A note on “branch-and-price approach for the multi-skill project scheduling problem”

In a recent paper published in Optimization Letters, Montoya et al. (Optim Lett 8:1721–1734, 2014) proposed a branch-and-price approach for a multi-skill project scheduling problem. In that paper, an integer linear programming formulation was first introduced which, unfortunately, has a number of inconsistences. At the interest of mathematical rigor, in this note, we refine such formulation.     

A note on “On Pareto optima, the Fermat-Weber problem, and polyhedral gauges”

Received September 3, 1997 / Revised version received March 20, 1998 Published online October 9, 1998     

A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type method to compute selected eigenpairs of the product eigenvalue problem _Am?_A1x=λx,$A_m?A_{1}x=\lambda x,$ where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.     

A note on Galvão's “A graph theoretical bound for the p-median problem”

Galvão (1981) proposed an algorithm for calculating a strong lower bound for the p-median problem on vertex-unweighted networks. It is shown that there is a mistake in the algorithm. An improved version of the algorithm is given.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

Nonsteady inverse problem for the multidimensional wave equation “in the large”

The nonsteady (dynamic) inverse problem of reconstructing the variable velocity of wave propagation in a multidimensional region with a smooth boundary is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 21–30, 1987.     

A note on “hierarchical models for multi-project planning and scheduling”

We consider the multi-mode resource-constrained project scheduling problem. The focus of our analysis is on an algorithm recently proposed by Speranza and Vercellis for finding makespan minimal solutions. The correctness of the algorithm is examined. By counterexamples we illustrate that the algorithm does not generally find (existing) optimal solutions.     

A note on “Reguralizers for structured sparsity”

In this note, the notion of admissible sets contained in the strictly positive orthant introduced in Micchelli et al. (Adv. Comp. Math. 38(3), 455–489 2013) is analyzed. This notion was used to generalize theoretical results and optimization methods for structured sparsity. Unfortunately, we will prove that there is no generalization using admissible sets.     

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ⁻¹ V((x − x ₀)ɛ⁻¹), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛ^δ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.     

A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

In this paper, we almost completely solve the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors where the number of vertices is a multiple of $8m$.     

A note on the second largest eigenvalue of the laplacian matrix of a graph ∗

In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.