The efficient frontier for bounded assets

This paper develops a closed form solution of the mean-variance portfolio selection problem for uncorrelated and bounded assets when an additional technical assumption is satisfied. Although the assumption of uncorrelated assets is unduly restrictive, the explicit determination of the efficient asset holdings in the presence of bound constraints gives insight into the nature of the efficient frontier. The mean-variance portfolio selection problem considered here deals with the budget constraint and lower bounds or the budget constraint and upper bounds. For the mean-variance portfolio selection problem dealing with lower bounds the closed form solution is derived for two cases: a universe of only risky assets and a universe of risky assets plus an additional asset which is risk free. For the mean-variance portfolio selection problem dealing with upper bounds, the results presented are for a universe consisting only of risky assets. In each case, the order in which the assets are driven to their bounds depends on the ordering of their expected returns.     

Computing closely matching upper and lower bounds on textile nesting problems

We consider an industrial cutting problem in textile manufacturing and report on algorithms for computing cutting images and lower bounds on waste for this problem. For the upper bounds we use greedy strategies based on hodographs and global optimization based on simulated annealing. For the lower bounds we use branch-and-bound methods for computing optimal solutions of placement subproblems that determine the performance of the overall subproblem. The upper bounds are computed in less than an hour on a common-day workstation and are competitive in quality with results obtained by human nesters. The lower bounds take a few hours to compute and are within 0.4% of the upper bound for certain types of clothing (e.g., for pants).     

On the number of integer polynomials with multiplicatively dependent roots

We give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.     

Error bounds for Newton's iterates derived from the Kantorovich theorem

Summary In this paper, it is shown that the upper and lower bounds of the errors in the Newton iterates recently obtained by Potra-Pták [11] and Miel [7], with the use of nondiscrete induction and majorizing sequence, respectively, follow immediately from the Kantorovich theorem and the Kantorovich recurrence relations. It is also shown that the upper and lower bounds of Miel are finer than those of Potra-Pták.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041     

A posteriori error estimators for convection-diffusion equations

Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received February 10, 1997 / Revised version received November 4, 1997     

Note on the estimation of the order of convergence of some iterative methods

In this paper we obtain upper and lower bounds for the unique positive roots of certain sequences of polynomials. The results are then applied to the determination of theR-order of iterative numerical processes.     

Rate Functions for Symmetric Markov Processes via Heat Kernel

By making full use of heat kernel estimates, we establish the integral tests on the zero-one laws of upper and lower bounds for the sample path ranges of symmetric Markov processes. In particular, these results concerning on upper rate bounds are applicable for local and non-local Dirichlet forms, while lower rate bounds are investigated in both subcritical setting and critical setting.     

Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation

We derive robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Here, robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter. The estimators are based either on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received June 5, 1996     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

Recursive lower and dual upper bounds for Bermudan-style options

Although Bermudan options are routinely priced by simulation and least-squares methods using lower and dual upper bounds, the latter are hardly optimized. In this paper, we optimize recursive upper bounds, which are more tractable than the original/nonrecursive ones, and derive two new results: (1) An upper bound based on (a martingale that depends on) stopping times is independent of the next-stage exercise decision and hence cannot be optimized. Instead, we optimize the recursive lower bound, and use its optimal recursive policy to evaluate the upper bound as well. (2) Less time-intensive upper bounds that are based on a continuation-value function only need this function in the continuation region, where this continuation value is less nonlinear and easier to fit (than in the entire support). In the numerical exercise, both upper bounds improve over state-of-the-art methods (including standard least-squares and pathwise optimization). Specifically, the very small gap between the lower and the upper bounds derived in (1) implies the recursive policy and the associated martingale are near optimal, so that these two specific lower/upper bounds are hard to improve, yet the upper bound is tighter than the lower bound.     

On inverses of Vandermonde and confluent Vandermonde matrices III

Summary We derive lower bounds for the norm of the inverse Vandermonde matrix and the norm of certain inverse confluent Vandermonde matrices. They supplement upper bounds which were obtained in previous papers.Sponsored in part by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under grant MCS 76-00842A01     

Quantifying the error of convex order bounds for truncated first moments

The concepts of convex order and comonotonicity have become quite popular in risk theory, essentially since Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M.J., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168] constructed bounds in the convex order sense for a sum S of random variables without imposing any dependence structure upon it. Those bounds are especially helpful, if the distribution of S cannot be calculated explicitly or is too cumbersome to work with. This will be the case for sums of lognormally distributed random variables, which frequently appear in the context of insurance and finance.In this article we quantify the maximal error in terms of truncated first moments, when S is approximated by a lower or an upper convex order bound to it. We make use of geometrical arguments; from the unknown distribution of S only its variance is involved in the computation of the error bounds. The results are illustrated by pricing an Asian option. It is shown that under certain circumstances our error bounds outperform other known error bounds, e.g. the bound proposed by Nielsen and Sandmann [Nielsen, J.A., Sandmann, K., 2003. Pricing bounds on Asian options. J. Financ. Quant. Anal. 38, 449-473].     

Lower and upper bounds of large deviation for sums of subexponential claims in a multi-risk model

In view of the actual condition of the insurance company, a multi-risk model is proposed. The lower and upper bounds for the sums of subexponential claims in this model are given. The proof method is based on the results of the total claim amount under subexponential class.     

Large deviation bounds for matrix Brownian motion

We prove large deviation bounds for the convergence of Hermitian matrix valued Brownian motion towards free Brownian motion. As a consequence, we obtain upper and lower bounds on the microstates entropy introduced by Voiculescu [24]. Oblatum 5-VIII-2002 & 18-XI-2002?Published online: 24 February 2003     

Scaling estimates for solutions and dynamical lower bounds on wavepacket spreading

We establish quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators in situations where, in addition to power-law upper bounds on solutions corresponding to energies in the spectrum, one also has lower bounds following a scaling law. As a consequence, we obtain improved dynamical results for the Fibonacci Hamiltonian and related models.     

Integrating heuristic information into exact methods: The case of the vertex p-centre problem

We solve the vertex p-centre problem optimally using an exact method that considers both upper and lower bounds as part of its search engine. Tight upper bounds are generated quickly via an efficient three-level heuristic, which are then used to derive potential ‘lower bounds’ accordingly. These two pieces of information when used together make our chosen exact method more efficient at obtaining optimal solutions relatively quickly. The proposed implementation produced excellent results when tested on the OR Library data set. This integrated approach can be adopted for those exact methods that consider both upper and lower bounds within their search engine and hence provide a wider spectrum of applicability in other hard combinatorial problems.     

A note on functional a posteriori estimates for elliptic optimal control problems

In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two‐sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017     

Quadrature rule-based bounds for functions of adjacency matrices

Bounds for entries of matrix functions based on Gauss-type quadrature rules are applied to adjacency matrices associated with graphs. This technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities (Estrada index, subgraph centrality, communicability) that describe properties of networks.     

图的代数连通度的界(英文)

本文首先给出了简单图的度序列的平方和的上界,利用这些结果,求出了简单图的代数连通度的几个上下界并确定了它们的临界图。另外,文章也给出了加权图的代数连通度的一个下界。     

Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.