an interior point constrained trust region method for a special class of nonlinear semidefinite programming problems in this paper an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming sdp problems is described and analyzed such nonlinear and nonconvex programs arise e g in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional hilbert space the algorithm treats the abstract states and controls as independent variables in particular an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear sdp condition a positive definiteness condition and a nonlinear matrix equation is considered the algorithm is designed to take advantage of the structure of the problem it is an extension of an interior point trust region method to nonlinear and nonconvex sdps with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration some convergence results are given and finally several numerical examples demonstrate the applicability of the considered algorithm

nonlinear programming algorithms using trust regions and augmented lagrangians with nonmonotone penalty parameters the strategy for obtaining global convergence is based on the trust region approach the merit function is a type of augmented lagrangian a new updating scheme is introduced for the penalty parameter by means of which monotone increase is not necessary global convergence results are proved and numerical experiments are presented

solving the richardson equations close to the critical points we study the richardson equations close to the critical values of the pairing strength gc where the occurrence of divergences precludes numerical solutions we derive a set of equations for determining the critical g values and the non collapsing pair energies studying the behaviour of the solutions close to the critical points we develop a procedure to solve numerically the richardson equations for arbitrary coupling strength

integrability and superintegrability of the generalized n level many mode jaynes cummings and dicke models we analyze symmetries of the integrable generalizations of jaynes cummings and dicke models associated with simple lie algebras g and their reductive subalgebras gk t skrypnyk generalized n level jaynes cummings and dicke models classical rational r matrices and nested bethe ansatz j phys a math theor 41 475202 2008 we show that their symmetry algebras contain commutative subalgebras isomorphic to the cartan subalgebras of g which can be added to the commutative algebras of quantum integrals generated with the help of the quantum lax operators we diagonalize additional commuting integrals and constructed with their help the most general integrable quantum hamiltonian of the generalized n level many mode jaynes cummings and dicke type models using nested algebraic bethe ansatz

the extended heine stieltjes polynomials associated with a special lmg model the extended heine stieltjes polynomials associated with a special lipkin meshkov glick lmg model corresponding to the standard two site bose hubbard model are derived based on the stieltjes correspondence it is shown that there is a one to one correspondence between zeros of this new polynomial and solutions of the bethe ansatz equations for the lmg model a one dimensional classical electrostatic analog corresponding to the special lmg model is established according to stieltjes early work it shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the bethe ansatz equations for the lmg model and the number of possible configurations of equilibrium positions of the charges equals exactly the number of energy levels in the lmg model some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the lmg model are derived

trust region methods for solving the optimal output feedback design problem we consider the problem of designing feedback control laws when a complete set of state variables is not available for linear autonomous control systems with quadratic performance criterion the design problem consists of choosing an appropriate static output feedback sof gain matrix according to a certain objective function the corresponding non linear matrix optimization problem can be interpreted as an equality constrained minimization problem for solving this problem we propose a constrained trust region ctr algorithm which presents a new and efficient numerical approach for this problem class on the other hand based on the formulation used in the past the sof problem can be also interpreted as an unconstrained programming problem thus based on this interpretation we also develop an unconstrained trust region utr method finally several numerical examples for optimal sof problems demonstrate the applicability of the considered algorithm

a robust trust region method for nonlinear optimization with inequality constraint a new trust region algorithm for inequality constrained optimization is presented which solves two linear programming subproblems and a serious of quadratic subproblems at each successful iteration to obtain a acceptable trial step the algorithm can circumvent the difficulties associated with the possible inconsistency of trust region subproblem moreover the algorithm can converge to a point which satisfies a certain first order necessary condition even when the original problems itself is infeasible some global convergence properties are proved without regularity assumption and local superlinear convergence of the algorithm is obtained under standard conditions preliminary numerical results are reported on some classic problems

a robust trust region algorithm with a nonmonotonic penalty parameter scheme for constrained optimization an algorithm for solving the problem of minimizing a nonlinear function subject to equality constraints is introduced this algorithm is a trust region algorithm in computing the trial step a projected hessian technique is used that converts the trust region subproblem to one similar to that for the unconstrained case to force global convergence the augmented lagrangian is employed as a merit function one of the main advantages of this algorithm is the way that the penalty parameter is updated we introduce an updating scheme that allows for the first time to the best of our knowledge the penalty parameter to be decreased whenever it is warranted the behavior of this penalty parameter is studied a convergence theory for this algorithm is presented it is shown that this algorithm is globally convergent and that the globalization strategy will not disrupt fast local convergence the local rate of convergence is also discussed this theory is sufficiently general so that it holds for any algorithm t

on the convergence theory of trust region based algorithms for equality constrained optimization in a recent paper dennis el alem and maciel proved global convergence to a stationary point for a general trust region based algorithm for equality constrained optimization this general algorithm is based on appropriate choices of trust region subproblems and seems particularly suitable for large problems this paper shows global convergence to a point satisfying the second order necessary optimality conditions for the same general trust region based algorithm the results given here can be seen as a generalization of the convergence results for trust regions methods for unconstrained optimization obtained by more and sorensen the behavior of the trust radius and the local rate of convergence are analyzed some interesting facts concerning the trust region subproblem for the linearized constraints the quasi normal component of the step and the hard case are presented it is shown how these results can be applied to a class of discretized optimal control problems

