the convergence of subspace trust region methods the trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence however the subproblem in the trust region method is difficult or time consuming to solve in practical computation especially in large scale problems in this paper we consider a new class of trust region methods specifically subspace trust region methods the subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace it is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially we investigate the global convergence and convergence rate of these methods

global optimality conditions and optimization methods for constrained polynomial programming problems the general constrained polynomial programming problem gpp is considered in this paper problem gpp has a broad range of applications and is proved to be np hard necessary global optimality conditions for problem gpp are established then a new local optimization method for this problem is proposed by exploiting these necessary global optimality conditions a global optimization method is proposed for this problem by combining this local optimization method together with an auxiliary function some numerical examples are also given to illustrate that these approaches are very efficient

a global optimization algorithm using parametric linearization relaxation in this paper a global optimization algorithm based on a parametric linearizing method for generalized quadratic programming gqp i e the quadratic programming problem with nonconvex quadratic constraints is proposed by utilizing the linearizing method initial nonconvex nonlinear problem gqp is reduced to a sequence of linear programming problems through the successive refinement of a linear relaxation of feasible region and of the objective function the proposed algorithm is convergent to the global minimum of gqp by means of the subsequent solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve global minimum of gqp on a microcomputer

a note on polynomial solvability of the cdt problem recent research has focused on the complexity of extensions of the classical trust region subproblem which addresses the minimization of a quadratic function over a unit ball in mathbb r n

formulas for hankel singular values and vectors for a class of input delay systems this paper gives explicit formulas for hankel singular values and vectors for a class of multiple input delay systems the class consists of systems whose gramian is reducible to an integral operator with semi separable kernel this structure is exploited in deriving a transcendental equation for singular values

a simpler formula for the singular values of a certain hankel operator this paper deals with the problem of computing the singular values and vectors of a hankel operator with symbol m w where m h is arbitrary inner and w h is rational a simplified version of the formula given in 6 is obtained for computing the singular values of the hankel operator this result is applied to the one block h optimal control of siso stable infinite dimensional plants and rational weights using this new formula a simple expression is derived for the h optimal controller whose structure was observed in 9

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

weighted sensitivity minimization general plants in h and rational weights abstract for general plants in h and rational weighting functions the problem of computing the weighted h optimal sensitivity is considered a general formula is derived for a function whose singularities determine the optimal sensitivity 6 a determinantal expression for the optimal sensitivity is then deduced and related to a conjecture of zhou and khargonekar

approximation of infinite dimensional systems a fourier series based method for approximation of stable infinite dimensional linear time invariant system models is discussed the basic idea is to compute the fourier series coefficients of the associated transfer function t sub d z and then take a high order partial sum two results on h sup infinity convergence and associated error bounds of the partial sum approximation are established it is shown that the fourier coefficients can be replaced by the discrete fourier transform coefficients while maintaining h sup infinity convergence thus a fast fourier transform algorithm can be used to compute the high order approximation this high order finite dimensional approximation can then be reduced using balanced truncation or optimal hankel approximation leading to the final finite dimensional approximation to the original infinite dimensional model this model has been tested on several transfer functions of the time delay type with promising results

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on spectral geometry of minimal surfaces in ⁿ by employing the standard isometric imbedding of cpn into the euclidean space a classification theorem for full minimal 2 type surfaces in cpn that are not holomorphic is given all such compact minimal surfaces are either totally real minimal surfaces in cp2 or totally real superminimal surfaces in cp3 and cp4 in the latter case they are locally unique moreover some eigenvalue inequalities for compact minimal surfaces of cpn with constant kaehler angle are shown

on the solution of a two ball trust region subproblem in this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution the method decomposes the subproblem and allows the application of efficient well studied methods for the solution of trust region subproblems arising in unconstrained optimization in the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem

strong duality for the cdt subproblem a necessary and sufficient condition in this paper we consider the problem of minimizing a nonconvex quadratic function subject to two quadratic inequality constraints as an application such a quadratic program plays an important role in the trust region method for nonlinear optimization such a problem is known as the celis dennis and tapia cdt subproblem in the literature the lagrangian dual of the cdt subproblem is a semidefinite program sdp hence convex and solvable however a positive duality gap may exist between the cdt subproblem and its lagrangian dual because the cdt subproblem itself is nonconvex in this paper we present a necessary and sufficient condition to characterize when the cdt subproblem and its lagrangian dual admits no duality gap i e the strong duality holds this necessary and sufficient condition is easy verifiable and involves only one any optimal solution of the sdp relaxation for the cdt subproblem moreover the condition reveals that it is actually rare to render a positive duality gap for the cdt subproblems in general moreover if the strong duality holds then an optimal solution for the cdt problem can be retrieved from an optimal solution of the sdp relaxation by means of a matrix rank one decomposition procedure the same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the lagrangian multipliers at a kkt point furthermore we show that the condition is numerically easy to work with approximatively

some remarks on hamiltonians and the infinite dimensional one block h problem in this note a simple proof is given for the hamiltonian solution to the h optimal sensitivity for plants with arbitrary inner transfer function the approach combines skew toeplitz theory and state space representations and gives rise to a straightforward and basis free methodology

singular values and vectors of a class of hankel operators abstract this paper considers the problem of determining the singular values and vectors of the class of hankel operators whose symbol can be written in the form w where is inner and w is rational we will obtain a new and simplified proof of the fact conjectured in 17 and first proved in 9 that the zeros of a certain determinantal expression coincide with the hankel singular values we will also show how this expression can be used to generate the singular vectors

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

hankel norm of sampled data systems generalizes the concept of hankel norm for purely continuous time systems to sampled data systems the hankel norm of a sampled data system is defined by taking into account intersample behaviors the formula of the hankel norm of a sampled data system is given by a hybrid state space model approach the authors show that the hankel norm is equal to that of an equivalent discrete time system two simple examples including sampled data controller reduction are demonstrated

narrowing the difficulty gap for the celis dennis tapia problem we study the celis dennis tapia cdt problem minimize a non convex quadratic function over the intersection of two ellipsoids in contrast to the well studied trust region problem where the feasible set is just one ellipsoid the cdt problem is not yet fully understood our main objective in this paper is to narrow the difficulty gap that occurs when the hessian of the lagrangian is indefinite at all karush kuhn tucker points we prove new sufficient and necessary conditions both for local and global optimality based on copositivity giving a complete characterization in the degenerate case

optimality conditions for trust region subproblems involving a conic model the problem of minimizing a conic function subject to an ellipsoidal constraint is discussed in this paper the feasible set of this problem is divided into three different cases necessary and sufficient optimality conditions are obtained for different cases where there is a characterization of optimality for the basic case these conditions provide a sound basis for the further development of efficient methods for trust region subproblems of a conic model

computing a celis dennis tapia trust region step for equality constrained optimization we study an approach for minimizing a convex quadratic function subject to two quadratic constraints this problem stems from computing a trust region step for an sqp algorithm proposed by celis dennis and tapia 1985 for equality constrained optimization our approach is to reformulate the problem into a univariate nonlinear equation 0 where the function is continuous at least piecewise differentiable and monotone well established methods then can be readily applied we also consider an extension of our approach to a class of non convex quadratic functions and show that our approach is applicable to reduced hessian sqp algorithms numerical results are presented indicating that our algorithm is reliable robust and has the potential to be used as a building block to construct trust region algorithms for small sized problems in constrained optimization

optimality conditions for the minimization of a quadratic with two quadratic constraints the trust region method has been proven to be very successful in both unconstrained and constrained optimization it requires the global minimum of a general quadratic function subject to ellipsoid constraints in this paper we generalize the trust region subproblem by allowing two general quadratic constraints conditions and properties of its solution are discussed

