13,274 research outputs found
Fast Distributed Algorithms for LP-Type Problems of Bounded Dimension
In this paper we present various distributed algorithms for LP-type problems
in the well-known gossip model. LP-type problems include many important classes
of problems such as (integer) linear programming, geometric problems like
smallest enclosing ball and polytope distance, and set problems like hitting
set and set cover. In the gossip model, a node can only push information to or
pull information from nodes chosen uniformly at random. Protocols for the
gossip model are usually very practical due to their fast convergence, their
simplicity, and their stability under stress and disruptions. Our algorithms
are very efficient (logarithmic rounds or better with just polylogarithmic
communication work per node per round) whenever the combinatorial dimension of
the given LP-type problem is constant, even if the size of the given LP-type
problem is polynomially large in the number of nodes
Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory
We present an information-theoretic approach to lower bound the oracle
complexity of nonsmooth black box convex optimization, unifying previous lower
bounding techniques by identifying a combinatorial problem, namely string
guessing, as a single source of hardness. As a measure of complexity we use
distributional oracle complexity, which subsumes randomized oracle complexity
as well as worst-case oracle complexity. We obtain strong lower bounds on
distributional oracle complexity for the box , as well as for the
-ball for (for both low-scale and large-scale regimes),
matching worst-case upper bounds, and hence we close the gap between
distributional complexity, and in particular, randomized complexity, and
worst-case complexity. Furthermore, the bounds remain essentially the same for
high-probability and bounded-error oracle complexity, and even for combination
of the two, i.e., bounded-error high-probability oracle complexity. This
considerably extends the applicability of known bounds
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
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
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