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

on secant updates for use in general constrained optimization in this paper we present two new classes of successive quadratic programming sqp secant methods for the equality constrained optimization problem one class of methods uses the sqp augmented lagrangian formulation while the other class uses the sqp lagrangian formulation we demonstrate under the standard assumptions that in both cases the bfgs and dfp versions of the algorithm are locally q superlinearly convergent to our knowledge this is the first time that either local or q superlinear convergence has been established for an sqp lagrangian secant method which uses either the bfgs or dfp updating philosophy and assumes no more than the standard assumptions since the standard assumptions do not require positive definiteness of the hessian of the lagrangian at the solution it is no surprise that our bfgs and dfp updates possess the hereditary positive definiteness property only on a proper subspace

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 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

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

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

a combined integro modal approach for predicting acoustic properties of irregular shaped cavities an integro modal approach is presented in this paper for computing the acoustic properties of irregular shaped cavities the method consists of discretizing the whole cavity into a series of subcavities whose acoustic pressure is decomposed either over a modal basis of regular subcavities or over that of the bounding cavities in the case of irregular shaped boundaries an integral formulation is then established to ensure continuity of both pressure and velocity between adjacent subcavities using a membrane with zero mass and stiffness to some extent the method provides a combined approach retaining the advantages of both the acoustoelastic method aem and the green s function method gfm numerical and experimental results are presented demonstrating the efficiency and accuracy of the suggested technique comparisons with other existing methods are also made it is shown that good accuracy on the computation of cavity modes can be obtained using a very limited number of subcavities

a recursive quadratic programming algorithm that uses differentiable exact penalty functions in this paper a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied the line search functions used are approximations to fletcher s differentiable exact penalty function global convergence and local superlinear convergence results are proved and some numerical results are given

regenerative analysis and steady state distributions for markov chains we apply regenerative theory to derive certain relations between steady state probabilities of a markov chain these relations are then used to develop a numerical algorithm to find these probabilities the algorithm is a modification of the gauss jordan method in which all elements used in numerical computations are nonnegative as a consequence the algorithm is numerically stable

a trust region algorithm for equality constrained optimization via a reduced dimension approach a trust region algorithm is presented for solving optimization problem with equality constraints the algorithm uses the byrd omojokun scheme to compute the steps and decompose the trial steps into two components normal component and tangential component but it differs from the byrd omojokun algorithm with a reduced dimension approach in computing each tangential component global convergence of the proposed algorithm is proved under some mild assumptions three numerical examples are given to illustrate the efficiency of the algorithm

folding algorithm for policy evaluation for markov decision processes with quasi birth death structure this technical note presents a new numerical procedure for policy evaluation of stochastic shortest path markov decision processes mdps having a level independent quasi birth death structure the algorithm is derived using a method analogous to the folding method of ye and li 1994 the computational complexity is o m 3 log 2 n o m 2 n where the process has n levels and m phases a simple example involving the control of two queues is presented to illustrate the application of this efficient policy evaluation algorithm to compare and rank control policies

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

a global convergence theory for an active trust region algorithm for solving the general nonlinear programing problem a new trust region active set algorithm for solving minimizing a nonlinear function subject to nonlinear equality and inequality constraints is described in this algorithm an active set strategy is used together with a projected hessian technique to compute the trial step a convergence theory for this algorithm is presented under important assumptions it is shown that the algorithm is globally convergent in particular it is shown that a subsequence of the iteration sequence is not bounded away from either fritz john points or kkt points

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

acoustical cavity excitation a tutorial exposition is given of cavity excitations and it is shown that relatively simple analytical solutions for the excitation of acoustic fields in cavities with lossy walls can be obtained for a broad class of such cavities the standard analytic procedure whereby one expresses the acoustic field as a sum over the eigenmodes of the cavity is reviewed and its limitations are pointed out an alternate procedure is to seek to make use of green s functions depending on a single coordinate and to not express such green s functions as sums over eigenfunctions instead one expresses the green s function as a closed form function that has a discontinuous slope at the value of the coordinate corresponding to the source location although the class of cavities for which this approach is possible is limited it is sufficiently broad to be of intrinsic interest and the results may be helpful to those seeking benchmark solutions and analytic insight displayed examples are for cavities whose shapes are spheres and cylinders with interior monopole dipole line and distributed sources the formulation yields simple expressions for the total power absorbed by the cavity walls

on the global convergence of an slp filter algorithm that takes eqp steps a global convergence proof is presented for a class of trust region filter type methods for nonlinear programming such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the methods are based on successively solving linear programming subproblems for which effective software is readily available the methods also permit the use of steps calculated on the basis of an equality constrained quadratic programming model which enables rapid convergence to take place for problems in which second order information is important the proof technique is presented in a fairly general context allowing a range of specific algorithm choices associated with choosing the quadratic model updating the trust region radius and with feasibility restoration

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

iterative methods for nearly singular linear systems iterative methods are developed and studied for near singular linear systems cx b our approach called the transformed minimal residual algorithm tmres is derived from any convergent iterative scheme sxk 1 txk b associated with a splitting c s t in each step of tmres the transformed residual s 1 b cx is minimized over a krylov space generated by s 1 t the original iterative scheme typically converges slowly when c is nearly singular while a krylov space generated by s 1 t often contains a much better approximation to a solution tmres is algebraically equivalent to the generalized minimal residual algorithm gmres preconditioned by s 1 although there are numerical differences since a different matrix s 1 c is used to generate the krylov space in preconditioned gmres special attention is given to sparsity and convergence issues related to linear systems of the form bf aa tr sigma bf i bf x bf b where sigma ge 0

folding algorithm a computational method for finite qbd processes with level dependent transitions this paper presents a new computational method for steady state analysis of finite quasi birth death qbd processes with level dependent transitions the qbd state space is defined in two dimension with n phases and k levels instead of formulating solutions in matrix geometric form the folding algorithm provides a technique for direct computation of spl pi p 0 where p is the qbd generator which is an nk spl times nk matrix taking a finite sequence of fixed cost binary reduction steps the k level matrix p is eventually reduced to a single level matrix from which a boundary vector is obtained each step halves the matrix size but keeps the qbd form the solution spl pi is expressed as a product of the boundary vector and a finite sequence of expansion factors the time and space complexity for solving spl pi p 0 is therefore reduced from o n sup 3 k and o n sup 2 k to o n sup 3 log sub 2 k and o n sup 2 log sub 2 k respectively the folding algorithm has a number of highly desirable advantages when it is applied to queueing analysis first the algorithm handles the multilevel control problem in finite buffer systems second its total independence of the phase structure allows the algorithm to apply to more elaborate multiple state markovian sources its computational efficiency numerical stability and superior error performance are also distinctive advantages

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

new explicit solutions in acoustics of closed spaces on the basis of divergent series a new approach to the acoustics of closed spaces is developed that involves solutions for polygonal shapes in explicit form it is shown that exact solutions can be constructed for polygonal geometries where all the interior angles are equal to n n is an integer it is stated that the set of such polygons consists of the rectangle known result and three types of triangles some new explicit formulas are obtained for the eigenfrequencies of the triangles it is demonstrated that the proposed technique also permits an exact representation of the impulse response function for the geometries described

convergence properties of minimization algorithms for convex constraints using a structured trust region in this paper we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region this development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others a common occurrence in large scale nonlinear applications after describing the structured trust region mechanism we prove global convergence for all algorithms in our class

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

global convergence of a nonmonotone filter method for equality constrained optimization abstract in this paper we present a global convergence theory for a class of nonmonotone filter trust region methods at each iteration the trial step is decomposed into a quasi normal step and a tangential step comparable to the traditional filter and monotone methods the new approach is more flexible and less computational scale under some reasonable conditions we show that there exists at least one accumulate point of the sequence of iterates that is a kkt point

global convergence of ssm for minimizing a quadratic over a sphere in an earlier paper minimizing a quadratic over a sphere siam j optim 12 2001 188 208 we presented the sequential subspace method ssm for minimizing a quadratic over a sphere this method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed we showed in this earlier paper that when the subspace contains a vector obtained by applying one step of newton s method to the first order optimality system ssm is locally quadratically convergent even when the original problem is degenerate with multiple solutions and with a singular jacobian in the optimality system in this paper we prove nonlocal convergence of ssm to a global minimizer whenever each ssm subspace contains the following three vectors i the current iterate ii the gradient of the cost function evaluated at the current iterate and iii an eigenvector associated with the smallest eigenvalue of the cost function hessian for nondegenerate problems the convergence rate is at least linear when vectors i iii are included in the ssm subspace