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

robust stabilizability of normalized coprime factors the infinite dimensional case the problem of robustly stabilizing a linear system subject to h bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite dimensional systems this class has possible unbounded finite rank input and output operators which include many delay and distributed systems the optimal stability margin is expressed in terms of the solutions of the control and filter algebraic riccati equations

optimality conditions for the minimization of quadratic 0 1 problems in this paper we present some explicit local and global optimality conditions for unconstrained binary quadratic problems local optimality conditions are categorized by the neighborhood including 1 flip 2 flip and k flip neighborhoods after defining an evaluation vector and an evaluation matrix we propose algorithms that make it possible for us to obtain 1 flip and or 2 flip local solutions in polynomial time we provide a key insight into the relationship between the local optimality conditions and the global optimality conditions a discussion of global geometric optimality conditions is also presented in this paper

non convex quadratic minimization problems with quadratic constraints global optimality conditions in this paper we first examine how global optimality of non convex constrained optimization problems is related to lagrange multiplier conditions we then establish lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints we also obtain necessary global optimality conditions which are different from the lagrange multiplier conditions for special classes of quadratic optimization problems these classes include weighted least squares with ellipsoidal constraints and quadratic minimization with binary constraints we discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi extremal non convex quadratic optimization problems

trust region problems with linear inequality constraints exact sdp relaxation global optimality and robust optimization the trust region problem which minimizes a nonconvex quadratic function over a ball is a key subproblem in trust region methods for solving nonlinear optimization problems it enjoys many attractive properties such as an exact semi definite linear programming relaxation sdp relaxation and strong duality unfortunately such properties do not in general hold for an extended trust region problem having extra linear constraints this paper shows that two useful and powerful features of the classical trust region problem continue to hold for an extended trust region problem with linear inequality constraints under a new dimension condition first we establish that the class of extended trust region problems has an exact sdp relaxation which holds without the slater constraint qualification this is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range convexity whenever the dimension condition is fulfilled second we show that the dimension condition together with the slater condition ensures that a set of combined first and second order lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust region problem and consequently for strong duality through simple examples we also provide an insightful account of our development from sdp relaxation to strong duality finally we show that the dimension condition is easily satisfied for the extended trust region model that arises from the reformulation of a robust least squares problem lsp as well as a robust second order cone programming model problem socp as an equivalent semi definite linear programming problem this leads us to conclude that under mild assumptions solving a robust lsp or socp under matrix norm uncertainty or polyhedral uncertainty is equivalent to solving a semi definite linear programming problem and so their solutions can be validated in polynomial time

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

an algorithm for convex quadratic programming that requires o n 3 5 l arithmetic operations a new interior point method for minimizing a convex quadratic function over a polytope is developed we show that our method requires o n3 5l arithmetic operations in our algorithm we construct a sequence pz0 pz1 pzk of nested convex sets that shrink towards the set of optimal solution s during iteration k we take a partial newton step to move from an approximate analytic center of pzk 1 to an approximate analytic center of pzk a system of linear equations is solved at each iteration to find the step direction the solution that is available after o ml iterations can be converted to an optimal solution our analysis indicates that inexact solutions to the linear system of equations could be used in implementing this algorithm

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

on minimal surfaces in a kahler manifold of constant holomorphic sectional curvature this paper studies minimal surfaces in kshler manifolds of constant holomorphic sectional curvature using the technique of the moving frame in particular we provide a classification of the minimal two spheres in cpst complex projective sf space equipped with the fubini study metric this classification can be described as follows to each holomorphic curve in c pt1 classically there is associated a particular framing of c l l called the frenet frame each element of the frenet frame induces a minimal surface in cp the classification theorem states that all minimal surfaces of topological type of the two sphere occur in this manner the theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in kshler manifolds of constant holomorphic sectional curvature together with the riemann roch theorem

