necessary conditions for constrained optimization problems with semicontinuous and continuous data we consider nonsmooth constrained optimization problems with semicontinuous and continuous data in banach space and derive necessary conditions sl without constraint qualification in terms of smooth subderivatives and normal cones these results in different versions are set in reflexive and smooth banach spaces

variational analysis in nonreflexive spaces and applications to control problems with l 1 perturbations we provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is l 1 our underlying technique relies on a recent sequential description of both the generalized gradient of clarke and of the approximate g subdifferential of functions defined on a smooth banach space we also show that the proximal limit formula for the generalized gradient and its l p analogue are direct consequences of these sequential formulas related characterizations of lipschitzness of a function on a smooth space are given

on the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming we estimate the worst case complexity of minimizing an unconstrained nonconvex composite objective with a structured nonsmooth term by means of some first order methods we find that it is unaffected by the nonsmoothness of the objective in that a first order trust region or quadratic regularization method applied to it takes at most mathcal o epsilon 2 function evaluations to reduce the size of a first order criticality measure below epsilon specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective and constraint evaluation worst case complexity of nonconvex equality constrained optimization when the solution is computed using a first order exact penalty method we obtain that in the reasonable case when the penalty parameters are bounded the complexity of reaching within epsilon of a kkt point is at most mathcal o epsilon 2 problem evaluations which is the same in order as the function evaluation complexity of steepest descent methods applied to unconstrained nonconvex smooth optimization

a relaxed constant positive linear dependence constraint qualification and applications in this work we introduce a relaxed version of the constant positive linear dependence constraint qualification cpld that we call rcpld this development is inspired by a recent generalization of the constant rank constraint qualification by minchenko and stakhovski that was called rcrcq we show that rcpld is enough to ensure the convergence of an augmented lagrangian algorithm and that it asserts the validity of an error bound we also provide proofs and counter examples that show the relations of rcrcq and rcpld with other known constraint qualifications in particular rcpld is strictly weaker than cpld and rcrcq while still stronger than abadie s constraint qualification we also verify that the second order necessary optimality condition holds under rcrcq

the lagrange multiplier rule for multifunctions in banach spaces we study general constrained multiobjective optimization problems with objectives being closed multifunctions in banach spaces in terms of the coderivatives and normal cones we provide generalized lagrange multiplier rules as necessary optimality conditions of the above problems in an asplund space setting sharper results are presented

a framework of constraint preserving update schemes for optimization on stiefel manifold this paper considers optimization problems on the stiefel manifold x mathsf t x i p xtx ip where x in mathbb r n times p x rn p is the variable and i p ip is the p p by p p identity matrix a framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of x x and the null space of x mathsf t xt while this general framework can unify many existing schemes a new update scheme with low complexity cost is also discovered then we study a feasible barzilai borwein like method under the new update scheme the global convergence of the method is established with an adaptive nonmonotone line search the numerical tests on the nearest low rank correlation matrix problem the kohn sham total energy minimization and a specific problem from statistics demonstrate the efficiency of the new method in particular the new method performs remarkably well for the nearest low rank correlation matrix problem in terms of speed and solution quality and is considerably competitive with the widely used scf iteration for the kohn sham total energy minimization

optimality condition and complexity analysis for linearly constrained optimization without differentiability on the boundary in this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region by exploiting an interior point technique we present first and second order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available for this type of problems existing necessary conditions often rely on the notion of subdifferential or become non trivially weaker than the kkt condition in the twice differentiable counterpart problems in contrast this paper presents a new set of first and second order necessary conditions that are derived without the use of subdifferential and reduce to exactly the kkt condition when twice differentiability holds as a result these conditions are stronger than some existing ones considered for the discussed minimization problem when only non negativity constraints are present to solve for these optimality conditions in the special but important case of linearly constrained problems we present two novel interior point trust region algorithms and show that their worst case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds since this work considers a more general problem than those in the literature our results also indicate that best known existing complexity bounds are actually held for a wider class of nonlinear programming problems this new development is significant since optimality conditions play a fundamental role in computational optimization and more and more nonlinear and nonconvex problems need to be solved in practice

l_p norm regularization algorithms for optimization over permutation matrices optimization problems over permutation matrices appear widely in facility layout chip design scheduling pattern recognition computer vision graph matching etc since this problem is np hard due to the combinatorial nature of permutation matrices we relax the variable to be the more tractable doubly stochastic matrices and add an l p norm 0 p 1 regularization term to the objective function the optimal solutions of the l p regularized problem are the same as the original problem if the regularization parameter is sufficiently large a lower bound estimation of the nonzero entries of the stationary points and some connections between the local minimizers and the permutation matrices are further established then we propose an l p regularization algorithm with local refinements the algorithm approximately solves a sequence of l p regularization subproblems by the projected gradient method using a nonmonotone line search with the barzilai borwein step sizes its performance can be fu

accelerated gradient methods for nonconvex nonlinear and stochastic programming in this paper we generalize the well known nesterov s accelerated gradient ag method originally designed for convex smooth optimization to solve nonconvex and possibly stochastic optimization problems we demonstrate that by properly specifying the stepsize policy the ag method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first order information similarly to the gradient descent method we then consider an important class of composite optimization problems and show that the ag method can solve them uniformly i e by using the same aggressive stepsize policy as in the convex case even if the problem turns out to be nonconvex we demonstrate that the ag method exhibits an optimal rate of convergence if the composite problem is convex and improves the best known rate of convergence if the problem is nonconvex based on the ag method we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization to the best of our knowledge this is the first time that the convergence of the ag method has been established for solving nonconvex nonlinear programming in the literature

classes of modules with the exchange property in searching for isomorphic refinements of direct sum decompositions of modules9 crawley and jonsson introduced an exchange property for modules that is strongly reminiscent of steinitz exchange lemma for vector spaces an r module m where r is an associative ring with identity is said to have the finite exchange property if m can be fitted into any finite sum grid as follows whenever a4 occurs as a direct summand of a finite direct sum a bielai then a m ci for suitable submodules ci of the ai 2 this concept led to very general extensions of the classical refinement theorems see e g 2 3 171 many of the problems suggested by crawley and jonsson in their pioneering paper are still open the most salient one does the finite exchange property imply the unrestricted exchange property we show that the answer is yes for modules which possess decompositions into indecomposable summands in the background of this observation stands the following

parametric nonlinear programming problems under the relaxed constant rank condition the paper deals with perturbed nonlinear programming problems under the relaxed constant rank regularity condition we study the relation of the relaxed constant rank regularity condition with the error bound property the directional differentiability of the optimal value function and necessary and sufficient second order optimality conditions

adaptive cubic regularisation methods for unconstrained optimization part ii worst case function and derivative evaluation complexity an adaptive regularisation framework using cubics arc was proposed for unconstrained optimization and analysed in cartis gould and toint part i math program doi 10 1007 s10107 009 0286 5 2009 generalizing at the same time an unpublished method due to griewank technical report na 12 1981 damtp university of cambridge an algorithm by nesterov and polyak math program 108 1 177 205 2006 and a proposal by weiser deuflhard and erdmann optim methods softw 22 3 413 431 2007 in this companion paper we further the analysis by providing worst case global iteration complexity bounds for arc and a second order variant to achieve approximate first order and for the latter second order criticality of the iterates in particular the second order arc algorithm requires at most mathcal o epsilon 3 2 iterations or equivalently function and gradient evaluations to drive the norm of the gradient of the objective below the desired accuracy epsilon and mathcal o epsilon 3 iterations to reach approximate nonnegative curvature in a subspace the orders of these bounds match those proved for algorithm 3 3 of nesterov and polyak which minimizes the cubic model globally on each iteration our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate hessians

complexity of partially separable convexly constrained optimization with non lipschitzian singularities an adaptive regularization algorithm using high order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non l

two new weak constraint qualifications and applications we present two new constraint qualifications cqs that are weaker than the recently introduced relaxed constant positive linear dependence rcpld cq rcpld is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally a major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a cq this is done in the first new cq which we call the constant rank of the subspace component crsc cq this new cq also preserves many of the good properties of rcpld such as local stability and the validity of an error bound we also introduce an even weaker cq called the constant positive generator cpg which can replace rcpld in the analysis of the global convergence of algorithms we close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods sequential quadratic programming augmented lagrangians

large scale linear programming techniques for the design of protein folding potentials we present large scale optimization techniques to model the energy function that underlies the folding process of proteins linear programming is used to identify parameters in the energy function model the objective being that the model predict the structure of known proteins correctly such trained functions can then be used either for ab initio prediction or for recognition of unknown structures in order to obtain good energy models we need to be able to solve dense linear programming problems with tens possibly hundreds of millions of constraints in a few hundred parameters which we achieve by tailoring and parallelizing the interior point code pcx

minimizing nonconvex functions for sparse vector reconstruction in this paper we develop a novel methodology for minimizing a class of nonconvex concave on the non negative orthant functions for solving an underdetermined linear system of equations as x when the solution vector s is known a priori to be sparse the proposed technique is based on locally replacing the original objective function by a quadratic convex function which is easily minimized the resulting algorithm is iterative and is absolutely converging to a fixed point of the original objective function for a certain selection of convex objective functions the class of algorithms called iterative reweighted least squares irls is shown to be a special case of the proposed methodology thus the proposed algorithms are a generalization and unification of the previous methods in addition we also propose a new class of algorithms with better convergence properties compared to the regular irls algorithms and hence can be considered as enhancements to these algorithms since the original objective functions are nonconvex the proposed algorithm is susceptible to convergence to a local minimum to alleviate this difficulty we propose a random perturbation technique that enhances the performance of the proposed algorithm the numerical results show that the proposed algorithms outperform some of the well known algorithms that are usually utilized for solving the same problem

optimality conditions and a smoothing trust region newton method for nonlipschitz optimization regularized minimization problems with nonconvex nonsmooth perhaps non lipschitz penalty functions have attracted considerable attention in recent years owing to their wide applications in image restoration signal reconstruction and variable selection in this paper we derive affine scaled second order necessary and sufficient conditions for local minimizers of such minimization problems moreover we propose a global convergent smoothing trust region newton method which can find a point satisfying the affine scaled second order necessary optimality condition from any starting point numerical examples are given to demonstrate the effectiveness of the smoothing trust region newton method

decomposition properties of strict mittag leffler modules we show that a not necessarily unitary ring with enough idempotents is left perfect if and only if there exists a cardinal number such that every flat strict mittag leffler module is a direct sum of generated modules several applications are given to the decomposition properties of modules into direct summands

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

necessary conditions for constrained optimization problems in smooth banach spaces and applications we derive necessary optimality conditions for constrained minimization problems with lower semicontinuous inequality constraints continuous equality constraints and a set constraint in smooth banach spaces then we apply these necessary optimality conditions to derive subdifferential characterizations of singular normal vectors to the epigraph and the graph of a function subdifferential calculus and necessary optimality conditions for mathematical programs with equilibrium constraints

every ℵ1 σ cs module is σ cs abstract it is proved that if m is a uniform module such that m 1 is cs then its endomorphism ring is local as a consequence it is shown that every 1 cs module is cs this solves open questions asked in contemp math 259 2000 467 j algebra 262 2003 194 on the way to these results it is also shown that the endomorphism ring of a uniform module m is local provided that every local summand of m 0 is a direct summand

necessary optimality conditions and exact penalization for non lipschitz nonlinear programs when the objective function is not locally lipschitz constraint qualifications are no longer sufficient for karush kuhn tucker kkt conditions to hold at a local minimizer let alone ensuring an exact penalization in this paper we extend quasi normality and relaxed constant positive linear dependence condition to allow the non lipschitzness of the objective function and show that they are sufficient for kkt conditions to be necessary for optimality moreover we derive exact penalization results for the following two special cases when the non lipschitz term in the objective function is the sum of a composite function of a separable lower semi continuous function with a continuous function and an indicator function of a closed subset we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function when the non lipschitz term is the sum of a continuous function and an indicator function of a closed subset we also show that our problem admits an exact penalization under an extended quasi normality involving the coderivative of the continuous function

preprocessing for quadratic programming techniques for the preprocessing of not necessarily convex quadratic programs are discussed most of the procedures extend known ones from the linear to quadratic cases but a few new preprocessing techniques are introduced the implementation aspects are also discussed numerical results are finally presented to indicate the potential of the resulting code both for linear and quadratic problems the impact of insisting that bounds of the variables in the reduced problem be as tight as possible rather than allowing some slack in these bounds is also shown to be numerically significant

