an affine scaling interior trust region method for lc1 minimization subject to bounds on variables in this literature we extend the classical affine scaling interior trust region algorithm for smooth bounded constrained nonlinear programming to the nonsmooth case where the objective function is only locally lipschitzian we propose and analyze a new affine scaling trust region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained lc 1 optimization subject to bounds on variables at each iteration the trust region subproblem is defined by minimizing a quadratic function which is required both first and second order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints the second order derivative of the objective function is explicitly required to be lipschitzian each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required a nonmonotonic criterion should bring about speeding up the convergence progress in some ill conditioned cases applications of the algorithm to various nonsmooth optimization problems are discussed

three positive solutions to an indefinite neumann problem a shooting method abstract we deal with the neumann boundary value problem u a t a t g u 0 0 u t 1 t 0 t u 0 u t 0 where the weight term has two positive humps separated by a negative one and g 0 1 r is a continuous function such that g 0 g 1 0 g s 0 for 0 s 1 and lim s 0 g s s 0 we prove the existence of three solutions when and are positive and sufficiently large

on the stability of the l 2 projection in h 1 ω we prove the stability in h1 of the l2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions we give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions in particular stability of the l2 projection in h1 holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically

nonuniqueness of an indefinite nonlinear diffusion problem in population genetics abstract we study the following neumann problem in one dimension space arising from population genetics u t d u x x h x u 2 1 u in 1 1 0 0 u 1 in 1 1 0 u 1 t u 1 t 0 in 0 where h changes sign in 1 1 and d is a positive parameter lou and nagylaki 2002 6 conjectured that if 1 1 h x d x 0 then this problem has at most one nontrivial steady state i e a time independent solution which is not identically equal to zero or one nakashima 2018 15 proved this uniqueness under some additional conditions on h x unexpectedly in this paper we discover 3 nontrivial steady states for some h x satisfying 1 1 h x d x 0 moreover bi stable phenomenon occurs in this scenario one with two layers is stable two with one layer each are ordered with the smaller one being stable and the larger one being unstable

sharp error estimate of the nonuniform l1 formula for linear reaction subdiffusion equations stability and convergence of the l1 formula on nonuniform time grids are studied for solving linear reaction subdiffusion equations with the caputo derivative a discrete fractional gronwall inequality is developed for the nonuniform l1 formula by introducing a discrete convolution kernel of riemann liouville fractional integral to simplify the consistency analysis of the nonuniform l1 formula we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete riemann liouville integral kernel with the help of discrete fractional gronwall inequality and global consistency error analysis a sharp error estimate reflecting the regularity of solution is obtained for a simple l1 scheme numerical examples are provided to verify the sharpness of the error analysis

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

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

error analysis of a finite difference method on graded meshes for a time fractional diffusion equation a reaction diffusion problem with a caputo time derivative of order alpha in 0 1 is considered the solution of such a problem is shown in general to have a weak singularity near the initial time t 0 and sharp pointwise bounds on certain derivatives of this solution are derived a new analysis of a standard finite difference method for the problem is given taking into account this initial singularity this analysis encompasses both uniform meshes and meshes that are graded in time and includes new stability and consistency bounds the final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme so one can choose an optimal mesh grading numerical results are presented that confirm the sharpness of the error analysis

a projective quasi newton method for nonlinear optimization abstract a trust region method for nonlinear optimization problems with equality constraints is proposed in this paper this method incorporates quadratic subproblems in which orthogonal projective matrices of the jacobian of constraint functions are used to replace qr decompositions as qr decomposition does not ensure continuity but projective matrix does convergence behaviour of the new method can be discussed under more reasonable assumptions the method maintains a two step feature one movement in the range space of the jacobian whereas the other one in the null space it is proved that all accumulation points of iterates are kkt karush kuhn tucker points and the method has a one step superlinear convergence rate

