on the generalized lanczos trust region method the so called trust region subproblem gets its name in the trust region method in optimization and also plays a vital role in various other applications several numerical algorithms have been proposed in the literature for solving small to medium size dense problems as well as for large scale sparse problems the generalized lanczos trust region gltr method proposed by n i m gould s lucidi m roma and p l toint siam j optim 9 1999 pp 561 580 is a natural extension of the classical lanczos method for the linear system to the trust region subproblem in this paper we first analyze the convergence of gltr to reveal its convergence behavior in theory and then propose new stopping criteria that can be integrated into gltr for better numerical performance specifically we develop a priori upper bounds for the convergence to both the optimal objective value as well as the optimal solution and argue that these bounds can be efficiently estimated numerically and serve as stopping criteria f

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

on the global convergence of trust region algorithms for unconstrained minimization many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large 2 we consider how large these approximations have to be if they prevent convergence when the objective function is bounded below and continuously differentiable thus we obtain a useful convergence result in the case when there is a bound on the second derivative approximations that depends linearly on the iteration number

a shamanskii like self adaptive levenberg marquardt method for nonlinear equations abstract in this paper we are concerned with the shamanskii like self adaptive levenberg marquardt methods for nonlinear equations we consider two choices of levenberg marquardt parameter one of them is the standard self adaptive levenberg marquardt parameter the other is nonmonotone self adaptive levenberg marquardt parameter by using the nonmonotone technique of grippo lampariello and lucidi under the error bound condition which is weaker than nonsingularity we show that the shamanskii like self adaptive levenberg marquardt methods converge with q order m 1 numerical experiments show the new algorithms are efficient and promising

a new matrix free algorithm for the large scale trust region subproblem we present a new method for the large scale trust region subproblem the method is matrix free in the sense that only matrix vector products are required we recast the trust region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter we then find the solution of the trust region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter the new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so called hard case we show that the new iteration is well defined and convergent at a superlinear rate we present computational results to illustrate convergence properties and robustness of the method

a filter trust region method for unconstrained optimization a new filter trust region algorithm for solving unconstrained nonlinear optimization problems is introduced based on the filter technique introduced by fletcher and leyffer it extends an existing technique of gould leyffer and toint siam j optim 15 2004 pp 17 38 for nonlinear equations and nonlinear least squares to the fully general unconstrained optimization problem the new algorithm is shown to be globally convergent to at least one second order critical point and numerical experiments indicate that it is very competitive with more classical trust region algorithms

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

a nonmonotone adaptive trust region method and its convergence abstract in this paper we combine the new trust region subproblem proposed in 1 with the nonmonotone technique to propose a new algorithm for unconstrained optimization the nonmonotone adaptive trust region method the local and global convergence properties of the nonmonotone adaptive trust region method are proved its efficiency is tested by numerical results

solving the trust region subproblem by a generalized eigenvalue problem the state of the art algorithms for solving the trust region subproblem trs are based on an iterative process involving solutions of many linear systems eigenvalue problems subspace optimization or line search steps a relatively underappreciated fact due to gander golub and von matt linear algebra appl 114 1989 pp 815 839 is that trss can be solved by one generalized eigenvalue problem with no outer iterations in this paper we rediscover this fact and discover its great practicality which exhibits good performance both in accuracy and efficiency moreover we generalize the approach in various directions namely by allowing for an ellipsoidal constraint dealing with the so called hard case and obtaining approximate solutions efficiently when high accuracy is unnecessary we demonstrate that the resulting algorithm is a general purpose trs solver effective both for dense and large sparse problems including the so called hard case our algorithm is easy to implement its essence i