non lipschitz ell_ p regularization and box constrained model for image restoration nonsmooth nonconvex regularization has remarkable advantages for the restoration of piecewise constant images constrained optimization can improve the image restoration using a priori information in this paper we study regularized nonsmooth nonconvex minimization with box constraints for image restoration we present a computable positive constant for using nonconvex nonsmooth regularization and show that the difference between each pixel and its four adjacent neighbors is either 0 or larger than in the recovered image moreover we give an explicit form of for the box constrained image restoration model with the non lipschitz nonconvex lp norm 0 p 1 regularization our theoretical results show that any local minimizer of this imaging restoration problem is composed of constant regions surrounded by closed contours and edges numerical examples are presented to validate the theoretical results and show that the proposed model can recover image restoration results very well

modules with indecomposable decompositions that complement maximal direct summands generalizing a fundamental property of semisimple modules anderson w x and fuller 1 introduced the following important concept for direct decompositions of modules a decomposition m s m is said to ig i i complement direct summands in case for each direct summand a of m w there is a subset h i such that m s a m cf also 2 p ig h i x 141 such a decomposition is necessarily an indecomposable decomposition and it is an interesting problem to study under which conditions an indecomposable decomposition complements direct summands there is w x an extensive literature concerning this problem by 1 theorem 6 right perfect rings can be characterized by the property that all projective right modules have decompositions that complement direct summands fuller w x 7 showed that over a generalized uniserial ring every module has a decomposition that complements direct summands thus providing a first class of non semisimple rings satisfying this property more generally w x tachikawa 14 proved that every left and right module over a ring of finite representation type has a decomposition that complements direct summands and that the converse is also true was established by fuller and w x w x reiten 9 restricting just to one side fuller 8 showed that rings over which every right module has a decomposition that complements direct summands are precisely the rings over which every right module is a direct sum of finitely generated modules they are also called right pure semisim

iterative reweighted minimization methods for l_p lp regularized unconstrained nonlinear programming in this paper we study general l p l p regularized unconstrained minimization problems in particular we derive lower bounds for nonzero entries of the first and second order stationary points and hence also of local minimizers of the l p l p minimization problems we extend some existing iterative reweighted l 1 l 1 mathrm irl 1 irl 1 and l 2 l 2 mathrm irl 2 irl 2 minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed form solution also we provide a unified convergence analysis for these methods in addition we propose a novel lipschitz continuous epsilon approximation to vert x vert p p x p p using this result we develop new mathrm irl 1 irl 1 methods for the l p l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first order stationary point provided that the approximation parameter epsilon is below a computable threshold value this is a remarkable result since all existing iterative reweighted minimization methods require that epsilon be dynamically updated and approach zero our computational results demonstrate that the new mathrm irl 1 irl 1 method and the new variants generally outperform the existing mathrm irl 1 irl 1 methods chen and zhou in 2012 foucart and lai in appl comput harmon anal 26 395 407 2009

joint power and admission control via linear programming deflation we consider the joint power and admission control problem for a wireless network consisting of multiple interfering links the goal is to support a maximum number of links at their specified signal to interference plus noise ratio sinr targets while using a minimum total transmission power in this work we first reformulate this np hard problem as a sparse l0 minimization problem and then relax it to a linear program furthermore we derive two easy to check necessary conditions for all links in the network to be simultaneously supported at their target sinr levels and use them to iteratively remove strong interfering links deflation an upper bound on the maximum number of supported links is also given numerical simulations show that the proposed approach compares favorably with the existing approaches in terms of the number of supported links the total transmission power and the execution time

a smoothing sqp framework for a class of composite l_q lq minimization over polyhedron the composite l q 0 q 1 minimization problem over a general polyhedron has received various applications in machine learning wireless communications image restoration signal reconstruction etc this paper aims to provide a theoretical study on this problem first we derive the karush kuhn tucker kkt optimality conditions for local minimizers of the problem second we propose a smoothing sequential quadratic programming framework for solving this problem the framework requires a approximate solution of a convex quadratic program at each iteration finally we analyze the worst case iteration complexity of the framework for returning an epsilon kkt point i e a feasible point that satisfies a perturbed version of the derived kkt optimality conditions to the best of our knowledge the proposed framework is the first one with a worst case iteration complexity guarantee for solving composite l q minimization over a general polyhedron

a cone continuity constraint qualification and algorithmic consequences every local minimizer of a smooth constrained optimization problem satisfies the sequential approximate karush kuhn tucker akkt condition this optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms it is natural to ask for conditions on the constraints under which akkt implies kkt these conditions will be called strict constraint qualifications scqs in this paper we define a cone continuity property ccp that will be shown to be the weakest possible scq its relation to other constraint qualifications will also be clarified in particular it will be proved that ccp is strictly weaker than the constant positive generator constraint qualification

