modeling and control of multilayer incinerator abstract a method to design a fuzzy controller based on a fuzzy model is suggested the identification method of a fuzzy model is also discussed the suggested method is applied for the modelling and control of a multilayer incinerator the performance of the designed fuzzy controller is tested by computer simulation using the identified process model the results are satisfactory

stability of quasilinear fuzzy system abstract this paper deals with the stability of a system described by a quasilinear fuzzy model the stability analysis is considered by investigation of the transition process of the system two cases are discussed when all subsystems are stable and when one or more subsystems are unstable a non conventional decision of stabilization of a locally unstable system is proposed

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

lyapunov based exact stability analysis and synthesis for linear single parameter dependent systems we propose a class of polynomially parameter dependent quadratic ppdq lyapunov functions for assessing the stability of single parameter dependent linear time invariant s pdlti systems in a non conservative manner it is shown that the stability of s pdlti systems is equivalent to the existence of a ppdq lyapunov function a bound on the degree of the polynomial dependence of the lyapunov function in the system parameter is given explicitly the resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex finite dimensional optimisation problem extensions of the approach to state feedback controller synthesis are also provided

computation of the structured singular value via moment lmi relaxations the structured singular value ssv provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties unfortunately computing the ssv is an np hard problem and the polynomial time algorithms available in the literature are only able to provide except for some special cases upper and lower bounds on the exact value of the ssv in this work we present a new algorithm to compute an upper bound on the ssv in case of mixed real complex uncertainties the underlying idea of the developed approach is to formulate the ssv computation as a nonconvex polynomial optimization problem which is relaxed into a sequence of convex optimization problems through moment based relaxation techniques two heuristics to compute a lower bound on the ssv are also discussed the analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the robust control toolbox in matlab and it provides in most of the cases coincident lower and upper bounds on the structured singular value

on the combinatorial and algebraic complexity of quantifier elimination in this paper a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given this algorithm improves the complexity of the asymptotically fastest algorithm for this problem known to this data a new feature of this algorithm is that the role of the algebraic part the dependence on the degrees of the imput polynomials and the combinatorial part the dependence on the number of polynomials are sparated another new feature is that the degrees of the polynomials in the equivalent quantifier free formula that is output are independent of the number of input polynomials as special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields and also for solving the existential problem in the first order theory over real closed fields are obtained

on the complexity of numerical analysis we study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis a the blum shub smale model of computation over the reals and b a problem we call the generic task of numerical computation which captures an aspect of doing numerical computation in floating point similar to the long exponent model that has been studied in the numerical computing community we show that both of these approaches hinge on the question of understanding the complexity of the following problem which we call posslp given a division free straight line program producing an integer n decide whether n 0 in the blum shub smale model polynomial time computation over the reals on discrete inputs is polynomial time equivalent to posslp when there are only algebraic constants we conjecture that using transcendental constants provides no additional power beyond nonuniform reductions to posslp and we present some preliminary results supporting this conjecture the generic task of numerical computation is also polynomial time equivalent to posslp we prove that posslp lies in the counting hierarchy combining this with work of tiwari we obtain that the euclidean traveling salesman problem lies in the counting hierarchy the previous best upper bound for this important problem in terms of classical complexity classes being pspace in the course of developing the context for our results on arithmetic circuits we present some new observations on the complexity of the arithmetic circuit identity testing acit problem in particular we show that if n is not ultimately easy then acit has subexponential complexity

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

brief paper local stability analysis using simulations and sum of squares programming the problem of computing bounds on the region of attraction for systems with polynomial vector fields is considered invariant subsets of the region of attraction are characterized as sublevel sets of lyapunov functions finite dimensional polynomial parametrizations for lyapunov functions are used a methodology utilizing information from simulations to generate lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed the suitability of lyapunov function candidates is assessed solving linear sum of squares optimization problems qualified candidates are used to compute invariant subsets of the region of attraction and to initialize various bilinear search strategies for further optimization we illustrate the method on small examples from the literature and several control oriented systems

