primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

the inverse spectral problem for self adjoint hankel operators in 8 5 we have considered a geometric problem in the theory of stationary gaussian processes and we have reduced this problem to the problem of the description of the bounded linear operators on hilbert space that are unitarily equivalent to moduli of hankel operators in this chapter we are going to solve the latter problem which in turn will lead to a solution of the above geometric problem in prediction theory

a probabilistic analytic center cutting plane method for feasibility of uncertain lmis many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution x that satisfies a set of inequalities of the form f f x d 0 d d this set typically contains an infinite and uncountable number of inequalities and it has been proved that the related robust feasibility problem is numerically hard to solve in general in this paper we discuss a family of cutting plane methods that solve efficiently a probabilistically relaxed version of the problem specifically under suitable hypotheses we show that an analytic center cutting plane scheme based on a probabilistic oracle returns in a finite and pre specified number of iterations a solution x which is feasible for most of the members of f except possibly for a subset having arbitrarily small probability measure

operator spaces and residually finite dimensional c algebras abstract for every operator space x the c algebra containing it in a universal way is residually finite dimensional that is has a separating family of finite dimensional representations in particular the free c algebra on any normed space is residually finite dimensional this is an extension of an earlier result by goodearl and menal and our short proof is based on a criterion due to exel and loring

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

hyers ulam rassias stability of a jensen type functional equation abstract in this paper we solve the jensen type functional equation 1 1 likewise we investigate the hyers ulam rassias stability of this equation

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

semidefinite programming for uncertain linear equations in static analysis of structures abstract this paper presents a method for computing a minimal bound that contains the solution set to the uncertain linear equations particularly we aim at finding a bounding ellipsoid for static response of structures where both external forces and elastic moduli of members are assumed to be imprecisely known and bounded by using the s lemma we formulate a semidefinite programming sdp problem which provides an outer approximation of the minimal bounding ellipsoid bounding ellipsoids are computed for nodal displacements of uncertain braced frames as the solutions of the presented sdp problems by using the primal dual interior point method

interval robust multi objective algorithm abstract this paper introduces a method for solving multi objective optimization problems in uncertain environment when the uncertainty factors of the optimization problem can be included into the mathematical model through bounded intervals i rmoa interval robust multi objective algorithm can find an enclosure of the robust pareto frontier in this approach the robust pareto solutions are the ones that have the best performance when the worst case scenario characterized by the uncertainty parameters is considered i rmoa has some positive aspects it does not require the calculus of derivatives it has only two input parameters it is a reliable tool for solving different robust optimization problems which can be nonlinear and discontinuous or have nonconvex pareto frontier for instance the success of the method depends only on the quality of the objective inclusion functions and the precision parameters its disadvantage lies in the fact that it requires high computational effort when high dimensional problems are considered or when a very accurate enclosure is needed two analytical robust test functions are proposed to be treated by i rmoa and to validate the results provided by a stochastic multi objective optimization method the results are satisfactory

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

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

stability of a jensen type equation in the space of generalized functions abstract we reformulate and solve the stability problem of a jensen type functional equation 3 f x y z 3 f x f y f z 2 f x y 2 2 f y z 2 2 f z x 2 0 in the spaces of some generalized functions such as tempered distributions and fourier hyperfunctions

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

stability of jensen equations in the space of generalized functions abstract making use of heat kernel we prove stabilities of the jensen and jensen pexider equations in a space of generalized functions like the spaces of tempered distributions and fourier hyperfunctions

residually finite quantum group algebras abstract we show that provided n 3 the involutive hopf algebra a u n coacting universally on an n dimensional hilbert space has enough finite dimensional representations in the sense that every non zero element acts non trivially in some finite dimensional representation this implies that the discrete quantum group with group algebra a u n is maximal almost periodic i e it embeds in its quantum bohr compactification answering a question posed by p soltan in 21 we also prove analogous results for the involutive hopf algebra b u n coacting universally on an n dimensional hilbert space equipped with a non degenerate bilinear form

contractive completions of hankel partial contractions abstract a hankel partial contraction is a hankel matrix such that not all of its entries are determined but in which every well defined submatrix is a contraction we address the problem of whether a hankel partial contraction in which the upper left triangle is known can be completed to a contraction it is known that the 2 2 and 3 3 cases can be solved and that 4 4 hankel partial contractions cannot always be completed we introduce a technique that allows us to exhibit concrete examples of such 4 4 matrices and to analyze in detail the dependence of the solution set on the given data at the same time we obtain necessary and sufficient conditions on the given cross diagonals in order for the matrix to be completed we also study the problem of extending a contractive hankel block of size n to one of size n 1

low complexity polytopic invariant sets for linear systems subject to norm bounded uncertainty we propose a novel algorithm to compute low complexity polytopic robust control invariant rci sets along with the corresponding state feedback gain for linear discrete time systems subject to norm bounded uncertainty additive disturbances and state input constraints using a slack variable approach we propose new results to transform the original nonlinear problem into a convex lmi problem whilst introducing only minor conservatism in the formulation through numerical examples we illustrate that the proposed algorithm can yield improved maximal minimal volume rci set approximations in comparison with the schemes given in the literature

sensitivity analysis of optimization problems under second order regular constraints we present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition this is a technical condition that is always satisfied in the case of semi definite optimization we derive lipschitz and holder expansions of approximate optimal solutions under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem we show how the theory applies to semi infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds and discuss the differentiability of metric projections as well as the moreau yosida regularization finally we show how the theory applies to semi definite optimization

operator algebras of functions abstract we present some general theorems about operator algebras that are algebras of functions on sets including theories of local algebras residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector valued reproducing kernel hilbert space we use these to further develop a quantized function theory for various domains that extends and unifies agler s theory of commuting contractions and the arveson drury popescu theory of commuting row contractions we obtain analogous factorization theorems prove that the algebras that we obtain are dual operator algebras and show that for many domains supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices

on the l1 algebra of certain monoids the monoids considered are the free monoid m x and the free monoid with involution mi x on a nonempty set x in each case relative to a simply defined involution an explicit construction is given for a separating family of continuous star matrix representations of the l 1 algebra of the monoid and it is shown that this algebra admits a faithful trace the results are based on earlier work by m j crabb et al concerning the complex semigroup algebras of m x and mi x

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

extended matrix cube theorems with applications to µ theory in control in this paper we study semi infinite systems of linear matrix inequalities which are generically np hard for these systems we introduce computationally tractable approximations and derive quantitative guarantees of their quality as applications we discuss the problem of maximizing a hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value with the help of our complex matrix cube theorem we demonstrate that the standard scaling upper bound on m is a tight upper bound on the largest level of structured perturbations of the matrixm for which all perturbed matrices share a common lyapunov certificate for the discrete time stability

symmetric hankel operators minimal norm extensions and eigenstructures abstract the minimal norm extension problem for real partial hankel matrices is studied let x i i n 1 n be given real numbers find x i i n such that the finite hankel matrix a h x x 1 x 2 x n x 2 x 3 0 x n 0 0 has lowest possible norm as an operator on the eucledian space r n this min max problem is reduced to an unconstrained maximization problem it is close to a nonlinear eigenvalue problem the results suggest a new class of computer algorithms

a generalization of the hyers ulam rassias stability of the pexider equation in this paper we prove a generalization of the stability of the pexider equation f x y g x h y in the spirit of hyers ulam rassias and g vruta