6,284 research outputs found
Exact duality in semidefinite programming based on elementary reformulations
In semidefinite programming (SDP), unlike in linear programming, Farkas'
lemma may fail to prove infeasibility. Here we obtain an exact, short
certificate of infeasibility in SDP by an elementary approach: we reformulate
any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only
elementary row operations, and rotations. When (P) is infeasible, the
reformulated system is trivially infeasible. When (P) is feasible, the
reformulated system has strong duality with its Lagrange dual for all objective
functions.
As a corollary, we obtain algorithms to generate the constraints of {\em all}
infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed
rank maximal solution.Comment: To appear, SIAM Journal on Optimizatio
Worst-Case Linear Discriminant Analysis as Scalable Semidefinite Feasibility Problems
In this paper, we propose an efficient semidefinite programming (SDP)
approach to worst-case linear discriminant analysis (WLDA). Compared with the
traditional LDA, WLDA considers the dimensionality reduction problem from the
worst-case viewpoint, which is in general more robust for classification.
However, the original problem of WLDA is non-convex and difficult to optimize.
In this paper, we reformulate the optimization problem of WLDA into a sequence
of semidefinite feasibility problems. To efficiently solve the semidefinite
feasibility problems, we design a new scalable optimization method with
quasi-Newton methods and eigen-decomposition being the core components. The
proposed method is orders of magnitude faster than standard interior-point
based SDP solvers.
Experiments on a variety of classification problems demonstrate that our
approach achieves better performance than standard LDA. Our method is also much
faster and more scalable than standard interior-point SDP solvers based WLDA.
The computational complexity for an SDP with constraints and matrices of
size by is roughly reduced from to
( in our case).Comment: 14 page
Computational Complexity versus Statistical Performance on Sparse Recovery Problems
We show that several classical quantities controlling compressed sensing
performance directly match classical parameters controlling algorithmic
complexity. We first describe linearly convergent restart schemes on
first-order methods solving a broad range of compressed sensing problems, where
sharpness at the optimum controls convergence speed. We show that for sparse
recovery problems, this sharpness can be written as a condition number, given
by the ratio between true signal sparsity and the largest signal size that can
be recovered by the observation matrix. In a similar vein, Renegar's condition
number is a data-driven complexity measure for convex programs, generalizing
classical condition numbers for linear systems. We show that for a broad class
of compressed sensing problems, the worst case value of this algorithmic
complexity measure taken over all signals matches the restricted singular value
of the observation matrix which controls robust recovery performance. Overall,
this means in both cases that, in compressed sensing problems, a single
parameter directly controls both computational complexity and recovery
performance. Numerical experiments illustrate these points using several
classical algorithms.Comment: Final version, to appear in information and Inferenc
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