a subspace minimization method for the trust region step we consider methods for large scale unconstrained minimization based on finding an approximate minimizer of a quadratic function subject to a two norm trust region constraint the steihaug toint method uses the conjugate gradient method to minimize the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary however if the conjugate gradient method is used with a preconditioner the steihaug toint method requires that the trust region norm be defined in terms of the preconditioning matrix if a different preconditioner is used for each subproblem the shape of the trust region can change substantially from one subproblem to the next which invalidates many of the assumptions on which standard methods for adjusting the trust region radius are based in this paper we propose a method that allows the trust region norm to be defined independently of the preconditioner the method solves the inequality constrained trust region subproblem over a sequence of evolving low dimensional subspaces each subspace includes an accelerator direction defined by a regularized newton method for satisfying the optimality conditions of a primal dual interior method a crucial property of this direction is that it can be computed by applying the preconditioned conjugate gradient method to a positive definite system in both the primal and dual variables of the trust region subproblem numerical experiments on problems from the cuter test collection indicate that the method can require significantly fewer function evaluations than other methods in addition experiments with general purpose preconditioners show that it is possible to significantly reduce the number of matrix vector products relative to those required without preconditioning

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

newton type minimization via the lanczos method this paper discusses the use of the linear conjugate gradient method developed via the lanczos method in the solution of large scale unconstrained minimization problems at each iteration of a newton type method the direction of search is defined as the solution of a quadratic subproblem when the number of variables is very large this subproblem may be solved using the linear conjugate gradient method of hestenes and stiefel we show how the equivalent lanczos characterization of the linear conjugate gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive definite hessian matrices at all points of the region of interest this derivation also makes it possible to compute a negative curvature direction at a stationary point the idea of a truncated newton method is to perform only a limited number of iterations of the quadratic subproblem this effectively gives a search direction that interpolates between the steepest de

the trust region subproblem with non intersecting linear constraints this paper studies an extended trust region subproblem etrs in which the trust region intersects the unit ball with m m linear inequality constraints when m 0 m 1 m 0 m 1 or m 2 m 2 and the linear constraints are parallel it is known that the etrs optimal value equals the optimal value of a particular convex relaxation which is solvable in polynomial time however it is also known that when m ge 2 m 2 and at least two of the linear constraints intersect within the ball i e some feasible point of the etrs satisfies both linear constraints at equality then the same convex relaxation may admit a gap with etrs this paper shows that the convex relaxation has no gap for arbitrary m m as long as the linear constraints are non intersecting

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

automatic determination of an initial trust region in nonlinear programming this paper presents a simple but efficient way to find a good initial trust region radius itrr in trust region methods for nonlinear optimization the method consists of monitoring the agreement between the model and the objective function along the steepest descent direction computed at the starting point further improvements for the starting point are also derived from the information gleaned during the initializing phase numerical results on a large set of problems show the impact the initial trust region radius may have on trust region methods behavior and the usefulness of the proposed strategy

finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws we develop here a new class of finite volume schemes on unstructured meshes for scalar conservation laws with stiff source terms the schemes are of equilibrium type hence with uniform bounds on approximate solutions valid in cell entropy inequalities and exact for some equilibrium states convergence is investigated in the framework of kinetic schemes numerical tests show high computational efficiency and a significant advantage over standard cell centered discretization of source terms equilibrium type schemes produce accurate results even on test problems for which the standard approach fails for some numerical tests they exhibit exponential type convergence rate in two of our numerical tests an equilibrium type scheme with 441 nodes on a triangular mesh is more accurate than a standard scheme with 50002 grid points

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

positive solutions of positive linear systems easily verifiable sufficient conditions are obtained for the existence of a positive solution componentwise of a linear nonhomogeneous system of equations with positive coefficients

a second derivative sqp method global convergence sequential quadratic programming sqp methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems although second derivative information may often be calculated there is little practical theory that justifies exact hessian sqp methods in particular the resulting quadratic programming qp subproblems are often nonconvex and thus finding their global solutions may be computationally nonviable this paper presents a second derivative sqp method based on quadratic subproblems that are either convex and thus may be solved efficiently or need not be solved globally additionally an explicit descent constraint is imposed on certain qp subproblems which guides the iterates through areas in which nonconvexity is a concern global convergence of the resulting algorithm is established

