A Discrete Convex Min-Max Formula for Box-TDI Polyhedra

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function in which the minimum is taken over the set of integral elements of a box total dual integral polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in nonlinear optimization), but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.

Extension of ADMMAlgorithmin Solving Optimal Control Model Governed by Partial Differential Equation

This paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Partial Differential Equation using the Alternating Direction Method of Multipliers (ADMM) accelerated with a parameter factor in the sense of Nesterov. The ADMM tool wasapplied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type. The constraint and objective functions of the optimal control model were discretized using the Crank-Nicolson and Composite Simpson’s Methods respectively into a derived discrete convex optimization form amenable to the ADMM. The primal-dual residuals were derived to ascertain the rate of convergence of themodel for increasing iterates. An existing example was used to test the efficiency and degree of accuracy of the algorithm and the results were favorable when compared the existing method.

Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces

Identities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.

New discrete inequalities of Hermite–Hadamard type for convex functions

We introduce new time scales on $\mathbb{Z}$ Z . Based on this, we investigate the discrete inequality of Hermite–Hadamard type for discrete convex functions. Finally, we improve our result to investigate the discrete fractional inequality of Hermite–Hadamard type for the discrete convex functions involving the left nabla and right delta fractional sums.

Scattering of electromagnetic waves by a discrete octahedron from resonant spheres

A solution is given to the problem of scattering of electromagnetic waves by a discrete convex polyhedron – an octahedron of resonant magnetodielectric spheres based on a complex rhombic crystal lattice. Here we consider a case equivalent to the X-ray optics of crystals, when α / λ՛<<1 and can be α / λg ~ 1; d, h, l / λ՛ ~ 1, where α is the radius of the spheres; λ՛, λg are the lengths of the scattered wave outside and inside the spheres; d, h, l are constant lattices. The solution of the problem is obtained based on the Fredholm integral equations of electrodynamics of the second kind with nonlocal boundary conditions. The expressions found in this work for a metacrystal in the form of an octahedron can be used to study the fields scattered by the crystal in the Fresnel and Fraunhofer zones, as well as to study its internal field. The relations obtained in this work can find application in the study of the scattering of waves of various kinds by convex polyhedrons, the creation on their basis of new types of limited metacrystals, including nanocrystals with resonance properties, and in the study of their behavior in various external media. As well as in the development of methods for modeling electromagnetic phenomena that can occur in real crystals in resonance regions in the optical and X-ray wavelength ranges.