a semilinear parabolic system for migration and selection in population genetics abstract the semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated the population occupies a finite habitat of arbitrary dimensionality and shape the selection coefficients depend on position the drift and diffusion coefficients may do so for two alleles the scalar case the global analysis of d henry 1981 geometric theory of semilinear parabolic equations lecture notes in mathematics vol 840 springer verlag berlin is extended from homogeneous isotropic migration corresponding to the laplacian to arbitrary migration corresponding to an arbitrary elliptic operator for multiple alleles sufficient conditions are given for the global loss of an allele that is nowhere the fittest in the case of no dominance with at least one change in the direction of selection sufficient conditions are established for global convergence to a stable equilibrium with all the intermediate alleles absent and one or two extreme alleles present sufficient conditions on the migration pattern for casting the elliptic operator into variational form are proved in this case the above results become more explicit

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

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

multiple existence of indefinite nonlinear diffusion problem in population genetics abstract we study the following neumann problem in one dimension u t d u x x h x u 2 1 u in 1 1 0 0 u 1 in 1 1 0 u 1 t u 1 t 0 in 0 where h changes sign in 1 1 and is symmetric i e h x h x and d is a positive small parameter in 9 we showed that this equation has a nontrivial steady state which is unique under the condition 1 1 h x d x 0 on the other hand in this paper we will prove that this equation has at least 8 nontrivial steady states for some h x satisfying 1 1 h x d x 0 this is a counterexample to a conjecture by lou and nagylaki 2

optimal spatial h1 norm analysis of a finite element method for a time fractional diffusion equation abstract a time fractional initial boundary value problem d t u u f where d t is a caputo fractional derivative of order 0 1 is considered on the space time domain 0 t where r d d 1 is a bounded lipschitz domain typical solutions u x t of such problems have components that behave like a multiple of t as t 0 so the integer order temporal derivatives of u blow up at t 0 the numerical method of the paper uses a standard finite element method in space on a quasiuniform mesh and considers both the l1 discretisation and alikhanov s l2 1 discretisation of the caputo derivative on suitably graded temporal meshes optimal error bounds in h 1 are proved no previous analysis of a discretisation of this problem using finite elements in space has established such a bound furthermore the optimal grading of the temporal mesh can be deduced from our analysis numerical experiments show that our error bounds are sharp

a family of improved secant methods via nonmonotone curvilinear paths technique for equality constrained optimization abstract this paper presents a family of improved secant algorithms via two preconditional curvilinear paths the preconditional modified gradient path and preconditional optimal path for solving general nonlinear optimization problems with nonlinear equality constraints we employ the stable bunch parlett factorization method of symmetric matrices so that two preconditional curvilinear paths are very easily formed the nonmonotone curvilinear search technique by introducing a nonsmooth merit function and adopting a dogleg typed movement is used to speed up the convergence progress in the contours of objective function with large curvature global convergence of the proposed algorithms is obtained under some reasonable conditions furthermore the dogleg typed step overcomes the maratos effect to bring the local superlinear convergence rate the results of numerical experiments are reported to show the effectiveness of the proposed algorithms

the uniqueness of an indefinite nonlinear diffusion problem in population genetics part ii abstract we study the following neumann problem which models the complete dominance case of population genetics of two alleles u t d u g x u 2 1 u in 0 1 0 0 u 1 in 0 1 0 u 0 t u 1 t 0 in 0 where g changes sign in 0 1 it is known that this equation has a nontrivial steady state u d for d sufficiently small 5 it has been conjectured by nagylaki and lou 2 that u d is a unique nontrivial steady state if g x d x 0 this was proved in 6 if g changes sign only once in this paper under additional condition on g x we treat the case when g has multiple zeros

