The Collapsing 0–1 Knapsack Problem

The Collapsing 0–1 Knapsack Problem is a type of non-linear knapsack problem in which the knapsack size is a non-increasing function of the number of items included.An algorithm is developed and computational results included.     

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective

Journal of Optimization Theory and Applications - We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible...     

Hybrid Lévy Models: Design and Computational Aspects

ABSTRACT

A hybrid model is a model, where two markets are studied jointly such that stochastic dependence can be taken into account. Such a dependence is well known for equity and interest rate markets on which we focus here. Other pairs can be considered in a similar way. Two different versions of a hybrid approach are developed. Independent time-inhomogeneous Lévy processes are used as the drivers of the dynamics of interest rates and equity. In both versions, the dynamics of the interest rate side is described by an equation for the instantaneous forward rate. Dependence between the markets is generated by introducing the driver of the interest rate market as an additional term into the dynamics of equity in the first version. The second version starts with the equity dynamics and uses a corresponding construction for the interest rate side. Dependence can be quantified in both cases by a single parameter. Numerically efficient valuation formulas for interest rate and equity derivatives are developed. Using market quotes for liquidly traded assets we show that the hybrid approach can be successfully calibrated.     

Two-phases Method and Branch and Bound Procedures to Solve the Bi–objective Knapsack Problem

The classical 0–1 knapsack problem is considered with two objectives. Two methods of the two–phases type are developed to generate the set of efficient solutions. In the first phase, the set of supported efficient solutions is determined by optimizing a parameterized single-objective knapsack problem. Two versions are proposed for a second phase, determining the non-supported efficient solutions: both versions are Branch and Bound approaches, but one is breadth first, while the other is depth first. Extensive numerical experiments have been realized to compare the results of both methods.     

A Tabu Search Algorithm for Quadratic 0—1 Programming Problem

1.IntroductionThemathematicalmodelofaquaduatico-1programmingproblemisasfollows:MinimizesubjecttwhereI,AsfaraspaperL1'2Jcanseemedel(I)(fordu=O)isveryimPOrtantinthemarshallingofsinglegrouptrainbetweenmarshallingstationsinrailwaynetworkandthemarshallingoftraininnetw0rkwiththetw0types0fvehiclefl0w,butproblem(I)isNP-C.C0nsiderarelax-ationproblemasf0llows:MinimizeIngeneral,solvingrelaxati0nproblemiseasierthansolvingcombinatiorial0ptimalpr0b-lem,thesameaslinearpr0grammingproblemissolvableinPOly…     

Solving a Class of Multiplicative Programs with 0–1 Knapsack Constraints

We develop a branch-and-bound algorithm to solve a nonlinear class of 0–1 knapsack problems. The objective function is a product of m2 affine functions, whose variables are mutually exclusive. The branching procedure in the proposed algorithm is the usual one, but the bounding procedure exploits the special structure of the problem and is implemented through two stages: the first stage is based on linear programming relaxation; the second stage is based on Lagrangian relaxation. Computational results indicate that the algorithm is promising.     

The Prediction Expression of Multidimensional Stationary Sequence and Rosenblatt''''s Speculation

In this paper we obtain a linear predictor and its error for a multidimensional,discrete parameter, full-rank and regular stationary stochastic process (S.P.) having a spectral density matrix f(λ), which mets some conditions. An example which shows that Szeg(?)-Kolmogorov's Theorem for the prediction error with lag 1 in terms f(λ): (1) ean't be formally kept true in matrix form is given.     

A Route to Routh—The Parametric Problem

There is a well known principle in classical mechanic stating that a variational problem independent of a space variable w (so called cyclic variable), but dependent on the velocity w′ can be expressed without both w and w′. This is the Routh reduction principle. We develop a geometrical approach to the problem and deal with general first order variational integrals admitting a Lie symmetry group of point transformations. In the classical setting the Routh reduction is applied only to the symmetries that preserve the independent variable. In this article we remove such restriction.     

The Forward–Backward Algorithm and the Normal Problem

The forward–backward splitting technique is a popular method for solving monotone inclusions that have applications in optimization. In this paper, we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward–backward operator. We also provide a formula for the range of the displacement map of the forward–backward operator. Several examples illustrate our theory.     

The Monge Problem of “Piles and Holes” on the Torus and the Problem of Small Denominators

We discuss the problem of existence of a smooth endomorphism of a closed n-dimensional manifold carrying a differential n-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the n-dimensional torus reduces to the problem of small denominators well known in analysis.     

The Nevanlinna—Pick Interpolation Problem in Multiply Connected Domains

We simplify and strengthen Abrahamse's result on the Nevanlinna–Pick interpolation problem in a finitely connected planar domain, according to which the problem has a solution if and only if the Pick matrices associated with character-automorphic Hardy spaces are positive semidefinite for all characters in ^n–1/^n–1, where n is the connectivity of the domain. The main aim of the paper is to reduce the indicated procedure (verification of the positive semidefiniteness) for the entire real (n–1)-torus ^n–1/^n–1 to a part of it, whose dimension is, possibly, less than n–1. Bibliography: 14 titles.     

The Existence and Multiplicity of Positive Solutions for a Third—order Three—point Boundary Value Problem

The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated,where n is an arbitrary natural number,The main tool is Krasnosel‘skii fixed point theorem on the cone.     

RESEARCH ANNOUNCEMENTS——The Construction of Eigenvalue Problem Equivalent to Multivariate Polynomial System and the Groebner Basis

In this paper we will show that one cau build up a joint eigenvalue problem eq-uivalent to the. given system. By this way, finding the solutions of the given systemis equivalent to finding all eigenvalues and eigenvectors of one matrix or matrix pen-cil. For the special case that the system has finite isolated solutions, we can obtainall solutions through computing the eigenvalues and eigenvectors of a matrix whichcan Le obtained by Gauss-Jordan elimination. Furthermore, we also find that one canget Groebner Basis for the ideal geuerated by the given system iu this way. For any polynomial f(x)∈K[x_1,x_2,…,x_n],f(x) can be written as     

The Presentation Problem of the Conjugate Cone of the Hardy Space Hp (0 < p ≤ 1)

The Hardy space Hpis not locally convex if 0 < p < 1, even though its conjugate space(Hp) separates the points of Hp. But then it is locally p-convex, and its conjugate cone(Hp) p is large enough to separate the points of Hp. In this case, the conjugate cone can be used to replace its conjugate space to set up the duality theory in the p-convex analysis. This paper deals with the representation problem of the conjugate cone(Hp) p of Hpfor 0 < p ≤ 1, and obtains the subrepresentation theorem(Hp) p L∞(T, C p).     

The Cauchy Problem of the Nonlinear Schrodinger Equations in R^1＋1

In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.