This paper proposes a new method to determine the viability of switched system on cone and an unbounded polyhedron. First, we investigate condition cone. Then, sufficient criterion for polyhedron, which is expressed by convex hull finite number extreme points nonnegative linear combination directions, presented using nonsmooth analysis. Based this criterion, instead all boundary points, just se...

This research aims to explain the intrinsic difficulty of Karp’s list twenty-one problems through use empirical complexity measures based on ellipsoidal width polyhedron generated by constraints relaxed linear programming problem. The variables used as are number nodes visited B&B and CPU time spent solving problems. measurements explanatory correspond Dikin ellipse eigenvalues within polyh...

Given vertex valencies admissible for a self-dual polyhedral graph, we describe an algorithm to explicitly construct such polyhedron. Inputting in the permutations of degree sequence can give rise non-isomorphic graphs. As application, find as function n ≥ 3 minimal number vertices polyhedron with at least one i each ≤ n, and polyhedra. Moreover, construction non-self-dual graphs order i-gonal ...

A path following algorithm is proposed for finding a solution to the nonlinear stationary point problem on an unbounded, convex and pointed polyhedron. The algorithm can start at an arbitrary point of the polyhedron. The path to be foliowed by the algorithm is described as the path of zeros of some piecewise continuously differentiable function defined on an appropriate subdivided manifold. Thi...

A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of R. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved t...

We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q “lives on a cone” to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new gen...

In this paper we discuss an algorithm to perform the conversion from the interior to the boundary of d-dimensional polyhedra, where both the d-polyhedron and its (d 1) boundary faces are represented as BSP trees. This approach allows also to compute the BSP tree of the set intersection between any hyperplane in Ed and the BSP representation of a d-polyhedron. If such section hyperplane is the a...

Quite recently, Murota introduced an M-convex function as a quantitative generalization of the set of integral points in a base polyhedron, as well as an extension of valuated matroid over base polyhedron. Just as a base polyhedron can be transformed through a network, an M-convex function can be induced through a network. This paper gives an algorithmic proof for the induction of an M-convex f...

The first reference to the rigidity of frameworks in the mathematical literature occurs in a problem posed by Euler in 1776. Consider a polyhedron P in 3-space. We view P as a ‘ panel-and-hinge framework’ in which the faces are 2-dimensional panels and the edges are 1-dimensional hinges. The panels are free to move continuously in 3space, subject to the constraints that the shapes of the panels...

AD-polyhedron is a polyhedron P such that if x, y are in P then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, Dpolyhedra are a unifying genera...