MIRHA: multi-start biased randomization of heuristics with adaptive local search for solving non-smooth routing problems

This paper discusses the use of probabilistic or randomized algorithms for solving vehicle routing problems with non-smooth objective functions. Our approach employs non-uniform probability distributions to add a biased random behavior to the well-known savings heuristic. By doing so, a large set of alternative good solutions can be quickly obtained in a natural way and without complex configuration processes. Since the solution-generation process is based on the criterion of maximizing the savings, it does not need to assume any particular property of the objective function. Therefore, the procedure can be especially useful in problems where properties such as non-smoothness or non-convexity lead to a highly irregular solution space, for which the traditional optimization methods—both of exact and approximate nature—may fail to reach their full potential. The results obtained so far are promising enough to suggest that the idea of using biased probability distributions to randomize classical heuristics is a powerful one that can be successfully applied in a variety of cases.     

Modifications of the Hurwicz’s decision rule

The Hurwicz’s criterion is one of the classical decision rules applied in decision making under uncertainty as a tool enabling to find an optimal pure strategy both for interval and scenarios uncertainty. The interval uncertainty occurs when the decision maker knows the range of payoffs for each alternative and all values belonging to this interval are theoretically probable (the distribution of payoffs is continuous). The scenarios uncertainty takes place when the result of a decision depends on the state of nature that will finally occur and the number of possible states of nature is known and limited (the distribution of payoffs is discrete). In some specific cases the use of the Hurwicz’s criterion in the scenarios uncertainty may lead to quite illogical and unexpected results. Therefore, the author presents two new procedures combining the Hurwicz’s pessimism-optimism index with the Laplace’s approach and using an additional parameter allowing to set an appropriate width for the ranges of relatively good and bad payoffs related to a given decision. The author demonstrates both methods on the basis of an example concerning the choice of an investment project. The methods described may be used in each decision making process within which each alternative (decision, strategy) is characterized by only one criterion (or one synthetic measure).     

Homogeneous and non-homogeneous boundary value problems for first order linear hyperbolic systems arising in fluid-mechanics (Part I)

We prove t h e existence and the uniqueness of differentiable and strong solutions for aclass of non-homogeneous boundary value problems for first order linear hyperbolic systems arising from the dynamics of compressible non-viscous fluids . The method provides.the existence of differentiable solutions without resorting to strong or weak solutions. A necessary and sufficient condition for the existence of solutions for the non-homogeneous problem is proved. I t consists of an explicitrelationship between the boundary values of u and those of the data f . Strong solutions are obtained without this supplementary assumption. See Theorems 3.1, 4.1, 4 . 2 , 4.3 and Corollary 4.4; see also Remarks 2.1 and 2.4. In this paper we consider equation (3.1) below. In the forthcoming part II we prove similar results for the corresponding evolution problem.     

Absence of wandering domains for some real entire functions with bounded singular sets

Let $f$ be a real entire function whose set $S(f)$ of singular values is real and bounded. We show that, if $f$ satisfies a certain function-theoretic condition (the “sector condition”), then $f$ has no wandering domains. Our result includes all maps of the form $z\mapsto \lambda \frac{\sinh (z)}{z} + a$ with $\lambda >0$ and $a\in \mathbb{R }$ . We also show the absence of wandering domains for certain non-real entire functions for which $S(f)$ is bounded and $f^n|_{S(f)}\rightarrow \infty $ uniformly. As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the exponential map $f(z)=e^z$ is the entire complex plane. Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.     

Homogeneous and non-homogeneous boundary value problems for first order linear hyperbolic systems arising in fluid mechanics (Part II)

We prove the existence and the uniqueness of differentiable and strong solutions for a class of boundary value problems for first order linear hyperbolic systems arising from the dynamics of compressible non-viscous fluids. In particular necessary and sufficient conditions for the existence of solutions for the non-homogeneous problem are studied; strong solutions are obtained without this supplementary condition. See Theorems3.2, 3.9, 4.1, 4.2 and Corollary 4.3; see also the discussion after Theorem 4.1. In particular we don't assume the boundary space to be maximal non-positive and the boundary matrix to be of constant rank on the boundary. In this paper we prove directly the existence of differentiable solutions without resort to weak or strong solutions. An essential tool will be the introduction of a space Z of regular functions verifying not only the assigned boundary conditions but also some suitable complementary boundary conditions; see also the introduction of Part I of this work [I].     

Higher order symmetrizer and application to an unusual transport equation

We study a class of stationary transport equation with nonlocal low-order tems We obtain the existence and uniqueness of a solution in sobolev spaces     

Oscillations of linear integro-differential equations

Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established.     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.