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

on the determinant representations of gaudin models scalar products and form factors we propose alternative determinant representations of certain form factors and scalar products of states in rational gaudin models realized in terms of compact spins we use alternative pseudo vacuums to write overlaps in terms of partition functions with domain wall boundary conditions contrarily to slavnov s determinant formulas this construction does not require that any of the involved states be solutions to the bethe equations a fact that could prove useful in certain non equilibrium problems moreover by using an atypical determinant representation of the partition functions we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges these eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual bethe equations consequently the current work allows important simplifications to numerical procedures addressing decoherence in gaudin models

robust regulation with h 2 performance abstract we consider the problem of minimizing the h 2 norm of a feedback system subject to the constraint that the controller robustly regulate against constants and sinusoids we show that for any 0 there exists a controller that both achieves robust regulation and renders the closed loop h 2 norm within of the optimal norm the norm achievable without the robust regulation constraint we also provide conditions that are both necessary and sufficient for there to exist a robustly regulating controller that achieves the optimal h 2 norm all proofs are constructive

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

a penalty free type nonmonotone trust region method for nonlinear constrained optimization in this paper we propose a penalty free type nonmonotone trust region method for solving general nonlinear programming problem the algorithmic framework yields global convergence without using any penalty function we analyze the global convergence of the main algorithm for the degenerate problems and also give the globally convergent results under the linear independence constraint qualification the preliminary numerical tests are reported

a new penalty free type algorithm based on trust region techniques abstract in this paper we propose a new penalty free type method for nonlinear equality constrained problems the new algorithm uses trust region framework and feasibility safeguarding technique moreover it has no choice of penalty parameter and penalty function as a merit function and it does not use the filter technique to avoid the penalty function either we analyze the global convergence of the main algorithm under the standard assumptions the preliminary numerical tests are reported

determinant representation of the domain wall boundary condition partition function of a richardson gaudin model containing one arbitrary spin in this work we present a determinant expression for the domain wall boundary condition partition function of rational xxx richardson gaudin models which in addition to spins contains one arbitrarily large spin s the proposed determinant representation is written in terms of a set of variables which from previous work are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic bethe equations such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the usual highly nonlinear bethe equations it can therefore offer significant gains in stability and computation speed

an lmi approach to h synthesis subject to almost asymptotic regulation constraints a multi objective controller synthesis problem is considered in which an output is to be regulated approximately by assuring a bound on the steady state peak amplification in response to an infinite energy disturbance while also guaranteeing a desired level of performance measured in terms of the worst case energy gain from a finite energy input to a performance output relying on a characterization of the controllers with which almost asymptotic regulation is accomplished the problem of guaranteeing the desired level of performance is reduced to solving a system of linear matrix inequalities subject to a set of linear equality constraints based on the solution of this system a procedure is outlined for the construction of a suitable controller whose order is equal to the order of the plant plus the order of the exogenous system

brief paper algebraic and topological aspects of the regulator problem for lumped linear systems the regulator problem with robustness is solved for systems modelled by rational transfer matrices a topology for possibly unstable plants is presented a necessary and sufficient condition is derived for the existence of a proper controller which provides internal stability and output regulation throughout an open neighbourhood of the plant a characterization of all such controllers is determined

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

performance with regulation constraints we study the problem where we have a regulation asymptotic tracking requirement together with a performance requirement for continuous time systems typically we measure performance by the h 2 or h norm of a chosen transfer function matrix although any other norm such as the l 1 norm could be used in the case when the performance is measured by the h 2 norm there is no loss in performance due to the regulation constraint on the other hand when the performance is measured by the h norm there exists in general certain loss of performance due to the regulation constraint and we explicitly characterize such a loss determining suitable controllers for a given system with both regulation and performance constraints requires solving only an unconstrained optimization problem for an auxiliary system which is explicitly constructed here

robust regulation in the presence of norm bounded uncertainty we consider robust regulation against steps and sinusoids in the presence of unstructured uncertainty the unstructured uncertainty is norm bounded by a constant that is given a priori this problem is equivalent to a certain multiobjective problem where one objective is robust regulation and the other is the standard objective of h sub spl infin suboptimal control it is shown that a solution to this problem exists if and only if the standard h sub spl infin problem admits a solution and certain matrix inequalities are satisfied these solvability conditions are readily computable controller synthesis is also addressed

