Solutions Of Degenerate Problems Research Articles

Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation

The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues
 are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.

First asymptotic approximations to a solution of singularly perturbed optimal control problem with intersecting solutions of degenerate problem

Using the direct scheme method for constructing asymptotic solution of optimal control problems with small parameters, consisting of immediate substituting a postulated asymptotic expansion of a solution into the problem condition and receiving problems for finding asymptotic terms, we construct formally the zero order asymptotic approximation to an optimal control and the first order approximation to an optimal trajectory of a singularly perturbed optimal control problem with a weakly controllable state equation, a cheap control, a not uniquely solvable degenerate state equation, fixed endpoints and with intersecting trajectories of the degenerate state equation corresponding to slow trajectories of the perturbed problem. Together with boundary-layer functions in the vicinities of both ends of the considered interval, the constructed asymptotics contains inner boundary-layer functions. An illustrative example is given.

The positive edge pricing rule for the dual simplex

In this paper, we develop the two-dimensional positive edge criterion for the dual simplex. This work extends a similar pricing rule implemented by Towhidi et al. (2014) [24] to reduce the negative effects of degeneracy in the primal simplex. In the dual simplex, degeneracy occurs when nonbasic variables have a zero reduced cost, and it may lead to pivots that do not improve the objective value. We analyze dual degeneracy to characterize a particular set of dual compatible variables such that if any of them is selected to leave the basis the pivot will be nondegenerate. The dual positive edge rule can be used to modify any pivot selection rule so as to prioritize compatible variables. The expected effect is to reduce the number of pivots during the solution of degenerate problems with the dual simplex. For the experiments, we implement the positive edge rule within the dual simplex of the COIN-OR LP solver, and combine it with both the dual Dantzig and the dual steepest edge criteria. We test our implementation on 62 instances from four well-known benchmarks for linear programming. The results show that the dual positive edge rule significantly improves on the classical pricing rules.