a modified nearly exact method for solving low rank trust region subproblem in this paper we first discuss how the nearly exact ne method proposed by more and sorensen 14 for solving trust region tr subproblems can be modified to solve large scale low rank tr subproblems efficiently our modified algorithm completely avoids computation of cholesky factorizations by instead relying primarily on the sherman morrison woodbury formula for computing inverses of diagonal plus low rank type matrices we also implement a specific version of the modified log barrier mlb algorithm proposed by polyak 17 where the generated log barrier subproblems are solved by a trust region method the corresponding direction finding tr subproblems are of the low rank type and are then solved by our modified ne method we finally discuss the computational results of our implementation of the mlb method and its comparison with a version of lancelot 5 based on a collection extracted from cuter 12 of nonlinear programming problems with simple bound constraints

an assessment of nonmonotone linesearch techniques for unconstrained optimization the purpose of this paper is to discuss the potential of nonmonotone techniques for enforcing convergence of unconstrained minimization algorithms from starting points distant from the solution linesearch based algorithms are considered for both small and large problems and extensive numerical experiments show that this potential is sometimes considerable a new variant is introduced in order to limit some of the identified drawbacks of the existing techniques this variant is again numerically tested and appears to be competitive finally the impact of preconditioning on the considered methods is examined

low complexity rls algorithms using dichotomous coordinate descent iterations in this paper we derive low complexity recursive least squares rls adaptive filtering algorithms we express the rls problem in terms of auxiliary normal equations with respect to increments of the filter weights and apply this approach to the exponentially weighted and sliding window cases to derive new rls techniques for solving the auxiliary equations line search methods are used we first consider conjugate gradient iterations with a complexity of operations per sample being the number of the filter weights to reduce the complexity and make the algorithms more suitable for finite precision implementation we propose a new dichotomous coordinate descent dcd algorithm and apply it to the auxiliary equations this results in a transversal rls adaptive filter with complexity as low as multiplications per sample which is only slightly higher than the complexity of the least mean squares lms algorithm multiplications simulations are used to compare the performance of the proposed algorithms against the classical rls and known advanced adaptive algorithms fixed point fpga implementation of the proposed dcd based rls algorithm is also discussed and results of such implementation are presented

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

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

stability of linear quadratic minimization over euclidean balls stability properties of the problem of minimizing a nonconvex linear quadratic function over a euclidean ball known as the trust region subproblem are studied in this paper we investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper lower semicontinuity of the karush kuhn tucker kkt point set map and the global solution map explicit formulas for computing the directional derivative and the frechet derivative of the optimal value function stability of the karush kuhn tucker point set under the perturbation of the quadratic form is also studied

permanence of species in nonautonomous discrete lotka volterra competitive system with delays and feedback controls a nonautonomous n species discrete lotka volterra competitive system of difference equations with delays and feedback controls is considered new sufficient conditions are obtained for the permanence of this discrete system the results indicate that one can choose suitable controls to make the species coexistence in the long run moreover we give some examples to illustrate the feasibility of our result which can be well suited for computational purposes

an interior point method for large constrained discrete ill posed problems ill posed problems are numerically underdetermined it is therefore often beneficial to impose known properties of the desired solution such as nonnegativity during the solution process this paper proposes the use of an interior point method in conjunction with truncated iteration for the solution of large scale linear discrete ill posed problems with box constraints an estimate of the error in the data is assumed to be available numerical examples demonstrate the competitiveness of this approach

finding optimal algorithmic parameters using derivative free optimization the objectives of this paper are twofold we devise a general framework for identifying locally optimal algorithmic parameters algorithmic parameters are treated as decision variables in a problem for which no derivative knowledge or existence is assumed a derivative free method for optimization seeks to minimize some measure of performance of the algorithm being fine tuned this measure is treated as a black box and may be chosen by the user examples are given in the text the second objective is to illustrate this framework by specializing it to the identification of locally optimal trust region parameters in unconstrained optimization the derivative free method chosen to guide the process is the mesh adaptive direct search a generalization of pattern search methods we illustrate the flexibility of the latter and in particular make provision for surrogate objectives locally optimal parameters with respect to overall computational time on a set of test problems are identified each function call may take several hours and may not always return a predictable result a tailored surrogate function is used to guide the search towards a local solution the parameters thus identified differ from traditionally used values and allow one to solve a problem that remained otherwise unsolved in a reasonable time using traditional values

