Abstract We study the minimization of a rank-one quadratic with indicators and show that underlying set function obtained by projecting out continuous variables is supermodular. Although supermodular is, in general, difficult, specific for can be minimized linear time. convex hull epigraph from inequalities lifting them into nonlinear original space variables. Explicit forms convex-hull descrip...

Given a set S of n distinct points { (xi'Yi) 1 0 5 i < n] 9 the convex hull problem is to determine the vertices of the convex hull H(S) . All the known algorithms for solving this problem have a worst-case running time of cn log n or higher, and employ only quadratic tests, i.e., tests of ‘the form f(Xo’YO ⌧19 Yl> l l l ☺ ⌧nBl’ Y, 1> : 0 with f being any polynomial of degree not exceeding 2 . ...

We introduce the concept of sos-convex Lyapunov functions for stability analysis of discrete time switched systems. These are polynomial Lyapunov functions that have an algebraic certificate of convexity, and can be efficiently found by semidefinite programming. We show that sos-convex Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear sy...

Let N be a lattice and P ⊂ N ⊗ Z R a lattice polytope, i.e., the convex hull of finitely many points in N. Ehrhart's theorem asserts that the lattice-point counting function f P (m) := # (mP ∩ N) is a polynomial, and thus 1 + P m≥1 f P (m) t m = δ P (t) (1−t) d+1 for some polynomial δ P (t), the δ-vector of P. Motivated by the Knudsen–Mumford–Waterman Conjecture about the existence of unimodula...

Consider an operator dQ(f) = d dxk (Q(x)f(x)) where Q(x) is some fixed polynomial of degree k. One can easily see that dQ has exactly one polynomial eigenfunction pn(x) in each degree n ≥ 0 and its eigenvalue λn,k equals (n+k)! n! . A more intriguing fact is that all zeros of pn(x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each pn(x) enj...

When using the convex hull approach in the boundary modeling process, ModelBased Calibration (MBC) software suites – such as Model-Based Calibration Toolbox from MathWorks – can be computationally intensive depending on the amount of data modeled. The reason for this is that the half-space representation of the convex hull is used. We discuss here another representation of the convex hull, the ...

This paper describes the procedure developed at MARINTEK for the automatic optimisation of hull lines to reduce the ship resistance as well as the wake wash generated by a ship within the given constraints. The methodology, developed computer programs and used constraints are described. The methodology is exercised using lines of a fast ship. A numerical comparison is performed between the resi...

a r t i c l e i n f o a b s t r a c t Keywords: Graph algorithms Simple-path convexity Simple-path convex hull Totally balanced hypergraphs In a connected hypergraph a vertex set X is simple-path convex (sp-convex, for short) if either |X| 1 or X contains every vertex on every simple path between two vertices in X (Faber and Jamison, 1986 [7]), and the sp-convex hull of a vertex set X is the mi...

A celebrated theorem of Balas gives a linear mixed-integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union. The number of inequalities in Balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled. In this paper we show that this is best possible: in every dimension the...