robust volume minimization based matrix factorization for remote sensing and document clustering this paper considers volume minimization volmin based structured matrix factorization volmin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum volume simplex that encloses all the columns of the data matrix recent work showed that volmin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications this paper focuses on both theoretical and practical aspects of volmin on the theory side exact equivalence of two independently developed sufficient conditions for volmin identifiability is proven here thereby providing a more comprehensive understanding of this aspect of volmin on the algorithm side computational complexity and sensitivity to outliers are two key challenges associated with real world applications of volmin these are addressed here via a new volmin algorithm that handles volume regularization in a computationally simple way and automatically detects and iteratively downweights outliers simultaneously simulations and real data experiments using a remotely sensed hyperspectral image and the reuters document corpus are employed to showcase the effectiveness of the proposed algorithm

spectral galerkin methods for the primary two point boundary value problem in modelling viscoelastic flows the essential character of the primary elliptic hyperbolic operator encountered in the modelling of viscoelastic flows is encaptured in a non linear fifth order two point boundary value problem in one dimension expansions in terms of beam functions and chebyshev polynomials are used to compute solutions to the primary two point boundary value problem within a spectral galerkin formulation an investigation of the performance of the methods with respect to the level of non linearity is carried out accurate results are obtained with chebyshev polynomials for high levels of non linearity whereas the behaviour of beam function expansions proves far more sensitive to the level of non linearity

sigma cotorsion rings abstract it is proved that a ring r is right perfect if and only if it is cotorsion as a right module over itself several other conditions are shown to be equivalent for example that every pure submodule of a free right r module is strongly pure essential in a direct summand or that the countable direct sum c r 0 of the cotorsion envelope of r r is cotorsion if c r is a flat cotorsion module then c r admits a decomposition into a direct sum of indecomposable modules with a local endomorphism ring the jacobson radical j s of the endomorphism ring s end r c is characterized as the maximum ideal that acts locally t nilpotently on c r if r is semilocal and c c r then the radical consists of those endomorphisms f c c whose image is contained in cj

joint power and admission control non convex l_ q approximation and an effective polynomial time deflation approach in an interference limited network joint power and admission control jpac aims at supporting a maximum number of links at their specified signal to interference plus noise ratio sinr targets while using minimum total transmission power various convex approximation deflation approaches have been developed for the jpac problem in this paper we propose an effective polynomial time non convex approximation deflation approach for solving the problem the approach is based on the non convex ell q 0 approximation of an equivalent sparse ell 0 reformulation of the jpac problem we show that for any instance of the jpac problem there exists a bar q in 0 1 such that it can be exactly solved by solving its ell q approximation problem with any q in 0 bar q we also show that finding the global solution of the ell q approximation problem is np hard then we propose a potential reduction interior point algorithm which can return an epsilon kkt solution of the np hard ell q approximation problem in polynomial time the returned solution can be used to check the simultaneous supportability of all links in the network and to guide an iterative link removal procedure resulting in the polynomial time non convex approximation deflation approach for the jpac problem numerical simulations show that the proposed approach outperforms the existing convex approximation approaches in terms of the number of supported links and the total transmission power particularly exhibiting a quite good performance in selecting which subset of links to support

a group bridge approach for variable selection in multiple regression problems when covariates can be naturally grouped it is important to carry out feature selection at the group and within group individual variable levels simultaneously the existing methods including the lasso and group lasso are designed for either variable selection or group selection but not for both we propose a group bridge approach that is capable of simultaneous selection at both the group and within group individual variable levels the proposed approach is a penalized regularization method that uses a specially designed group bridge penalty it has the oracle group selection property in that it can correctly select important groups with probability converging to one in contrast the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods

worst case complexity of smoothing quadratic regularization methods for non lipschitzian optimization in this paper we propose a smoothing quadratic regularization sqr algorithm for solving a class of nonsmooth nonconvex perhaps even non lipschitzian minimization problems which has wide applications in statistics and sparse reconstruction the proposed sqr algorithm is a first order method at each iteration the sqr algorithm solves a strongly convex quadratic minimization problem with a diagonal hessian matrix which has a simple closed form solution that is inexpensive to calculate we show that the worst case complexity of reaching an epsilon scaled stationary point is o epsilon 2 moreover if the objective function is locally lipschitz continuous the sqr algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an epsilon clarke stationary point in at most o epsilon 3 iterations

