8,964 research outputs found
Introducing the sequential linear programming level-set method for topology optimization
The authors would like to thank Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk/). Dr H Alicia Kim acknowledges the support from Engineering and Physical Sciences Research Council, grant number EP/M002322/1Peer reviewedPublisher PD
Bad semidefinite programs: they all look the same
Conic linear programs, among them semidefinite programs, often behave
pathologically: the optimal values of the primal and dual programs may differ,
and may not be attained. We present a novel analysis of these pathological
behaviors. We call a conic linear system {\em badly behaved} if the
value of is finite but the dual program has no
solution with the same value for {\em some} We describe simple and
intuitive geometric characterizations of badly behaved conic linear systems.
Our main motivation is the striking similarity of badly behaved semidefinite
systems in the literature; we characterize such systems by certain {\em
excluded matrices}, which are easy to spot in all published examples.
We show how to transform semidefinite systems into a canonical form, which
allows us to easily verify whether they are badly behaved. We prove several
other structural results about badly behaved semidefinite systems; for example,
we show that they are in in the real number model of computing.
As a byproduct, we prove that all linear maps that act on symmetric matrices
can be brought into a canonical form; this canonical form allows us to easily
check whether the image of the semidefinite cone under the given linear map is
closed.Comment: For some reason, the intended changes between versions 4 and 5 did
not take effect, so versions 4 and 5 are the same. So version 6 is the final
version. The only difference between version 4 and version 6 is that 2 typos
were fixed: in the last displayed formula on page 6, "7" was replaced by "1";
and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by
"A_3 - A_2 - A_1
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
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