Work supported by an Alfred P. Sloan Doctoral Dissertation Fellowship. We consider the pentagram map on the space of plane convex pentagons obtained by drawing a pentagon’s diagonals, recovering unpublished results of Conway and proving new ones. We generalize this to a ‘‘pentagram map’’ on convex polygons of more than five sides, showing that iterated images of any initial polygon converge exp...

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the pr...

We will be concerned with linear positive maps φ : Mm(C) → Mn(C). To fix notation we begin with setting up the notation and the relevant terminology (cf. [7]). We say that φ is positive if φ(A) is a positive element in Mn(C) for every positive matrix from Mm(C). If k ∈ N, then φ is said to be k-positive (respectively k-copositive) whenever [φ(Aij)] k i,j=1 (respectively [φ(Aji)] k i,j=1) is pos...

We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an n-sided polygon in O(n) time. Achieving this running time requires proving several new structural properties of the MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm...

We prove some results on the farthest points in convex metric spaces and in linear metric spaces. The continuity of the farthest point map and characterization of strictly convex linear metric spaces in terms of farthest points are also discussed.

According to atmospheric observations new particle formation seems to be a function of sulphuric acid concentration to the power from one to two. The nucleation theorem then predicts that the critical cluster contains one to two sulphuric acid molecules. However, existing nucleation theories predicts that the power is more (or equal) than 2. Here we present an activation theory, which can expla...

We study point set topological properties of the moment map. In particular, we introduce the notion of a convex Hamiltonian manifold. This notion combines convexity of the momentum image and connectedness of moment map fibers with a certain openness requirement for the moment map. We show that convexity rules out many pathologies for moment maps. Then we show that the most important classes of ...

In this paper, -subgradients for convex set-valued maps are defined. We prove an existence theorem for -subgradients of convex set-valued maps. Also, we give necessary optimality conditions for an -solution of a convex set-valued optimization problem (CSP). Moreover, using the single-valued function induced from the set-valued map, we obtain theorems describing the -subgradient sum formula for ...

Coordination and harmony between different departments of a company can be an important factor in achieving competitive advantage if the company corrects alignment between strategies of different departments. This paper presents an integrated decision model based on recent advances of geometric programming technique. The demand of a product considers as a power function of factors such as produ...

Recently, Combined Heat and Power (CHP) systems have been utilized increasingly in power systems. With the addition penetration of CHP-based co-generation of electricity and heat, the determination of economic dispatch of power and heat becomes a more complex and challenging issue. The optimal operation of CHP-based systems is inherently a nonlinear and non-convex optimization problem with a lo...