fault tolerant cycles embedding in hypercubes with faulty edges let q n be an n dimensional hypercube with f e 5 in this paper we consider the faulty hypercube under the following two additional conditions 1 each vertex is incident to at least two fault free edges and 2 every 4 cycle does not have any pair of non adjacent vertices whose degrees are both two after removing the faulty edges we prove that there exists a fault free cycle of every even length from 4 to 2 n in q n our result improves the result by liu and wang 2014 in terms of the lengths of embedding cycles where under the same conditions a fault free hamiltonian cycle was constructed

worst case evaluation complexity for unconstrained nonlinear optimization using high order regularized models the worst case evaluation complexity for smooth possibly nonconvex unconstrained optimization is considered it is shown that if one is willing to use derivatives of the objective function up to order p for p ge 1 p 1 and to assume lipschitz continuity of the p th derivative then an epsilon approximate first order critical point can be computed in at most o epsilon p 1 p o p 1 p evaluations of the problem s objective function and its derivatives this generalizes and subsumes results known for p 1 p 1 and p 2 p 2

newton type methods for non convex optimization under inexact hessian information we consider variants of trust region and adaptive cubic regularization methods for non convex optimization in which the hessian matrix is approximated under certain condition on the inexact hessian and using approximate solution of the corresponding sub problems we provide iteration complexity to achieve varepsilon approximate second order optimality which have been shown to be tight our hessian approximation condition offers a range of advantages as compared with the prior works and allows for direct construction of the approximate hessian with a priori guarantees through various techniques including randomized sampling methods in this light we consider the canonical problem of finite sum minimization provide appropriate uniform and non uniform sub sampling strategies to construct such hessian approximations and obtain optimal iteration complexity for the corresponding sub sampled trust region and adaptive cubic regularization methods

a survey of the theory of hypercube graphs abstract we present a comprehensive survey of the theory of hypercube graphs basic properties related to distance coloring domination and genus are reviewed the properties of the n cube defined by its subgraphs are considered next including thickness coarseness hamiltonian cycles and induced paths and cycles finally various embedding and packing problems are discussed including the determination of the cubical dimension of a given cubical graph

hamiltonian path embedding and pancyclicity on the mobius cube with faulty nodes and faulty edges a graph g v e is said to be pancyclic if it contains fault free cycles of all lengths from 4 to v in g let f sub v and f sub e be the sets of faulty nodes and faulty edges of an n dimensional mobius cube mq sub n respectively and let f f sub v spl cup f sub e a faulty graph is pancyclic if it contains fault free cycles of all lengths from 4 to v f sub v in this paper we show that mq sub n f contains a fault free hamiltonian path when f spl les n 1 and n spl ges 1 we also show that mq sub n f is pancyclic when f spl les n 2 and n spl ges 2 since mq sub n is regular of degree n both results are optimal in the worst case

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

a trust region algorithm with a worst case iteration complexity of mathcal o epsilon 3 2 o 3 2 for nonconvex optimization we propose a trust region algorithm for solving nonconvex smooth optimization problems for any overline epsilon in 0 infty 0 the algorithm requires at most mathcal o epsilon 3 2 o 3 2 iterations function evaluations and derivative evaluations to drive the norm of the gradient of the objective function below any epsilon in 0 overline epsilon 0 this improves upon the mathcal o epsilon 2 o 2 bound known to hold for some other trust region algorithms and matches the mathcal o epsilon 3 2 o 3 2 bound for the recently proposed adaptive regularisation framework using cubics also known as the arc algorithm our algorithm entitled trace follows a trust region framework but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst case global complexity bound importantly we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms we also prove a worst case upper bound on the number of iterations function evaluations and derivative evaluations that the algorithm requires to obtain an approximate second order stationary point

on the evaluation complexity of cubic regularization methods for potentially rank deficient nonlinear least squares problems and its relevance to constrained nonlinear optimization we propose a new termination criterion suitable for potentially singular zero or nonzero residual least squares problems with which cubic regularization variants take at most mathcal o epsilon 3 2 residual and jacobian evaluations to drive either the euclidean norm of the residual or its gradient below epsilon this is the best known bound for potentially rank deficient nonlinear least squares problems we then apply the new optimality measure and cubic regularization steps to a family of least squares merit functions in the context of a target following algorithm for nonlinear equality constrained problems this approach yields the first evaluation complexity bound of order epsilon 3 2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first order criticality

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

on high order model regularization for constrained optimization in two recent papers regularization methods based on taylor polynomial models for minimization were proposed that only rely on holder conditions on the higher order employed derivatives grapiglia and nesterov considered cubic regularization with a sufficient descent condition that uses the current gradient and resembles the classical armijo s criterion cartis gould and toint used taylor models with arbitrary order regularization and defined methods that tackle convex constraints employing the descent criterion that compares actual reduction with predicted reduction the methods presented in this paper consider general not necessarily taylor models of arbitrary order employ a very mild descent criterion and handle general nonnecessarily convex constraints complexity results are compatible with the ones presented in the papers mentioned above

a decoupled first second order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds in order to be provably convergent towards a second order stationary point optimization methods applied to nonconvex problems must necessarily exploit both first and second order information however as revealed by recent complexity analyses of some of these methods the overall effort to reach second order points is significantly larger when compared to the one of approaching first order ones on the other hand there are other algorithmic schemes initially designed with first order convergence in mind that do not appear to maintain the same first order performance when modified to take second order information into account in this paper we propose a technique that separately computes first and second order steps and that globally converges to second order stationary points it consists in better connecting the steps to be taken and the stationarity criteria potentially guaranteeing larger steps and decreases in the objective our approach is shown to lead to an improvement of the corresponding complexity bound with respect to the first order optimality tolerance while having a positive impact on the practical behavior although the applicability of our ideas is wider we focus the presentation on trust region methods with and without derivatives and motivate in both cases the interest of our strategy

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

bipanconnectivity of faulty hypercubes with minimum degree abstract in this paper we consider the conditionally faulty hypercube q n with n 2 where each vertex of q n is incident with at least m fault free edges 2 m n 1 we shall generalize the limitation m 2 in all previous results of edge bipancyclicity we also propose a new edge fault tolerant bipanconnectivity called k edge fault tolerant bipanconnectivity a bipartite graph is k edge fault tolerant bipanconnected if g f remains bipanconnected for any f e g with f k for every integer m under the same hypothesis we show that q n is n 2 edge fault tolerant edge bipancyclic and bipanconnected and the results are optimal with respect to the number of edge faults tolerated this not only improves some known results on edge bipancyclicity and bipanconnectivity of hypercubes but also simplifies the proof

accelerated methods for non convex optimization we present an accelerated gradient method for nonconvex optimization problems with lipschitz continuous first and second derivatives in a time o epsilon 7 4 log 1 epsilon the method f

accelerating the cubic regularization of newton s method on convex problems in this paper we propose an accelerated version of the cubic regularization of newton s method nesterov and polyak in math program 108 1 177 205 2006 the original version used for minimizing a convex function with lipschitz continuous hessian guarantees a global rate of convergence of order o big 1 over k 2 big where k is the iteration counter our modified version converges for the same problem class with order o big 1 over k 3 big keeping the complexity of each iteration unchanged we study the complexity of both schemes on different classes of convex problems in particular we argue that for the second order schemes the class of non degenerate problems is different from the standard class

multiple regular graph embeddings into a hypercube with unbounded expansion as topological structures interconnection networks play an important role in parallel and distributed computer systems particularly in multicomputer systems which provide an effective mechanism for exchanging data between processors in this paper we study the node fault tolerant capability of an n dimensional hypercube with respect to multiple regular graph embeddings into a hypercube with unbounded expansion we present a fault tolerant method for multiple regular graph embeddings into a hypercube with dilation 3 congestion 1 and load 1 these results show that o n email protected log email protected 2 faults can be tolerated where n is the number of dimensions in a hypercube and l is the number of the nodes of the regular graph the presented embedding methods are mainly optimized for balancing the processor loads while minimizing load and congestion as far as possible furthermore our method expands on some previously known findings also this technology can be applied in grid computing and cloud computing

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

on the complexity of finding first order critical points in constrained nonlinear optimization the complexity of finding epsilon approximate first order critical points for the general smooth constrained optimization problem is shown to be no worse that o epsilon 2 in terms of function and constraints evaluations this result is obtained by analyzing the worst case behaviour of a first order short step homotopy algorithm consisting of a feasibility phase followed by an optimization phase and requires minimal assumptions on the objective function since a bound of the same order is known to be valid for the unconstrained case this leads to the conclusion that the presence of possibly nonlinear nonconvex inequality equality constraints is irrelevant for this bound to apply

an interior affine scaling cubic regularization algorithm for derivative free optimization subject to bound constraints abstract in this paper we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function we first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method at each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique the global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions preliminary numerical results are reported to show the effectiveness of the proposed algorithm

matrix concentration inequalities via the method of exchangeable pairs this paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix the analysis requires a matrix extension of the scalar concentration theory developed by sourav chatterjee using stein s method of exchangeable pairs when applied to a sum of independent random matrices this approach yields matrix generalizations of the classical inequalities due to hoeffding bernstein khintchine and rosenthal the same technique delivers bounds for sums of dependent random matrices and more general matrix valued functions of dependent random variables

global convergence rate analysis of unconstrained optimization methods based on probabilistic models we present global convergence rates for a line search method which is based on random first order models and directions whose quality is ensured only with certain probability we show that in terms of the order of the accuracy the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models the use of probabilistic models only increases the complexity by a constant which depends on the probability of the models being good we particularize and improve these results in the convex and strongly convex case we also analyse a probabilistic cubic regularization variant that allows approximate probabilistic second order models and show improved complexity bounds compared to probabilistic first order methods again as a function of the accuracy the probabilistic cubic regularization bounds are of the same optimal order as for the deterministic case

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

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

panconnectivity and pancyclicity of the 3 ary n cube network under the path restrictions the k ary n cube denoted by q n k is one of the most important interconnection networks for parallel computing in this paper we consider the problem of embedding paths and cycles into 3 ary n cubes under the path restrictions let p be a path in q n 3 we show that when v p 2n 3 there exists a path of any length from n 1 to v q n 3 p 1 between two arbitrary nodes in q n 3 p we also prove that when e p 2n 1 there exists a cycle of any length from e p n to v q n 3 in q n 3 passing through p our results are best possible in some sense

embedding of cycles in arrangement graphs arrangement graphs have been proposed as an attractive interconnection topology for large multiprocessor systems the authors study these graphs by proving the existence of hamiltonian cycles in any arrangement graph they also prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph they show that an arrangement graph can be decomposed into node disjoint cycles in many different ways

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

solving large scale cubic regularization by a generalized eigenvalue problem cubic regularization methods have several favorable properties in particular under mild assumptions they are globally convergent towards critical points with second order necessary conditions sat

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

recursive trust region methods for multiscale nonlinear optimization a class of trust region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems like those arising in systems governed by partial differential equations the algorithms in this class make use of the discretization level as a means of speeding up the computation of the step this use is recursive leading to true multilevel multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations a simple algorithm of the class is then described and its numerical performance is shown to be numerically promising this observation then motivates a proof of global convergence to first order stationary points on the fine grid that is valid for all algorithms in the class

complexity analysis of second order line search algorithms for smooth nonconvex optimization there has been much recent interest in finding unconstrained local minima of smooth functions due in part to the prevalence of such problems in machine learning and robust statistics a particular focus is algorithms with good complexity guarantees second order newton type methods that make use of regularization and trust regions have been analyzed from such a perspective more recent proposals based chiefly on first order methodology have also been shown to enjoy optimal iteration complexity rates while providing additional guarantees on computational cost in this paper we present an algorithm with favorable complexity properties that differs in two significant ways from other recently proposed methods first it is based on line searches only each step involves computation of a search direction followed by a backtracking line search along that direction second its analysis is rather straightforward relying for the most part on the standard technique for demonstrating sufficient decrease in t

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