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

an interior affine scaling projective algorithm for nonlinear equality and linear inequality constrained optimization in this paper we propose a new nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving optimization subject to nonlinear equality and linear inequality constraints the general full trust region subproblem for solving the nonlinear equality and linear inequality constrained optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints the horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced hessian function subject only to an ellipsoidal trust region constraint in a null subspace of the tangential space while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint by introducing the fletcher s penalty function as the merit function trust region strategy with interior point backtracking technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems the global convergence of the proposed algorithm while maintaining fast local convergence rate of the proposed algorithm are established under some reasonable conditions a nonmonotonic criterion should bring about speeding up the convergence progress in some high nonlinear function conditioned cases

an affine scaling projective reduced hessian algorithm for minimum optimization with nonlinear equality and linear inequality constraints abstract in this paper we propose a nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving minimum optimization subject to both nonlinear equality and linear inequality constraints the general full trust region subproblem for solving the minimum optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints by qr decomposition of an affine scaling matrix and an orthonormal basis on the null subspace the horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced hessian function subject only to an ellipsoidal trust region constraint while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint combining trust region strategy with line search technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems by adopting the l1 penalty function as the merit function the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions the second order correction step and a nonmonotonic criterion are used to overcome maratos effect and speed up the convergence progress in some ill conditioned cases respectively

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

conditions for convergence of trust region algorithms for nonsmooth optimization this paper discusses some properties of trust region algorithms for nonsmooth optimization the problem is expressed as the minimization of a functionh f x whereh is convex andf is a continuously differentiable mapping from to bounds for the second order derivative approximation matrices are discussed it is shown that powel s 7 8 results hold for nonsmooth optimization

a new difference scheme for the time fractional diffusion equation abstract in this paper we construct a new difference analog of the caputo fractional derivative called the l 2 1 formula the basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered stability of the suggested schemes and also their convergence in the grid l 2 norm with the rate equal to the order of the approximation error are proved the obtained results are supported by the numerical calculations carried out for some test problems

the uniqueness of indefinite nonlinear diffusion problem in population genetics part i we study the following neumann problem in one dimension ut du g x u2 1 u in 0 1 0 0 u 1in 0 1 0 u 0 t u 1 t 0in 0 g changes sign in 0 1 this equation models the complete dominance case in population genetics of two alleles it is known that this equation has a nontrivial steady state ud for d sufficiently small we show that the steady state ud is linearly stable moreover under the condition 01g x dx 0 we show that ud is a unique nontrivial steady state a conjecture of nagylaki and lou in one dimensional case has been largely resolved

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

on the local convergence of quasi newton methods for constrained optimization we consider the application of a general class of quasi newton methods to the solution of the classical equality constrained nonlinear optimization problem specifically we develop necessary and sufficient conditions for the q superlinear convergence of such methods and present a companion linear convergence theorem the essential conditions relate to the manner in which the hessian of the lagrangian function is approximated

continuity of the null space basis and constrained optimization many constrained optimization algorithms use a basis for the null space of the matrix of constraint gradients recently methods have been proposed that enable this null space basis to vary continuously as a function of the iterates in a neighborhood of the solution this paper reports results from topology showing that in general there is no continuous function that generates the null space basis of all full rank rectangular matrices of a fixed size thus constrained optimization algorithms cannot assume an everywhere continuous null space basis we also give some indication of where these discontinuities must occur we then propose an alternative implementation of a class of constrained optimization algorithms that uses approximations to the reduced hessian of the lagrangian but is independent of the choice of null space basis this approach obviates the need for a continuously varying null space basis

multi layered stationary solutions for a spatially inhomogeneous allen cahn equation abstract we consider stationary solutions of a spatially inhomogeneous allen cahn type nonlinear diffusion equation in one space dimension the equation involves a small parameter e and its nonlinearity has the form h x 2 f u where h x represents the spatial inhomogeneity and f u is derived from a double well potential with equal well depth when e is very small stationary solutions develop transition layers we first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h x and that at most a single layer can appear near each local minimum point of h x we then discuss the stability of layered stationary solutions and prove that the morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h x we also show the existence of stationary solutions having clustering layers at the local maximum points of h x