an interior point trust region algorithm for general symmetric cone programming an interior point trust region algorithm is proposed for minimizing a general non convex quadratic objective function in the intersection of a symmetric cone and an affine subspace the algorithm uses a trust region model to ensure descent on a suitable merit function global first order and second order convergence results are proved numerical results are presented

the generalized trust region subproblem solution complexity and convex hull results we consider the generalized trust region subproblem gtrs of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint a lifting of this problem recasts the gtrs as minimizing a linear objective subject to two nonconvex quadratic constraints our first main contribution is structural we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil this result may be of interest in building relaxations for nonconvex quadratic programs moreover this result allows us to reformulate the gtrs as the minimization of two convex quadratic functions in the original space our next set of contributions is algorithmic we present an algorithm for solving the gtrs up to an epsilon additive error based on this reformulation we carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms notably our algorithms run in linear in the size of the input time furthermore our algorithm for computing an epsilon optimal solution has a slightly improved running time dependence on epsilon over the state of the art algorithm our analysis shows that the dominant cost in solving the gtrs lies in solving a generalized eigenvalue problem establishing a natural connection between these problems finally generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality interval and hollow constrained variants of the gtrs this gives the first linear time algorithm in the literature for these variants of the gtrs

nonmonotone trust region method for solving optimization problems in this paper a trust region tr method with nonmonotone technique for optimization is proposed we construct a new ratio of actual descent and predicted descent which is a simple and natural generalization of the modified armijo line search rule the paper exposes the relationship between the trust region method and line search approach since this method possesses the robust properties of trust region subproblem it is globally convergent although we employ the nonmonotone sequence of function values instead of the monotone sequence in addition the proof of convergence is obviously simpler than one of newton type method with nonmonotone line search finally applications of the nonmonotone tr algorithm to some optimization problems are discussed

rankine hugoniot riemann solver considering source terms and multidimensional effects a new approach for a flux solver is introduced which takes into account source terms viscous terms and multidimensional effects the basic idea is to distribute the source terms which also contain the viscous terms and multidimensional effects from the cells to the cell interfaces then the fluxes on both sides of a cell interface are determined by the rankine hugoniot conditions and a linearized riemann solver the resulting rankine hugoniot riemann rhr solver yields much more accurate results than conventional riemann solvers for steady premixed laminar flames in 1d and 2d and a steady 2d inviscid channel flow with injection unsteady flow simulations of two colliding flames producing sound and of acoustic oscillations flattening a 2d bunsen flame demonstrate that the new flux solver is able to compute acoustic effects in flames accurately this approach for a flux solver is more general and can also be applied to solve other partial differential equations which can be expressed as hyperbolic systems with source terms ex or including higher spatial derivatives e g for the shallow water equations and for the magnetohydrodynamical equations

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

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

on the complexity of steepest descent newton s and regularized newton s methods for nonconvex unconstrained optimization problems it is shown that the steepest descent and newton s methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to o epsilon 2 to drive the norm of the gradient below epsilon this shows that the upper bound of o epsilon 2 evaluations known for the steepest descent is tight and that newton s method may be as slow as the steepest descent method in the worst case the improved evaluation complexity bound of o epsilon 3 2 evaluations known for cubically regularized newton s methods is also shown to be tight

an inexact and nonmonotone proximal method for smooth unconstrained minimization an implementable proximal point algorithm is established for the smooth nonconvex unconstrained minimization problem at each iteration the algorithm minimizes approximately a general quadratic by a truncated strategy with step length control the main contributions are i a framework for updating the proximal parameter ii inexact criteria for approximately solving the subproblems iii a nonmonotone criterion for accepting the iterate the global convergence analysis is presented together with numerical results that validate and put into perspective the proposed approach

