876 research outputs found
Approximate Convex Optimization by Online Game Playing
Lagrangian relaxation and approximate optimization algorithms have received
much attention in the last two decades. Typically, the running time of these
methods to obtain a approximate solution is proportional to
. Recently, Bienstock and Iyengar, following Nesterov,
gave an algorithm for fractional packing linear programs which runs in
iterations. The latter algorithm requires to solve a
convex quadratic program every iteration - an optimization subroutine which
dominates the theoretical running time.
We give an algorithm for convex programs with strictly convex constraints
which runs in time proportional to . The algorithm does NOT
require to solve any quadratic program, but uses gradient steps and elementary
operations only. Problems which have strictly convex constraints include
maximum entropy frequency estimation, portfolio optimization with loss risk
constraints, and various computational problems in signal processing.
As a side product, we also obtain a simpler version of Bienstock and
Iyengar's result for general linear programming, with similar running time.
We derive these algorithms using a new framework for deriving convex
optimization algorithms from online game playing algorithms, which may be of
independent interest
Linear Convergence of a Frank-Wolfe Type Algorithm over Trace-Norm Balls
We propose a rank- variant of the classical Frank-Wolfe algorithm to solve
convex optimization over a trace-norm ball. Our algorithm replaces the top
singular-vector computation (-SVD) in Frank-Wolfe with a top-
singular-vector computation (-SVD), which can be done by repeatedly applying
-SVD times. Alternatively, our algorithm can be viewed as a rank-
restricted version of projected gradient descent. We show that our algorithm
has a linear convergence rate when the objective function is smooth and
strongly convex, and the optimal solution has rank at most . This improves
the convergence rate and the total time complexity of the Frank-Wolfe method
and its variants.Comment: In NIPS 201
On the solution existence and stability of polynomial optimization problems
This paper introduces and investigates a regularity condition in the
asymptotic sense for the optimization problems whose objective functions are
polynomial. We prove two sufficient conditions for the existence of solutions
for polynomial optimization problems. Further, when the constraint sets are
semi-algebraic, we show results on the stability of the solution map of
polynomial optimization problems. At the end of the paper, we discuss the
genericity of the regularity condition.Comment: The old title: A regularity condition in polynomial optimizatio
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