2,170 research outputs found
Stochastic Programming with Probability
In this work we study optimization problems subject to a failure constraint.
This constraint is expressed in terms of a condition that causes failure,
representing a physical or technical breakdown. We formulate the problem in
terms of a probability constraint, where the level of "confidence" is a
modelling parameter and has the interpretation that the probability of failure
should not exceed that level. Application of the stochastic Arrow-Hurwicz
algorithm poses two difficulties: one is structural and arises from the lack of
convexity of the probability constraint, and the other is the estimation of the
gradient of the probability constraint. We develop two gradient estimators with
decreasing bias via a convolution method and a finite difference technique,
respectively, and we provide a full analysis of convergence of the algorithms.
Convergence results are used to tune the parameters of the numerical algorithms
in order to achieve best convergence rates, and numerical results are included
via an example of application in finance
Nonconvex Generalization of ADMM for Nonlinear Equality Constrained Problems
The ever-increasing demand for efficient and distributed optimization
algorithms for large-scale data has led to the growing popularity of the
Alternating Direction Method of Multipliers (ADMM). However, although the use
of ADMM to solve linear equality constrained problems is well understood, we
lacks a generic framework for solving problems with nonlinear equality
constraints, which are common in practical applications (e.g., spherical
constraints). To address this problem, we are proposing a new generic ADMM
framework for handling nonlinear equality constraints, neADMM. After
introducing the generalized problem formulation and the neADMM algorithm, the
convergence properties of neADMM are discussed, along with its sublinear
convergence rate , where is the number of iterations. Next, two
important applications of neADMM are considered and the paper concludes by
describing extensive experiments on several synthetic and real-world datasets
to demonstrate the convergence and effectiveness of neADMM compared to existing
state-of-the-art methods
Super-Linear Convergence of Dual Augmented-Lagrangian Algorithm for Sparsity Regularized Estimation
We analyze the convergence behaviour of a recently proposed algorithm for
regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is
based on a new interpretation of DAL as a proximal minimization algorithm. We
theoretically show under some conditions that DAL converges super-linearly in a
non-asymptotic and global sense. Due to a special modelling of sparse
estimation problems in the context of machine learning, the assumptions we make
are milder and more natural than those made in conventional analysis of
augmented Lagrangian algorithms. In addition, the new interpretation enables us
to generalize DAL to wide varieties of sparse estimation problems. We
experimentally confirm our analysis in a large scale -regularized
logistic regression problem and extensively compare the efficiency of DAL
algorithm to previously proposed algorithms on both synthetic and benchmark
datasets.Comment: 51 pages, 9 figure
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
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