discontinuous solutions using the method of manufactured solutions on finite volume solvers when applying the method of manufactured solutions on computational fluid dynamic software it is a requirement that all solutions be continuous on the computational domain this stipulation is limiting for the verification and validation of numerical solutions where discontinuities are frequent in an effort to adapt the standard method of manufactured solutions procedure we propose a piecewise approach for modeling solutions with discontinuities linearly and quadratically exact transformations are used for determining the exact solutions and source terms for cells split by discontinuities upwind manufactured solutions are initialized and a least squares fit is used to solve for solutions downwind of the discontinuity such that the rankine hugoniot conditions are satisfied the codes used throughout this research are finite volume 1st and 2nd order inviscid schemes combined with uniform structured grids this study shows that a modified method of manufactured solutions procedure can be performed wit

a nonmonotone trust region method with adaptive radius for unconstrained optimization problems in this paper we incorporate a nonmonotone technique with the new proposed adaptive trust region radius shi and guo 2008 4 in order to propose a new nonmonotone trust region method with an adaptive radius for unconstrained optimization both the nonmonotone techniques and adaptive trust region radius strategies can improve the trust region methods in the sense of global convergence the global convergence to first and second order critical points together with local superlinear and quadratic convergence of the new method under some suitable conditions numerical results show that the new method is very efficient and robustness for unconstrained optimization problems

multivariate spectral gradient method for unconstrained optimization multivariate spectral gradient method is proposed for solving unconstrained optimization problems combined with some quasi newton property multivariate spectral gradient method allows an individual adaptive stepsize along each coordinate direction which guarantees that the method is finitely convergent for positive definite quadratics especially it converges no more than two steps for positive definite quadratics with diagonal hessian and quadratically for objective functions with positive definite diagonal hessian moreover based on a nonmonotone line search global convergence is established for multivariate spectral gradient algorithm at last numerical results are reported which show that this method is promising and deserves further discussing

a class of gradient unconstrained minimization algorithms with adaptive stepsize in this paper the development convergence theory and numerical testing of a class of gradient unconstrained minimization algorithms with adaptive stepsize are presented the proposed class comprises four algorithms the first two incorporate techniques for the adaptation of a common stepsize for all coordinate directions and the other two allow an individual adaptive stepsize along each coordinate direction all the algorithms are computationally efficient and possess interesting convergence properties utilizing estimates of the lipschitz constant that are obtained without additional function or gradient evaluations the algorithms have been implemented and tested on some well known test cases as well as on real life artificial neural network applications and the results have been very satisfactory

adaptive regularization with cubics on manifolds adaptive regularization with cubics arc is an algorithm for unconstrained non convex optimization akin to the trust region method its iterations can be thought of as approximate safe guarded newton steps for cost functions with lipschitz continuous hessian arc has optimal iteration complexity in the sense that it produces an iterate with gradient smaller than varepsilon in o 1 varepsilon 1 5 iterations for the same price it can also guarantee a hessian with smallest eigenvalue larger than sqrt varepsilon in this paper we study a generalization of arc to optimization on riemannian manifolds in particular we generalize the iteration complexity results to this richer framework our central contribution lies in the identification of appropriate manifold specific assumptions that allow us to secure these complexity guarantees both when using the exponential map and when using a general retraction a substantial part of the paper is devoted to studying these assumptions relevant beyond arc and providing user friendly sufficient conditions for them numerical experiments are encouraging

numerical integration techniques for discontinuous manufactured solutions when applying the method of manufactured solutions mms on computational fluid dynamic software determining the exact solutions and source terms for finite volume codes where the stored value is an integrated average over the control volume is non trivial and not frequently discussed mms with discontinuities further complicates the problem of determining these values in an effort to adapt the standard mms procedure to solutions that contain discontinuities we show that newton cotes and gauss quadrature numerical integration methods exhibit high error first order limitations we propose a new method for determining the exact solutions and source terms on a uniform structured grid containing shock discontinuities by performing linearly and quadratically exact transformations on split cells transformations are performed on triangular and quadrilateral elements of a systematically divided discontinuous cell using a quadratic transformation in conjunction with a nine point gauss quadrature method a minimum of 4th order accuracy is achieved for fully general solutions and shock shapes a linear approximation of curved shocks is also experimentally shown to be 2nd order accurate the numerical integration method is then applied to a cfd code using simple discontinuous manufactured solutions which return consistent 1st order convergence values the result is an important step towards being able to use mms to verify solutions with discontinuities this work also highlights the use of higher order numerical integration techniques for continuous and discontinuous solutions that are required for mms on higher order finite volume codes

