Practical Volume Estimation by a New Annealing Schedule for Cooling Convex Bodies

We study the problem of estimating the volume of convex polytopes, focusing on H- and V-polytopes, as well as zonotopes. Although a lot of effort is devoted to practical algorithms for H-polytopes there is no such method for the latter two representations. We propose a new, practical algorithm for all representations, which is faster than existing methods. It relies on Hit-and-Run sampling, and combines a new simulated annealing method with the Multiphase Monte Carlo (MMC) approach. Our method introduces the following key features to make it adaptive: (a) It defines a sequence of convex bodies in MMC by introducing a new annealing schedule, whose length is shorter than in previous methods with high probability, and the need of computing an enclosing and an inscribed ball is removed; (b) It exploits statistical properties in rejection-sampling and proposes a better empirical convergence criterion for specifying each step; (c) For zonotopes, it may use a sequence of convex bodies for MMC different than balls, where the chosen body adapts to the input. We offer an open-source, optimized C++ implementation, and analyze its performance to show that it outperforms state-of-the-art software for H-polytopes by Cousins-Vempala (2016) and Emiris-Fisikopoulos (2018), while it undertakes volume computations that were intractable until now, as it is the first polynomial-time, practical method for V-polytopes and zonotopes that scales to high dimensions (currently 100). We further focus on zonotopes, and characterize them by their order (number of generators over dimension), because this largely determines sampling complexity. We analyze a related application, where we evaluate methods of zonotope approximation in engineering.Comment: 20 pages, 12 figures, 3 table

Optimization Algorithms for Faster Computational Geometry

We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined by $m$ hyperplanes, and the minimum enclosing ball (MinEB) of a set of $n$ points, both in $d$-dimensional space. We improve the running time of iterative algorithms on MaxIB from $\tilde{O}(m d \alpha^3 / \varepsilon^3)$ to $\tilde{O}(md + m \sqrt{d} \alpha / \varepsilon)$, a speed-up up to $\tilde{O}(\sqrt{d} \alpha^2 / \varepsilon^2)$, and MinEB from $\tilde{O}(n d / \sqrt{\varepsilon})$ to $\tilde{O}(nd + n \sqrt{d} / \sqrt{\varepsilon})$, a speed-up up to $\tilde{O}(\sqrt{d})$. Our improvements are based on a novel saddle-point optimization framework. We propose a new algorithm $\mathtt{L1L2SPSolver}$ for solving a class of regularized saddle-point problems, and apply a randomized Hadamard space rotation which is a technique borrowed from compressive sensing. Interestingly, the motivation of using Hadamard rotation solely comes from our optimization view but not the original geometry problem: indeed, it is not immediately clear why MaxIB or MinEB, as a geometric problem, should be easier to solve if we rotate the space by a unitary matrix. We hope that our optimization perspective sheds lights on solving other geometric problems as well.Comment: An abstract of this paper is going to appear in the conference proceedings of ICALP 201

A Novel Approach for Ellipsoidal Outer-Approximation of the Intersection Region of Ellipses in the Plane

In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional (2D) space is proposed. First, the vertices of a tight polygon that contains the convex intersection of the ellipses are found in an efficient manner. To do so, the intersection points of the ellipses that fall on the boundary of the intersection region are determined, and a set of points is generated on the elliptic arcs connecting every two neighbouring intersection points. By finding the tangent lines to the ellipses at the extended set of points, a set of half-planes is obtained, whose intersection forms a polygon. To find the polygon more efficiently, the points are given an order and the intersection of the half-planes corresponding to every two neighbouring points is calculated. If the polygon is convex and bounded, these calculated points together with the initially obtained intersection points will form its vertices. If the polygon is non-convex or unbounded, we can detect this situation and then generate additional discrete points only on the elliptical arc segment causing the issue, and restart the algorithm to obtain a bounded and convex polygon. Finally, the smallest area ellipse that contains the vertices of the polygon is obtained by solving a convex optimization problem. Through numerical experiments, it is illustrated that the proposed technique returns a tighter outer-approximation of the intersection of multiple ellipses, compared to conventional techniques, with only slightly higher computational cost

Practical Volume Computation of Structured Convex Bodies, and an Application to Modeling Portfolio Dependencies and Financial Crises

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies. We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions, which are the method of choice for intersections with a single hyperplane; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple with additional structure; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed in the given body, such as a second-order cone program; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.Comment: 28 pages, 10 figure

Statistical mechanics for metabolic networks during steady-state growth

Which properties of metabolic networks can be derived solely from stoichiometric information about the network's constituent reactions? Predictive results have been obtained by Flux Balance Analysis (FBA), by postulating that cells set metabolic fluxes within the allowed stoichiometry so as to maximize their growth. Here, we generalize this framework to single cell level using maximum entropy models from statistical physics. We define and compute, for the core metabolism of Escherichia coli, a joint distribution over all fluxes that yields the experimentally observed growth rate. This solution, containing FBA as a limiting case, provides a better match to the measured fluxes in the wild type and several mutants. We find that E. coli metabolism is close to, but not at, the optimality assumed by FBA. Moreover, our model makes a wide range of predictions: (i) on flux variability, its regulation, and flux correlations across individual cells; (ii) on the relative importance of stoichiometric constraints vs. growth rate optimization; (iii) on quantitative scaling relations for singe-cell growth rate distributions. We validate these scaling predictions using data from individual bacterial cells grown in a microfluidic device at different sub-inhibitory antibiotic concentrations. Under mild dynamical assumptions, fluctuation-response relations further predict the autocorrelation timescale in growth data and growth rate adaptation times following an environmental perturbation.Comment: 12 pages, 4 figure

Inner approximation of convex cones via primal-dual ellipsoidal norms

We study ellipsoids from the point of view of approximating convex sets. Our focus is on finding largest volume ellipsoids with specified centers which are contained in certain convex cones. After reviewing the related literature and establishing some fundamental mathematical techniques that will be useful, we derive such maximum volume ellipsoids for second order cones and the cones of symmetric positive semidefinite matrices. Then we move to the more challenging problem of finding a largest pair (in the sense of geometric mean of their radii) of primal-dual ellipsoids (in the sense of dual norms) with specified centers that are contained in their respective primal-dual pair of convex cones