h interpolation in systems with commensurate input lags h infty optimal control and system design problems are known to mathematically reduce to the framework of operator interpolation here a solution scheme in the context of systems with multiple input lags is provided the main result a linear algebraic characterization of eigenvalue eigenfunction pairs can be utilized also in finite dimensional approximations of the associated hankel operator a particular feature of the analysis is the heavy reliance on time domain techniques and utilization of explicit time domain properties of the transfer function

a global optimization algorithm using linear relaxation various local optimization approaches have been developed for solving the non convex quadratically constrained quadratic programs qp but up to now less work has been devoted to solving global optimization of qp problem due to the inherent difficulty this paper considers the global minimum of qp that arise in various practice problems by utilizing the inherent property of the quadratic function an approximations a linear relaxation of qp is then obtained thus initial qp is reduced to a sequence of linear programming problems through the successive refinement of a linear relaxation of feasible region of the objective function the proposed algorithm is convergent to the global minimum of qp by means of the subsequent solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of qp on a microcomputer

second order cone constraints for extended trust region subproblems the classical trust region subproblem trs minimizes a nonconvex quadratic objective over the unit ball in this paper we consider extensions of trs having extra constraints when two parallel cuts are added to trs we show that the resulting nonconvex problem has an exact representation as a semidenite program with additional linear and second order cone constraints for the case where an additional ellipsoidal constraint is added to trs resulting in the two trust region subproblem ttrs we provide a new relaxation including second order cone constraints that strengthens the usual sdp relaxation

model reduction of neutral linear and nonlinear time invariant time delay systems with discrete and distributed delays the problem of model reduction by moment matching for linear and nonlinear differential time delay systems is studied the class of models considered includes neutral differential time delay systems with discrete delays and distributed delays the description of moment is revisited by means of a sylvester like equation for linear time delay systems and by means of the center manifold theory for nonlinear time delay systems in addition the moments at infinity are characterized for both linear and nonlinear time delay systems parameterized families of models achieving moment matching are given the parameters can be exploited to derive delay free reduced order models or time delay reduced order models with additional properties e g interpolation at an arbitrary large number of points finally the problem of obtaining a reduced order model of an unstable system is discussed and solved

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

a global convergence theory for general trust region based algorithms for equality constrained optimization this work presents a global convergence theory for a broad class of trust region algorithms for the smooth nonlinear programming problem with equality constraints the main result generalizes powell s 1975 result for unconstrained trust region algorithms the trial step is characterized by very mild conditions on its normal and tangential components the normal component need not be computed accurately the theory requires a quasi normal component to satisfy a fraction of cauchy decrease condition on the quadratic model of the linearized constraints the tangential component then must satisfy a fraction of cauchy decrease condition on a quadratic model of the lagrangian function in the translated tangent space of the constraints determined by the quasi normal component estimates of the lagrange multipliers and the hessians are assumed only to be bounded the other main characteristic of this class of algorithms is that the step is evaluated by using the augmented lagrangian as a merit function with the penalty parameter updated using the el alem scheme the properties of the step and the way that the penalty parameter is chosen are sufficient to establish global convergence as an example an algorithm is presented that can be viewed as a generalization of the steihaug toint dogleg algorithm for the unconstrained case it is based on a quadratic programming algorithm that uses a step in a quasi normal direction to the tangent space of the constraints and then takes feasible conjugate reduced gradient steps to solve the reduced quadratic program this algorithm should cope quite well with large problems for which effective preconditioners are known

optimality conditions and optimization methods for quartic polynomial optimization abstract in this paper multivariate quartic polynomial optimization program qpop is considered quartic optimization problems arise in various practical applications and are proved to be np hard we discuss necessary global optimality conditions for quartic problem qpop and then we present a new strongly or e strongly local optimization method according to necessary global optimality conditions which may escape and improve some kkt points finally we design a global optimization method for problem qpop by combining the new strongly or e strongly local optimization method and an auxiliary function numerical examples show that our algorithms are efficient and stable

solving generalized cdt problems via two parameter eigenvalues we consider solving a nonconvex quadratic minimization problem with two quadratic constraints one of which being convex this problem is a generalization of the celis denis tapia cdt problem and thus we refer to it as gcdt generalized cdt the cdt problem has been widely studied but no polynomial time algorithm was known until bienstock s recent work his algorithm solves the cdt problem in polynomial time with respect to the number of bits in data and log epsilon 1 by admitting an epsilon error in the constraints the algorithm however appears to be difficult to implement in this paper we present another algorithm for gcdt which is guaranteed to find a global solution for almost all gcdt instances and slightly perturbed ones in some exceptionally rare cases in exact arithmetic including eigenvalue computation our algorithm is based on the approach proposed by iwata nakatsukasa and takeda 2015 for computing the signed distance between overlapping ellipsoids our algorithm computes all the lagrange multipliers of gcdt by solving a two parameter linear eigenvalue problem obtains the corresponding kkt points and finds a global solution as the kkt point with the smallest objective value in practice in finite precision arithmetic our algorithm requires o n 6 log log u 1 computational time where n is the number of variables and u is the unit roundoff although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed numerical experiments illustrate that our algorithm performs well in finite precision arithmetic

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

a global optimization using linear relaxation for generalized geometric programming many local optimal solution methods have been developed for solving generalized geometric programming ggp but up to now less work has been devoted to solving global optimization of ggp problem due to the inherent difficulty this paper considers the global minimum of ggp problems by utilizing an exponential variable transformation and the inherent property of the exponential function and some other techniques the initial nonlinear and nonconvex ggp problem is reduced to a sequence of linear programming problems the proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of ggp on a microcomputer

approximation of delay systems a case study abstract the approximation of delay systems is studied a numerically well behaved method for computing hankel optima rational approximations for delay systems is presented based on certain properties of hankel matrices and the theory of laguerre polynomials the cf method of trefethen and certain pade approximations of delay systems are also considered the importance of a certain wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the l norm and the hankel norm is shown completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions a case study and numerical comparison is presented for the approximation of the parametric family of first order stable delay systems g s exp s ts 1 numerical experience indicates that l optimized cf approximations based on short truncated maclaurin series give in general somewhat smaller l approximation errors than hankel optim

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

l approximation and nuclearity of delay systems abstract the l convergence of partial fraction decompositions of single input single output delay systems is studied conditions for the nuclearity of such systems are derived and examples given

approximate solution of the trust region problem by minimization over two dimensional subspaces the trust region problem minimization of a quadratic function subject to a spherical trust region constraint occurs in many optimization algorithms in a previous paper the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two dimensional trust region problem they showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem this paper reports computational results showing that the two dimensional minimization approach gives nearly optimal reductions in then dimension quadratic model over a wide range of test cases we also show that there is very little difference in efficiency and reliability between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization these results may encourage the application of similar approximate trust region techniques in other contexts

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

explicit formulas for optimally robust controllers for delay systems this paper considers single input single output systems whose transfer functions take the form of a strictly proper rational function times a delay a closed form expression is presented for the controller which is optimally robust with respect to perturbations measured in the gap metric the formula allows the h sub spl infin loop shaping procedure of glover mcfarlane to be carried out explicitly for this class of systems without the need to first find a rational approximation of the plant the form of the controller involves a certain algebra of pseudo derivation operators these operators and their matrix generalizations play a central role in the derivation of the controller a discussion of the main properties of these operators is given an example is presented of a controller design to achieve disturbance attenuation and robust set point following for a plant with two lightly damped poles and a nontrivial time delay the performance is compared and shown to be superior to that of a smith predictor

on the weighted sensitivity minimization problem for delay systems we consider the h optimal sensitivity problem for delay systems in particular we consider computation of inf w q q h jr where w s is any function in rh jr and in h jr is any inner function we derive a new explicit solution in the pure delay case where e sh h 0

h 0e sensitivity minimization for delay systems this paper announces results on the problem of feedback compensator design for x norm weighted sensitivity minimization when the plant contains a delay in the input a complete solution is presented for the case of one pole zero weighting function and a single input single output plant for stable minimum phase rational part generalizations and proofs will be published elsewhere

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

further results on approximating nonconvex quadratic optimization by semidefinite programming relaxation we study approximation bounds for the semidefinite programming sdp relaxation of quadratically constrained quadratic optimization min f 0 x subject to f k x le 0 k 1 dots m where fk x xtakx bk tx ck in the special case of ellipsoid constraints with interior feasible solution at 0 we show that the sdp relaxation coupled with a rank 1 decomposition result of sturm and zhang math oper res to appear yields a feasible solution of the original problem with objective value at most 1 gamma 2 sqrt m gamma 2 times the optimal objective value where gamma sqrt smash b max k f k 0 1 for the single trust region problem corresponding to m 1 this yields an exact optimal solution in the general case we extend some bounds derived by nesterov optim methods softw 9 1998 pp 141 160 working paper core universite catholique de louvain louvain la neuve belgium 1998 ye math program 84 1999 pp 219 226 and nesterov wolkowicz and ye in handbook of semidefinite programming h wolkowicz r saigal and l vandenberghe eds kluwer academic publishers dordrecht the netherlands 2000 pp 360 419 for the special case where a k is diagonal and bk 0 for k 1 m we also discuss the generation of approximate solutions with high probability

a two variable approach to the two trust region subproblem the trust region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite programming sdp relaxation intersecting the feasible set with a second ellipsoid results in the two trust region subproblem ttrs even though ttrs can also be solved in polynomial time existing algorithms do not use sdp in this paper we investigate the use of sdp for ttrs starting from the basic sdp relaxation of ttrs which admits a gap recent research has tightened the basic relaxation using valid second order cone inequalities even still closing the gap requires more for the special case of ttrs in dimension n 2 we fully characterize the remaining valid inequalities which can be viewed as strengthened versions of the second order cone inequalities just mentioned we also demonstrate that these valid inequalities can be used computationally even when n 2 to solve ttrs instances that were previously unsolved using sdp based techniques

an integral equation approach to the singular values of a system with distributed output delays abstract we present a state space approach for the analysis of the hankel singular values of a system with output delays and we derive asymptotic estimates for them

harmonic maps of the two sphere into a complex grassmann manifold ii where qt is a complex valued one form defined up to a factor of absolute value 1 this form qp defines a complex structure on m for x e m the space f x has an orthogonal space f x of dimension n k we denote by f x and f x their corresponding projective spaces of dimensions k 1 and n k 1

alternative theorems for quadratic inequality systems and global quadratic optimization we establish alternative theorems for quadratic inequality systems consequently we obtain lagrange multiplier characterizations of global optimality for classes of nonconvex quadratic optimization problems we present a generalization of dine s theorem to a system of two homogeneous quadratic functions with a regular cone the class of regular cones are cones k for which k cup k is a subspace as a consequence we establish a generalization of the powerful s lemma which paves the way to obtain a complete characterization of global optimality for a general quadratic optimization model problem involving a system of equality constraints in addition to a single quadratic inequality constraint we then present an alternative theorem for a system of three nonhomogeneous inequalities by way of establishing the convexity of the joint range of three homogeneous quadratic functions using a regular cone this yields lagrange multiplier characterizations of global optimality for classes of trust region type problems with two inequality constraints finally we establish an alternative theorem for systems involving an arbitrary finite number of quadratic inequalities involving z matrices which are matrices with nonpositive off diagonal elements and present necessary and sufficient conditions for global optimality for classes of nonconvex inequality constrained quadratic optimization problems