asymmetric lee distance codes for dna based storage we introduce a new family of codes termed asymmetric lee distance ald codes designed to correct errors arising in dna based storage systems and systems with parallel string transmission protocols ald codes are defined over a quaternary alphabet and analyzed in this particular setting but the derived results hold for other alphabet sizes as well our technical contributions are twofold first we derive upper bounds on the size of the codes under the ald metric based on linear programming techniques second we propose a number of code constructions which imply lower bounds

convergence of nonmonotone line search method in this paper we develop a new nonmonotone line search for general line search method and establish some global convergence theorems the new nonmonotone line search is a novel form of the nonmonotone armijo line search and allows one to choose a larger step size at each iteration which is available in constructing new line search methods and possibly reduces the function evaluations at each iteration moreover we analyze the convergence rate of some special line search methods with the new line search preliminary numerical results show that some line search methods with the new nonmonotone line search are available and efficient in practical computation

solving the trust region subproblem using the lanczos method the approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods when the number of variables is large the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust region boundary in this paper we investigate ways of continuing the process once the boundary has been encountered the key is to observe that the trust region problem within the currently generated krylov subspace has a very special structure which enables it to be solved very efficiently we compare the new strategy with existing methods the resulting software package is available as hsl vf05 within the harwell subroutine library

a nonmonotone line search technique and its application to unconstrained optimization a new nonmonotone line search algorithm is proposed and analyzed in our scheme we require that an average of the successive function values decreases while the traditional nonmonotone approach of grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 requires that a maximum of recent function values decreases we prove global convergence for nonconvex smooth functions and r linear convergence for strongly convex functions for the l bfgs method and the unconstrained optimization problems in the cute library the new nonmonotone line search algorithm used fewer function and gradient evaluations on average than either the monotone or the traditional nonmonotone scheme

graphical criteria for positive solutions to linear systems abstract we study linear systems of equations with coefficients in a generic partially ordered ring r and a unique solution and seek conditions for the solution to be nonnegative that is every component of the solution is a quotient of two nonnegative elements in r the requirement of a nonnegative solution arises typically in applications for example in biology and ecology where quantities of interest are concentrations and abundances we provide novel conditions on a labeled multidigraph associated with the linear system that guarantee the solution to be nonnegative furthermore we study a generalization of the first class of linear systems where the coefficient matrix has a specific block form and provide analogous conditions for nonnegativity of the solution similarly based on a labeled multidigraph the latter scenario arises naturally in chemical reaction network theory when studying full or partial parameterizations of the positive part of the steady state variety of a polynomial dynamical system in the concentrations of the molecular species

an efficient two step trust region algorithm for exactly determined consistent systems of nonlinear equations abstract in this paper we propose a modified two step trust region algorithm for solving nonlinear systems two step trust region algorithms at every iteration use a trust region step and an approximate step by saving the jacobian matrix to have fewer computations we introduce a convex combination to modify the trust region subproblems along with an additional criterion to verify whether the first step is accepted or not we establish global and quadratic convergence of the algorithm under some mild assumptions numerical results show that the modified algorithm is efficient and promising

the convergence of conjugate gradient method with nonmonotone line search abstract the conjugate gradient method is a useful and powerful approach for solving large scale minimization problems liu and storey developed a conjugate gradient method which has good numerical performance but no global convergence under traditional line searches such as armijo line search wolfe line search and goldstein line search in this paper we propose a new nonmonotone line search for liu storey conjugate gradient method ls in short the new nonmonotone line search can guarantee the global convergence of ls method and has a good numerical performance by estimating the lipschitz constant of the derivative of objective functions in the new nonmonotone line search we can find an adequate step size and substantially decrease the number of functional evaluations at each iteration numerical results show that the new approach is effective in practical computation

