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Polytopes and arrangements: diameter and curvature
"... By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the number of inequalities defining the polytope. By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diameter of ..."
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By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the number of inequalities defining the polytope. By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diameter of a bounded cell of an arrangement is less than the dimension. We prove continuous analogues of two results of HoltKlee and KleeWalkup: we construct a family of polytopes which attain the conjectured order of the largest total curvature, and we prove that the special case where the number of inequalities is twice the dimension is equivalent to the general case. We substantiate these conjectures in low dimensions and highlight additional links. 1 Continuous Analogue of the Conjecture of Hirsch Let P be a full dimensional convex polyhedron defined by m inequalities in dimension n. The diameter δ(P) is the smallest number such that any two vertices of the polyhedron P can be connected by a path with at most δ(P) edges. The conjecture of Hirsch, formulated in 1957 and reported in [2], states that the diameter of a polyhedron defined by m inequalities in dimension n is not greater than m − n. The conjecture does not hold for unbounded polyhedra. A polytope is a bounded polyhedron.
A continuous dstep conjecture for polytopes
 Discrete Comput. Geom
"... The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimensi ..."
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The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.
Hyperplane arrangements with large average diameter
, 2007
"... Let ∆A(n, d) denote the largest possible average diameter of a bounded cell of a simple arrangement defined by n hyperplanes in dimension d. We have ∆A(n, 2) ≤ 2 + 2 n−1 in the plane, and ∆A(n, 3) ≤ 3 + 4 n−1 in dimension 3. In general, the average diameter of a bounded cell of a simple arrangeme ..."
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Cited by 4 (3 self)
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Let ∆A(n, d) denote the largest possible average diameter of a bounded cell of a simple arrangement defined by n hyperplanes in dimension d. We have ∆A(n, 2) ≤ 2 + 2 n−1 in the plane, and ∆A(n, 3) ≤ 3 + 4 n−1 in dimension 3. In general, the average diameter of a bounded cell of a simple arrangement is conjectured to be less than the dimension; that is, ∆A(n, d) ≤ d. We propose an hyperplane arrangement with � � n−d d cubical cells for n ≥ 2d. It implies that the dimension d is an asymptotic lower bound for ∆A(n, d) for fixed d. In particular, we propose line and plane arrangements with large average diameter yielding ∆A(n, 2) ≥ 2 − and ∆A(n, 3) ≥ 3 − 6 n−1 + 6( ⌊ n
Computing MultiHomogeneous Bézout Numbers is Hard
, 2004
"... The multihomogeneous Bézout number is a bound for the number of solutions of a system of multihomogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multihomogeneous system, one can ask for the optimal multihomogenization that would mini ..."
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The multihomogeneous Bézout number is a bound for the number of solutions of a system of multihomogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multihomogeneous system, one can ask for the optimal multihomogenization that would minimize the Bézout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multihomogeneous Bézout number is actually NPhard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multihomogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multihomogeneous Bézout number up to a fixed factor cannot exist even in a randomized setting, unless BPP ⊇ NP. 1
An Information Geometric Approach to Polynomialtime Interiorpoint Algorithms — Complexity Bound via Curvature Integral —
, 2007
"... In this paper, we study polynomialtime interiorpoint algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for con ..."
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In this paper, we study polynomialtime interiorpoint algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for conic linear programs introduced by selfconcordant barrier functions, and develop a precise iterationcomplexity estimate of the polynomialtime interiorpoint algorithm based on an integral of (embedding) curvature of the central trajectory in a rigorous differential geometrical sense. We further study implication of the theory applied to classical linear programming, and establish a surprising link to the strong “primaldual curvature ” integral bound established by Monteiro and Tsuchiya, which is based on the work of Vavasis and Ye of the layeredstep interiorpoint algorithm. By using these results, we can show that the total embedding curvature of the central trajectory, i.e., the aforementioned integral over the whole central trajectory, is bounded by O(n3.5 log(¯χ ∗ A + n)) where ¯χ ∗ A is a condition number of the coefficient matrix A and n is the number of nonnegative variables. In particular, the integral is bounded by O(n4.5m) for combinatorial linear programs including network flow problems where m is the number of constraints. We also provide a complete differentialgeometric characterization of the primaldual curvature in the primaldual algorithm. Finally, in view of this integral bound, we observe that the primal (or dual) interiorpoint algorithm requires fewer number of iterations than the primaldual interiorpoint algorithm at least in the case of linear programming.
THE CENTRAL CURVE IN LINEAR PROGRAMMING
"... Abstract. The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangeme ..."
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Abstract. The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instancespecific bound on the total curvature of the central path, a quantity relevant for interior point methods. The global geometry of central curves is studied in detail. 1.
Shrinkwrapping trajectories for linear programming, preprint available at optimizationonline.org
 Cornell University
"... Hyperbolic Programming (HP) –minimizing a linear functional over an affine subspace of a finitedimensional real vector space intersected with the socalled hyperbolicity cone – is a class of convex optimization problems that contains wellknown Linear Programming (LP). In particular, for any LP one ..."
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Hyperbolic Programming (HP) –minimizing a linear functional over an affine subspace of a finitedimensional real vector space intersected with the socalled hyperbolicity cone – is a class of convex optimization problems that contains wellknown Linear Programming (LP). In particular, for any LP one can readily provide a sequence of HP relaxations. Based on these hyperbolic relaxations, a new ShrinkWrapping approach to solve LP has been proposed by Renegar. The resulting ShrinkWrapping trajectories, in a sense, generalize the notion of central path in interiorpoint methods. We study the geometry of ShrinkWrapping trajectories for Linear Programming. In particular, we analyze the geometry of these trajectories in the proximity of the socalled central line, and contrast the behavior of these trajectories with that of the central path for some pathological LP instances. In addition, we provide an elementary real proof of convexity of hyperbolicity cones. 1