amoebas of genus at most one abstract the amoeba of a laurent polynomial f c z 1 1 z n 1 is the image of its zero set v f under the log absolute value map understanding the space of amoebas i e the decomposition of the space of all polynomials say with given support or newton polytope with regard to the existing complement components is a widely open problem in this paper we investigate the class of polynomials f whose newton polytope new f is a simplex and whose support a contains exactly one point in the interior of new f amoebas of polynomials in this class may have at most one bounded complement component we provide various results on the space of these amoebas in particular we give upper and lower bounds in terms of the coefficients of f for the existence of this complement component and show that the upper bound becomes sharp under some extremal condition we establish connections from our bounds to purbhoo s lopsidedness criterion and to the theory of a discriminants finally we provide a complete classification of the space of amoebas for the case that the exponent of the inner monomial is the barycenter of the simplex newton polytope in particular we show that the set of all polynomials with amoebas of genus 1 is path connected in the corresponding space of amoebas which proves a special case of the question on connectivity for general newton polytopes stated by h rullgard

a frank wolfe type theorem for nondegenerate polynomial programs in this paper we study the existence of optimal solutions to a constrained polynomial optimization problem more precisely let f 0 f 0 and f 1 ldots f p mathbb r n rightarrow mathbb r f 1 f p r n r be convenient polynomial functions and let s x in mathbb r n f i x le 0 i 1 ldots p ne emptyset s x r n f i x 0 i 1 p under the assumption that the map f 0 f 1 ldots f p mathbb r n rightarrow mathbb r p 1 f 0 f 1 f p r n r p 1 is non degenerate at infinity we show that if f 0 f 0 is bounded from below on s s then f 0 f 0 attains its infimum on s s

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

local stability analysis for uncertain nonlinear systems we propose a method to compute provably invariant subsets of the region of attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems we consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field this uncertainty description is motivated by both incapabilities in modeling as well as bilinearity and dimension of the sum of squares programming problems whose solutions provide invariant subsets of the region of attraction we demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

linearization functors on real convex sets we prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets these operations are convex analogues of hom functors tensor products symmetric powers exterior powers and general schur functors on vector spaces and lead to novel constructions even for polyhedra we discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

equilibrium independent passivity we extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium independent passivity a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point we provide a necessary input output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems an example from network stability analysis is presented which demonstrates the utility of this new definition we then proceed to numerical certification of equilibrium independent passivity using sum of squares programming finally through numerical examples we show that equilibrium independent passivity is less restrictive than incremental passivity

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

brief paper exact determinations of the maximal output admissible set for a class of nonlinear systems this paper is concerned with obtaining necessary and sufficient conditions for fulfilling specified state and control pointwise in time constraints against a certain class of nonlinear dynamics the results are generalizations of the maximal output admissible set theory to the case of nonlinear systems main contribution of the present paper is to propose explicit algorithmic procedures to determine the maximal output admissible set another contribution is that we discuss on a finite determinability of the maximal output admissible set for nonlinear systems some relations between observability of nonlinear systems and finite characterizations of the maximal output admissible set are clarified

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

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

analysis of autocatalytic networks in biology autocatalytic networks in particular the glycolytic pathway constitute an important part of the cell metabolism changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can lead to perturbations from its nominal operating condition we investigate the effects of such perturbations on stability properties e g the extent of regions of attraction of a particular family of autocatalytic network models numerical experiments demonstrate that systems that are robust with respect to perturbations in the parameter space have an easily verifiable in terms of proof complexity region of attraction properties motivated by the computational complexity of optimization based formulations we take a compositional approach and exploit a natural decomposition of the system induced by the underlying biological structure into a feedback interconnection of two input output subsystems a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics this decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input output properties of the subsystems it enables numerical as well as analytical construction of block diagonal lyapunov functions for a large family of autocatalytic pathways

set membership estimation of fiber laser physical parameters from input output power measurements the problem of fiber laser parameters estimation from input output transient power measurements is considered in the paper the proposed procedure is based on a suitable laser model obtained from the rate and propagation equations by proper manipulation of the simplified laser equations the estimation problem is formulated in the bounded error estimation framework where a priori information on physical parameters can be properly taken into account first the so called feasible parameter set is defined then the problem of optimally picking a single model from the set is formulated in terms of central conditional estimation and solved by exploiting recent results by the authors properties of the proposed solution are discussed the proposed algorithm is tested on a set of simulated data obtained from a commercial laser simulator