60 research outputs found
Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier
This paper explores a surprising equivalence between two seemingly-distinct
convex optimization methods. We show that simulated annealing, a well-studied
random walk algorithms, is directly equivalent, in a certain sense, to the
central path interior point algorithm for the the entropic universal barrier
function. This connection exhibits several benefits. First, we are able improve
the state of the art time complexity for convex optimization under the
membership oracle model. We improve the analysis of the randomized algorithm of
Kalai and Vempala by utilizing tools developed by Nesterov and Nemirovskii that
underly the central path following interior point algorithm. We are able to
tighten the temperature schedule for simulated annealing which gives an
improved running time, reducing by square root of the dimension in certain
instances. Second, we get an efficient randomized interior point method with an
efficiently computable universal barrier for any convex set described by a
membership oracle. Previously, efficiently computable barriers were known only
for particular convex sets
On the Curvature of the Central Path of Linear Programming Theory
We prove a linear bound on the average total curvature of the central path of
linear programming theory in terms on the number of independent variables of
the primal problem, and independent on the number of constraints.Comment: 24 pages. This is a fully revised version, and the last section of
the paper was rewritten, for clarit
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