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

a numerical method based on polynomial sextic spline functions for the solution of special fifth order boundary value problems a second order accurate numerical scheme is presented for the solution of special fifth order boundary value problems with two point boundary conditions the polynomial sextic spline functions are applied to construct the numerical algorithm convergence of the method is discussed through standard convergence analysis a numerical illustration is given to show the pertinent features of the technique

sufficient conditions for error bounds for a lower semicontinuous l s c inequality system on a banach space it is shown that error bounds hold provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero a sufficient condition for a global error bound to exist is also given for an l s c inequality system on a real normed linear space it turns out that a global error bound closely relates to metric regularity which is useful for presenting sufficient conditions for an l s c system to be regular at sets under the generalized slater condition a continuous convex system on rn is proved to be metrically regular at bounded sets

spline approximate solution of fifth order boundary value problem we use non polynomial spline function to develop numerical methods for the solution of the fifth order linear boundary value problems end conditions of the spline are derived we compare our results with the results produced by other decomposition method local polynomial regression and spline methods however it is observed that our approach produce better numerical solutions in the same that max e i is a minimum

complexity analysis of interior point algorithms for non lipschitz and nonconvex minimization we propose a first order interior point algorithm for a class of non lipschitz and nonconvex minimization problems with box constraints which arise from applications in variable selection and regularized optimization the objective functions of these problems are continuously differentiable typically at interior points of the feasible set our first order algorithm is easy to implement and the objective function value is reduced monotonically along the iteration points we show that the worst case iteration complexity for finding an epsilon scaled first order stationary point is o epsilon 2 o 2 furthermore we develop a second order interior point algorithm using the hessian matrix and solve a quadratic program with a ball constraint at each iteration although the second order interior point algorithm costs more computational time than that of the first order algorithm in each iteration its worst case iteration complexity for finding an epsilon scaled second order stationary point is reduced to o epsilon 3 2 o 3 2 note that an epsilon scaled second order stationary point must also be an epsilon scaled first order stationary point

the numerical solution of fifth order boundary value problemsby the decomposition method in this paper we present a fast and accurate numerical scheme for the solution of fifth order boundary value problems with two point boundary conditions the adomian decomposition method and a modified form of this method are applied to construct the numerical solution the new approach provides the solution in the form of a rapidly convergent series and not at grid points two numerical illustrations are given to show the pertinent features of technique

mathematical programs with geometric constraints in banach spaces enhanced optimality exact penalty and sensitivity in this paper we study the mathematical program with geometric constraints such that the image of a mapping from a banach space is included in a nonempty and closed subset of a finite dimensional space we obtain the nonsmooth enhanced fritz john necessary optimality conditions in terms of the approximate subdifferential in the case where the banach space is a weakly compactly generated asplund space the optimality condition obtained can be expressed in terms of the limiting subdifferential while in the general case it can be expressed in terms of the clarke subdifferential one of the technical difficulties in obtaining such a result in an infinite dimensional space is that no compactness result can be used to show the existence of local minimizers of a perturbed problem in this paper we employ the celebrated ekeland s variational principle to obtain the results instead the enhanced fritz john condition allows us to obtain the enhanced karush kuhn tucker condition under the pseudo normality and th

an efficient computational method for linear fifth order two point boundary value problems in this paper we present a new algorithm to solve general linear fifth order boundary value problems bvps in the reproducing kernel space w 2 6 a b representation of the exact solution is given in the reproducing kernel space its approximate solution is obtained by truncating the n term of the exact solution some examples are displayed to demonstrate the computational efficiency of the method

improved iteratively reweighted least squares for unconstrained smoothed ell_q minimization in this paper we first study ell q minimization and its associated iterative reweighted algorithm for recovering sparse vectors unlike most existing work we focus on unconstrained ell q minimization for which we show a few advantages on noisy measurements and or approximately sparse vectors inspired by the results in daubechies et al comm pure appl math 63 2010 pp 1 38 for constrained ell q minimization we start with a preliminary yet novel analysis for unconstrained ell q minimization which includes convergence error bound and local convergence behavior then the algorithm and analysis are extended to the recovery of low rank matrices the algorithms for both vector and matrix recovery have been compared to some state of the art algorithms and show superior performance on recovering sparse vectors and low rank matrices

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem