Globally Convergent Cutting Plane Method for Nonconvex Nonsmooth Minimization

Nowadays, solving nonsmooth (not necessarily differentiable) optimization problems plays a very important role in many areas of industrial applications. Most of the algorithms developed so far deal only with nonsmooth convex functions. In this paper, we propose a new algorithm for solving nonsmooth optimization problems that are not assumed to be convex. The algorithm combines the traditional cutting plane method with some features of bundle methods, and the search direction calculation of feasible direction interior point algorithm (Herskovits, J. Optim. Theory Appl. 99(1):121–146, 1998). The algorithm to be presented generates a sequence of interior points to the epigraph of the objective function. The accumulation points of this sequence are solutions to the original problem. We prove the global convergence of the method for locally Lipschitz continuous functions and give some preliminary results from numerical experiments.     

The Pascoletti–Serafini Scalarization Scheme and Linear Vector Optimization

The Pascoletti–Serafini scalarization scheme for general vector optimization problems is studied. It is specified to linear vector optimization to give minimal representation formulae for the weakly efficient solution set and the efficient solution set. Several facts on connectedness of the solution sets of Pascoletti–Serafini’s scalar auxiliary problems, both for linear vector optimization and for nonlinear vector optimization, are established.     

On the Vector ε-Algorithm for Solving Linear Systems of Equations

The four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation and the topological epsilon algorithm, when applied to linearly generated vector sequences are Krylov subspace methods and it is known that they are equivalent to some well-known conjugate gradient type methods. However, the vector -algorithm is an extrapolation method, older than the four extrapolation methods above, and no similar results are known for it. In this paper, a determinantal formula for the vector -algorithm is given. Then it is shown that, when applied to a linearly generated vector sequence, the algorithm is also a Krylov subspace method and for a class of matrices the method is equivalent to a preconditioned Lanczos method. A new determinantal formula for the CGS is given, and an algebraic comparison between the vector -algorithm for linear systems and CGS is also given.     

The Attractor for a Nonlinear Reaction-Diffusion System in the Unbounded Domain and Kolmogorove's ε-Entropy

The autonomous and nonautonomous semilinear reaction-diffusion systems in unbounded domains are considered. It is proved that under some natural assumptions these systems possess locally compact attractors in the corresponding phase spaces. Moreover, the upper and lower bounds of the Kolmogorov's ε-entropy for these attractors are also obtained.     

Representation Formulas for Analytic Function Non－exponential in a Half Plane

ＲｅｐｒｅｓｅｎｔａｔｉｏｎＦｏｒｍｕｌａｓｆｏｒＡｎａｌｙｔｉｃＦｕｎｃｔｉｏｎＮｏｎ－ｅｘｐｏｎｅｎｔｉａｌｉｎａＨａｌｆＰｌａｎｅＤｅｎｇＧｕａｎｔｉｅ（邓冠铁）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕａｚｈｏｎｇＵｎｉｖｅｒｓｉ...     

N-Widths and ε-Dimensions for High-Dimensional Approximations

In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d _n (W,H ^γ ), and ε-dimensions n _ε (W,H ^γ ) of periodic d-variate function classes W with anisotropic smoothness, where d may be large. We are interested in finding the accurate dependence of d _n (W,H ^γ ) and n _ε (W,H ^γ ) as a function of two variables n, d and ε, d, respectively. Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonometric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H ^γ . Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths d _n (W,H ^γ ) and ε-dimensions n _ε (W,H ^γ ). Some of the received results imply that the curse of dimensionality can be broken in some relevant situations.     

Symplectic Transformations and a Reduction Method for Asymptotically Linear Hamiltonian Systems

In many fields of applications, especially in applications from mechanics, many equations of motion can be written as Hamiltonian systems. In this paper, we study a class of asymptotically linear Hamiltonian systems. We construct a symplectic transformation which reduces the linear systems of the Hamiltonian systems. This reduction method can be applied to study the existence of periodic solutions for a class of asymptotically linear Hamiltonian systems under weaker conditions on the linear systems of the Hamiltonian systems.     

Some Properties of A Lack-of-Fit Test for a Linear Errors in Variables Model

The relationship between the linear errors-in-variables model and the corresponding ordinary linear model in statistical inference is studied. It is shown that normality of the distribution of covariate is a necessary and sufficient condition for the equivalence. Therefore, testing for lack-of-fit in linear errors-in-variables model can be converted into testing for it in the corresponding ordinary linear model under normality assumption. A test of score type is constructed and the limiting chi-squared distribution is derived under the null hypothesis.Furthermore, we discuss the power of the test and the choice of the weight function involved in the test statistic.     

The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.     

An a Posteriori Error Estimate for a Semi-Lagrangian Scheme for Hamilton–Jacobi Equations

We present an a posteriori estimate for a first order semi-Lagrangian method for Hamilton–Jacobi equations. The result requires piecewise C ^1,1 regularity of the viscosity solution and is stated for the Bellman equation related to the infinite horizon problem, although it can be applied to more general Hamilton–Jacobi equations with convex Hamiltonians. This estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution.     

Basic Homogenization Results for a Biconnected ε-Periodic Structure

We present, in a bounded domain, a model of an l -periodic structure composed of two phases, both being connected but only one reaching the boundary of the domain, avoiding in this way the local type convergences of the homogenization process. In this framework we revise some basic tools of the homogenization theory in porous media: the extension and the restriction operators, the Ne ) as inequality. Moreover, we obtain some compacity properties which reduce the proof of the pressure type convergences from the homogenization of fluid flows through porous media to the expected procedure of a priori estimations and two-scale convergences. As all the properties can be proved without much technical difficulties, avoiding annoying hypotheses and the use of Kolmogorov's criterion of compacity, the present structure seems one of the most convenient realistic models of porous media that can be studied with the methods of homogenization.     

Criteria of stability for a Class of Linear Systems

In this paper,employing the geometric criteria of stabilty we obtain some stable results of linear differential equations with impulsive effects.     

Multiple Solutions for a Fourth-order Asymptotically Linear Elliptic Problem

Under simple conditions, we prove the existence of three solutions for a fourth-order asymptotically linear elliptic boundary value problem. For the resonance case at infinity, we do not need to assume any more conditions to ensure the boundedness of the （PS） sequence of the corresponding functional.     

The Intersection of a Triangular Bézier Patch and a Plane

In this paper, the problem of finding the intersection of a triangular Bézier patch and a plane is studied. For the degree that one frequently encounters in practice, i.e. $n=2,3$, an efficient and reliable algorithm is obtained, and computational steps are presented.     

Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme

Given a general primal problem and its Fenchel–Lagrange dual one, which is obtained by using a conjugation scheme based on coupling functions and the perturbational approach, the aim in this work is to establish conditions under which strong duality can be guaranteed. To this purpose, even convexity and properness are a compulsory requirement over the involved functions in the primal problem. Furthermore, two closedness-type regularity conditions and a characterization for strong duality are derived.     

Linear stability and positivity results for a generalized size-structured Daphnia model with inflow§

We employ semigroup and spectral methods to analyze the linear stability of positive stationary solutions of a generalized size-structured Daphnia model. Using the regularity properties of the governing semigroup, we are able to formulate a general stability condition, which permits an intuitively clear interpretation in a special case of model ingredients. Moreover, we derive a comprehensive instability criterion that reduces to an elegant instability condition for the classical Daphnia population model in terms of the inherent net reproduction rate of Daphnia individuals.