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

nonmonotone adaptive trust region method for unconstrained optimization problems in this paper an adaptive trust region method with nonmonotone technique for unconstrained optimization problems is presented and analyzed the global and local convergence results of the algorithm are established numerical results show that the new method is efficient

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

a preconditioning framework for sequences of diagonally modified linear systems arising in optimization we propose a framework for building preconditioners for sequences of linear systems of the form a delta k x k b k where a is symmetric positive semidefinite and delta k is diagonal positive semidefinite such sequences arise in several optimization methods e g in affine scaling methods for bound constrained convex quadratic programming and bound constrained linear least squares as well as in trust region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares for all the matrices of a sequence the preconditioners are obtained by updating any preconditioner for a available in the ldl t form the preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences furthermore under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of delta k are sufficiently large we present two low cost preconditioners sharing the above mentioned p

trust region algorithms and timestep selection unconstrained optimization problems are closely related to systems of ordinary differential equations odes with gradient structure in this work we prove results that apply to both areas we analyze the convergence properties of a trust region or levenberg marquardt algorithm for optimization the algorithm may also be regarded as a linearized implicit euler method with adaptive timestep for gradient odes from the optimization viewpoint the algorithm is driven directly by the levenberg marquardt parameter rather than the trust region radius this approach is discussed for example in r fletcher practical methods of optimization 2nd ed john wiley new york 1987 but no convergence theory is developed we give a rigorous error analysis for the algorithm establishing global convergence and an unusual extremely rapid type of superlinear convergence the precise form of superlinear convergence is exhibited the ratio of successive displacements from the limit point is bounded above and below by geometrically decreasing sequences we also show how an inexpensive change to the algorithm leads to quadratic convergence from the ode viewpoint this work contributes to the theory of gradient stability by presenting an algorithm that reproduces the correct global dynamics and gives very rapid local convergence to a stable steady state

a new trust region method for unconstrained optimization in this paper we propose a new trust region method for unconstrained optimization problems the new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions we also analyze the global linear convergence local superlinear and quadratic convergence rate of the new method numerical results show that the new trust region method is available and efficient in practical computation

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 inexact proximal method for quasiconvex minimization abstract in this paper we propose an inexact proximal point method to solve constrained minimization problems with locally lipschitz quasiconvex objective functions assuming that the function is also bounded from below lower semicontinuous and using proximal distances we show that the sequence generated for the method converges to a stationary point of the problem

a new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration convergence properties and their low memory requirements in this work we propose a new class of spectral conjugate gradient methods which ensures sufficient descent independent of the accuracy of the line search moreover an attractive property of our proposed methods is that they achieve a high order accuracy in approximating the second order curvature information of the objective function by utilizing the modified secant condition proposed by babaie kafaki et al s babaie kafaki r ghanbari n mahdavi amiri two new conjugate gradient methods based on modified secant equations journal of computational and applied mathematics 234 2010 1374 1386 further a global convergence result for general functions is established provided that the line search satisfies the wolfe conditions our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness

eigenvalue based algorithm and analysis for nonconvex qcqp with one constraint a nonconvex quadratically constrained quadratic programming qcqp with one constraint is usually solved via a dual sdp problem or more s algorithm based on iteratively solving linear systems in this work we introduce an algorithm for qcqp that requires finding just one eigenpair of a generalized eigenvalue problem and involves no outer iterations other than the usually black box iterations for computing the eigenpair numerical experiments illustrate the efficiency and accuracy of our algorithm we also analyze the qcqp solution extensively including difficult cases and show that the canonical form of a matrix pair gives a complete classification of the qcqp in terms of boundedness and attainability and explain how to obtain a global solution whenever it exists

a new family of conjugate gradient methods in this paper we develop a new class of conjugate gradient methods for unconstrained optimization problems a new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions in particular polak ribiere polyak and liu storey conjugate gradient methods are special cases of the new class of conjugate gradient methods by estimating the local lipschitz constant of the derivative of objective functions we can find an adequate step size and substantially decrease the function evaluations at each iteration numerical results show that these new conjugate gradient methods are effective in minimizing large scale non convex non quadratic functions