On the Cauchy problem for finitely degenerate hyperbolic equations of second order

This paper is devoted to the study of the Cauchy problem inC ^∞ and in the Gevrey classes for some second order degenerate hyperbolic equations with time dependent coefficients and lower order terms satisfying a suitable Levi condition.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.     

Certain nonlocal problem for a degenerate hyperbolic equation

Translated from Matematicheskie Zametki, Vol. 51, No. 3, pp. 91–96, March, 1992.     

The mixed problem for a degenerate second-order hyperbolic equation

We study conditions for uniqueness and existence of a solution for the mixed problem for a second-order hyperbolic equation that is degenerate at a finite instant of time.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 39–42.     

A hyperbolic smoothing approach to the Multisource Weber problem

The Multisource Weber problem, also known as the continuous location-allocation problem, or as the Fermat-Weber problem, is considered here. A particular case of the Multisource Weber problem is the minimum sum-of-distances clustering problem, also known as the continuous \(p\) -median problem. The mathematical modelling of this problem leads to a \(min-sum-min\) formulation which, in addition to its intrinsic bi-level nature, is strongly nondifferentiable. Moreover, it has a large number of local minimizers, so it is a typical global optimization problem. In order to overcome the intrinsic difficulties of the problem, the so called Hyperbolic Smoothing methodology, which follows a smoothing strategy using a special \( \, C^{\infty } \, \) differentiable class function, is adopted. The final solution is obtained by solving a sequence of low dimension \( \, C^{\infty } \, \) differentiable unconstrained optimization sub-problems which gradually approaches the original problem. For the purpose of illustrating both the reliability and the efficiency of the method, a set of computational experiments making use of traditional test problems described in the literature was performed. Apart from consistently presenting better results when compared to related approaches, the novel technique introduced here was able to deal with instances never tackled before in the context of the Multisource Weber problem.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

The degenerate hyperbolic problem for the two-dimensional nonlinear wave system

This paper constructs a local classical solution of degenerate hyperbolic problem for the two-dimensional nonlinear wave system. To deal with the parabolic degeneracy, we introduce a new set of coordinates to transform the nonlinear wave system to a new system that has explicitly singularity-regularity structures. By constructing a weighted metric space, we establish the existence of solution for the new system. Returning the solution to the original variables, we obtain the existence of the classical solution for the nonlinear wave system with degenerate boundary value problem.     

A decomposition approach to the two-stage stochastic unit commitment problem

The unit commitment problem has been a very important problem in the power system operations, because it is aimed at reducing the power production cost by optimally scheduling the commitments of generation units. Meanwhile, it is a challenging problem because it involves a large amount of integer variables. With the increasing penetration of renewable energy sources in power systems, power system operations and control have been more affected by uncertainties than before. This paper discusses a stochastic unit commitment model which takes into account various uncertainties affecting thermal energy demand and two types of power generators, i.e., quick-start and non-quick-start generators. This problem is a stochastic mixed integer program with discrete decision variables in both first and second stages. In order to solve this difficult problem, a method based on Benders decomposition is applied. Numerical experiments show that the proposed algorithm can solve the stochastic unit commitment problem efficiently, especially those with large numbers of scenarios.     

A degenerate hyperbolic equation under Levi conditions

Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions     

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators

We consider second order degenerate hyperbolic Cauchy problems, the degeneracy coming either from low regularity (less than Lipschitz continuity) of the coefficients with respect to time, or from weak hyperbolicity. In the weakly hyperbolic case, we assume an intermediate condition between effective hyperbolicity and the Levi condition. We construct the fundamental solution and study the propagation of singularities using an unified approach to these different kinds of degeneracy.     

A decomposition approach to solve a bilevel capacitated facility location problem with equity constraints

In this paper, we study a capacitated facility location problem with two decision makers. One (say, the leader) decides on which subset of facilities to open and the capacity to be installed in each facility with the goal of minimizing the overall costs; the second decision maker (say, the follower), once the facilities have been designed, aims at maximizing the profit deriving from satisfying the demands of a given set of clients beyond a certain threshold imposed by the leader. The leader can foresee but cannot control the follower’s behavior. The resulting mathematical formulation is a discrete–continuous bilevel optimization problem. We propose a decomposition approach to cope with the bilevel structure of the problem and the integrality of a subset of variables under the control of the leader. Such a proposal has been tested on a set of benchmark instances available in the literature.     

A decomposition approach for solving a broadcast domination network design problem

We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers f _i assigned to each node i, such that for any node j in the graph, there exists some node k having a positive f _k -value whose shortest distance to node j is no more than f _k . The cost of a broadcast domination solution is the sum of all node power values. The network design problem constructs edges that decrease the minimum broadcast domination cost on the graph. The overall problem we consider minimizes the sum of edge construction costs and broadcast domination costs. We show that this problem is NP-hard in the strong sense, even on unweighted graphs. We then propose a decomposition strategy, which iteratively adds valid inequalities based on optimal broadcast domination solutions corresponding to the first-stage network design solutions. We demonstrate that our decomposition approach is computationally far superior to the solution of a single large-scale mixed-integer programming formulation.     

A decomposition approach for a new test-scenario in complex problem solving

Over the last years, psychological research has increasingly used computer-supported tests, especially in the analysis of complex human decision making and problem solving. The approach is to use computer-based test scenarios and to evaluate the performance of participants and correlate it to certain attributes, such as the participant's capacity to regulate emotions. However, two important questions can only be answered with the help of modern optimization methodology. The first one considers an analysis of the exact situations and decisions that led to a bad or good overall performance of test persons. The second important question concerns performance, as the choices made by humans can only be compared to one another, but not to the optimal solution, as it is unknown in general.Additionally, these test-scenarios have usually been defined on a trial-and-error basis, until certain characteristics became apparent. The more complex models become, the more likely it is that unforeseen and unwanted characteristics emerge in studies. To overcome this important problem, we propose to use mathematical optimization methodology not only as an analysis and training tool, but also in the design stage of the complex problem scenario.We present a novel test scenario, the IWR Tailorshop, with functional relations and model parameters that have been formulated based on optimization results. We also present a tailored decomposition approach to solve the resulting mixed-integer nonlinear programs with nonconvex relaxations and show some promising results of this approach.