algebraic bethe ansatze and eigenvalue based determinants for dicke jaynes cummings gaudin quantum integrable models in this work we construct an alternative formulation to the traditional algebraic bethe ans tze for quantum integrable models derived from a generalized rational gaudin algebra realized in terms of a collection of spins 1 2 coupled to a single bosonic mode the ensemble of resulting models which we call dicke jaynes cummings gaudin models are particularly relevant for the description of light matter interaction in the context of quantum optics having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions we also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges since these eigenvalues obey a much simpler set of quadratic bethe equations the resulting expressions could then offer important simplifications for the numerical treatment of these models

a global convergence theory for the celis dennis tapia trust region algorithm for constrained optimization a global convergence theory for a class of trust region algorithms for solving the equality constrained optimization problem is presented this theory is sufficiently general that it holds for any algorithm that generates steps giving at least a fraction of cauchy decrease in the quadratic model of the constraints and that uses the augmented lagrangian as a merit function this theory is used to establish global convergence of the 1984 celis dennis tapia algorithm with a different scheme for updating the penalty parameter the behavior of the penalty parameter is also discussed

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

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

a global convergence theory for dennis el alem and maciel s class of trust region algorithms for constrained optimization without assuming regularity this work presents a convergence theory for dennis el alem and maciel s class of trust region based algorithms for solving the smooth nonlinear programming problem with equality constraints the results are proved under very mild conditions on the quasi normal and tangential components of the trial steps the lagrange multiplier estimates and the hessian estimates are assumed to be bounded no regularity assumption is made in particular linear independence of the gradients of the constraints is not assumed the theory proves global convergence for the class in particular it shows that a subsequence of the iteration sequence satisfies one of four types of mayer bliss stationary conditions in the limit

generalized n level jaynes cummings and dicke models classical rational r matrices and algebraic bethe ansatz we construct integrable n level generalizations of jaynes cummings and dicke models corresponding to the simple reductive lie algebras of rank n we show that for each such lie algebra there exist many integrable jaynes cummings and dicke type models each of which is associated with the reductive subalgebras containing cartan subalgebra and is obtained via reduction from jaynes cummings or dicke type models with a maximal number of bosons we diagonalize the constructed jaynes cummings and dicke type hamiltonians in the physically most interesting case of using a nested algebraic bethe ansatz

global and local convergence of a penalty free method for nonlinear programming we present a class of trust region algorithms that do not use any penalty function or a filter for nonlinear equality constrained optimization in each iteration the infeasibility is controlled by a progressively decreasing upper limit and trial steps are computed by a byrd omojokun type trust region strategy measures of optimality and infeasibility are computed whose relationship serves as a criterion on which the algorithm decides which one to focus on improving as a result the algorithm keeps a balance between the improvements on optimality and feasibility even if no restoration phase which is required by filter methods is used the framework of the algorithm ensures the global convergence without assuming regularity or boundedness on the iterative sequence by using a second order correction strategy marato s effect is avoided and fast local convergence is shown the preliminary numerical results are reported

brief paper generalized asymptotic regulation with guaranteed h2 performance an lmi solution a multi objective controller synthesis problem is considered in which a set of generalized asymptotic regulation constraints are to be satisfied while also guaranteeing a desired level of performance measured in terms of the asymptotic variance of a performance output in response to a white noise random process with zero mean and identity covariance matrix the generalized regulation constraints are expressed as bounds on the state steady peak in response to disturbances generated by an anti stable autonomous exogenous system with nonzero initial states the controllers that guarantee these constraints can be realized by replicating the dynamics of the exogenous system within a certain structure formed by gain matrices which are subject to convex constraints and an accompanying controller with which the feedback loop is to be stabilized formulating the design of the accompanying controller in a way to guarantee the additional performance objective the overall design is rendered tractable over a set of free variables in terms of which the synthesis of a suitable controller is also outlined the order of the controller is equal to the order of the plant plus the order of the exogenous system by an adaptation of the developed techniques a solution is also provided for the problem of generalized asymptotic regulation with suboptimal transient response

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

further results on the robust tracking problem in two degree of freedom control systems the robust tracking problem in two degree of freedom control systems is formulated in a general setting which includes the case where the controlled output is not available directly a complete solution of the problem which is an extension of previous results is derived in a considerably simplified and transparent manner furthermore we show what kind of information about the output is needed to achieve robust tracking

adaptive multilevel inexact sqp methods for pde constrained optimization we present a class of inexact adaptive multilevel trust region sqp methods for the efficient solution of optimization problems governed by nonlinear pdes the algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process 1 implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2 implementable accuracy requirements for iterative solvers of the linearized pde and adjoint pde on the current grid we prove global convergence to a stationary point of the infinite dimensional problem moreover we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equations numerical results are presented

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism