Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

\v{C}ech-Delaunay gradient flow and homology inference for self-maps

We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for \v{C}ech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive \v{C}ech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.Comment: 22 pages, 8 figure

On the geometric dilation of closed curves, graphs, and point sets

The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal submission; it includes additional material from a conference submission (ref. [6] in the paper