181 research outputs found
New approximations for the cone of copositive matrices and its dual
We provide convergent hierarchies for the cone C of copositive matrices and
its dual, the cone of completely positive matrices. In both cases the
corresponding hierarchy consists of nested spectrahedra and provide outer
(resp. inner) approximations for C (resp. for its dual), thus complementing
previous inner (resp. outer) approximations for C (for the dual). In
particular, both inner and outer approximations have a very simple
interpretation. Finally, extension to K-copositivity and K-complete positivity
for a closed convex cone K, is straightforward.Comment: 8
A Dual and Interior Point Approach to Solve Convex Min-Max Problems
Abstract In this paper we propose an interior point method for solving the dual form of minmax type problems The dual variables are updated by means of a scaling supergradient method The boundary of the dual feasible region is avoided by the use of a logarithmic barrier function A major dierence with other interior point methods is the nonsmoothness of the objective functio
Social welfare and profit maximization from revealed preferences
Consider the seller's problem of finding optimal prices for her
(divisible) goods when faced with a set of consumers, given that she can
only observe their purchased bundles at posted prices, i.e., revealed
preferences. We study both social welfare and profit maximization with revealed
preferences. Although social welfare maximization is a seemingly non-convex
optimization problem in prices, we show that (i) it can be reduced to a dual
convex optimization problem in prices, and (ii) the revealed preferences can be
interpreted as supergradients of the concave conjugate of valuation, with which
subgradients of the dual function can be computed. We thereby obtain a simple
subgradient-based algorithm for strongly concave valuations and convex cost,
with query complexity , where is the additive
difference between the social welfare induced by our algorithm and the optimum
social welfare. We also study social welfare maximization under the online
setting, specifically the random permutation model, where consumers arrive
one-by-one in a random order. For the case where consumer valuations can be
arbitrary continuous functions, we propose a price posting mechanism that
achieves an expected social welfare up to an additive factor of
from the maximum social welfare. Finally, for profit maximization (which may be
non-convex in simple cases), we give nearly matching upper and lower bounds on
the query complexity for separable valuations and cost (i.e., each good can be
treated independently)
Approximation of fractional Brownian motion by martingales
We study the problem of optimal approximation of a fractional Brownian motion
by martingales. We prove that there exist a unique martingale closest to
fractional Brownian motion in a specific sense. It shown that this martingale
has a specific form. Numerical results concerning the approximation problem are
given
On a Convex Set with Nondifferentiable Metric Projection
A remarkable example of a nonempty closed convex set in the Euclidean plane
for which the directional derivative of the metric projection mapping fails to
exist was constructed by A. Shapiro. In this paper, we revisit and modify that
construction to obtain a convex set with smooth boundary which possesses the
same property
Immobile indices and CQ-free optimality criteria for linear copositive programming problems
We consider problems of linear copositive programming where feasible sets consist of vectors
for which the quadratic forms induced by the corresponding linear matrix combinations
are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define
immobile indices of its constraints and a normalized immobile index set. We prove that the
normalized immobile index set is either empty or can be represented as a union of a finite
number of convex closed bounded polyhedra. We show that the study of the structure of
this set and the connected properties of the feasible set permits to obtain new optimality
criteria for copositive problems. These criteria do not require the fulfillment of any additional
conditions (constraint qualifications or other). An illustrative example shows that the
optimality conditions formulated in the paper permit to detect the optimality of feasible
solutions for which the known sufficient optimality conditions are not able to do this. We
apply the approach based on the notion of immobile indices to obtain new formulations of
regularized primal and dual problems which are explicit and guarantee strong duality.publishe
A change in the optical polarization associated with a gamma-ray flare in the blazar 3C 279
It is widely accepted that strong and variable radiation detected over all
accessible energy bands in a number of active galaxies arises from a
relativistic, Doppler-boosted jet pointing close to our line of sight. The size
of the emitting zone and the location of this region relative to the central
supermassive black hole are, however, poorly known, with estimates ranging from
light-hours to a light-year or more. Here we report the coincidence of a
gamma-ray flare with a dramatic change of optical polarization angle. This
provides evidence for co-spatiality of optical and gamma-ray emission regions
and indicates a highly ordered jet magnetic field. The results also require a
non-axisymmetric structure of the emission zone, implying a curved trajectory
for the emitting material within the jet, with the dissipation region located
at a considerable distance from the black hole, at about 10^5 gravitational
radii.Comment: Published in Nature issued on 18 February 2010. Corresponding
authors: Masaaki Hayashida and Greg Madejsk
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