11,981 research outputs found
On the local stability of semidefinite relaxations
We consider a parametric family of quadratically constrained quadratic
programs (QCQP) and their associated semidefinite programming (SDP)
relaxations. Given a nominal value of the parameter at which the SDP relaxation
is exact, we study conditions (and quantitative bounds) under which the
relaxation will continue to be exact as the parameter moves in a neighborhood
around the nominal value. Our framework captures a wide array of statistical
estimation problems including tensor principal component analysis, rotation
synchronization, orthogonal Procrustes, camera triangulation and resectioning,
essential matrix estimation, system identification, and approximate GCD. Our
results can also be used to analyze the stability of SOS relaxations of general
polynomial optimization problems.Comment: 23 pages, 3 figure
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
Regularization and Kernelization of the Maximin Correlation Approach
Robust classification becomes challenging when each class consists of
multiple subclasses. Examples include multi-font optical character recognition
and automated protein function prediction. In correlation-based
nearest-neighbor classification, the maximin correlation approach (MCA)
provides the worst-case optimal solution by minimizing the maximum
misclassification risk through an iterative procedure. Despite the optimality,
the original MCA has drawbacks that have limited its wide applicability in
practice. That is, the MCA tends to be sensitive to outliers, cannot
effectively handle nonlinearities in datasets, and suffers from having high
computational complexity. To address these limitations, we propose an improved
solution, named regularized maximin correlation approach (R-MCA). We first
reformulate MCA as a quadratically constrained linear programming (QCLP)
problem, incorporate regularization by introducing slack variables in the
primal problem of the QCLP, and derive the corresponding Lagrangian dual. The
dual formulation enables us to apply the kernel trick to R-MCA so that it can
better handle nonlinearities. Our experimental results demonstrate that the
regularization and kernelization make the proposed R-MCA more robust and
accurate for various classification tasks than the original MCA. Furthermore,
when the data size or dimensionality grows, R-MCA runs substantially faster by
solving either the primal or dual (whichever has a smaller variable dimension)
of the QCLP.Comment: Submitted to IEEE Acces
Regularizing Portfolio Optimization
The optimization of large portfolios displays an inherent instability to
estimation error. This poses a fundamental problem, because solutions that are
not stable under sample fluctuations may look optimal for a given sample, but
are, in effect, very far from optimal with respect to the average risk. In this
paper, we approach the problem from the point of view of statistical learning
theory. The occurrence of the instability is intimately related to over-fitting
which can be avoided using known regularization methods. We show how
regularized portfolio optimization with the expected shortfall as a risk
measure is related to support vector regression. The budget constraint dictates
a modification. We present the resulting optimization problem and discuss the
solution. The L2 norm of the weight vector is used as a regularizer, which
corresponds to a diversification "pressure". This means that diversification,
besides counteracting downward fluctuations in some assets by upward
fluctuations in others, is also crucial because it improves the stability of
the solution. The approach we provide here allows for the simultaneous
treatment of optimization and diversification in one framework that enables the
investor to trade-off between the two, depending on the size of the available
data set
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