185 research outputs found
Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
Dynamic evolution for risk-neutral densities
Option price data is often used to infer risk-neutral densities for future
prices of an underlying asset. Given the prices of a set of options on the same
underlying asset with different strikes and maturities, we propose a nonparametric
approach for estimating the evolution of the risk-neutral density in time. Our
method uses bicubic splines in order to achieve the desired smoothness for the
estimation and an optimization model to choose the spline functions that best fit the
price data. Semidefinite programming is employed to guarantee the nonnegativity of
the densities. We illustrate the process using synthetic option price data generated
using log-normal and absolute diffusion processes as well as actual price data for
options on the S&P500 index.
We also used the risk-neutral densities that we computed to price exotic options
and observed that this approach generates prices that closely approximate the
market prices of these options.FCT POCI/MAT/59442/2004, PTDC/MAT/64838/2006
Self-Calibrating Cameras Using Semidefinite Programming
Novel methods are proposed for self-calibrating a purerotating camera using semidefinite programming (SDP). Key to the approach is the use of the positive-definiteness requirement for the dual image of the absolute conic (DIAC). The problem is couched within a convex optimization framework and convergence to the global optimum is guaranteed. Experiments on various data sets indicate that the proposed algorithms more reliably deliver accurate and meaningful results. This work points the way to an alternative and more general approach to self-calibration using the advantageous properties of SDP. Algorithms are also discussed for cameras undergoing general motion
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
A semismooth newton method for the nearest Euclidean distance matrix problem
The Nearest Euclidean distance matrix problem (NEDM) is a fundamentalcomputational problem in applications such asmultidimensional scaling and molecularconformation from nuclear magnetic resonance data in computational chemistry.Especially in the latter application, the problem is often large scale with the number ofatoms ranging from a few hundreds to a few thousands.In this paper, we introduce asemismooth Newton method that solves the dual problem of (NEDM). We prove that themethod is quadratically convergent.We then present an application of the Newton method to NEDM with -weights.We demonstrate the superior performance of the Newton method over existing methodsincluding the latest quadratic semi-definite programming solver.This research also opens a new avenue towards efficient solution methods for the molecularembedding problem
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
An inexact SQP Newton method for convex SC1 minimization problems
Master'sMASTER OF SCIENC
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