4,560 research outputs found
A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization
In this paper we consider a general problem set-up for a wide class of convex
and robust distributed optimization problems in peer-to-peer networks. In this
set-up convex constraint sets are distributed to the network processors who
have to compute the optimizer of a linear cost function subject to the
constraints. We propose a novel fully distributed algorithm, named
cutting-plane consensus, to solve the problem, based on an outer polyhedral
approximation of the constraint sets. Processors running the algorithm compute
and exchange linear approximations of their locally feasible sets.
Independently of the number of processors in the network, each processor stores
only a small number of linear constraints, making the algorithm scalable to
large networks. The cutting-plane consensus algorithm is presented and analyzed
for the general framework. Specifically, we prove that all processors running
the algorithm agree on an optimizer of the global problem, and that the
algorithm is tolerant to node and link failures as long as network connectivity
is preserved. Then, the cutting plane consensus algorithm is specified to three
different classes of distributed optimization problems, namely (i) inequality
constrained problems, (ii) robust optimization problems, and (iii) almost
separable optimization problems with separable objective functions and coupling
constraints. For each one of these problem classes we solve a concrete problem
that can be expressed in that framework and present computational results. That
is, we show how to solve: position estimation in wireless sensor networks, a
distributed robust linear program and, a distributed microgrid control problem.Comment: submitted to IEEE Transactions on Automatic Contro
Model-free bounds for multi-asset options using option-implied information and their exact computation
We consider derivatives written on multiple underlyings in a one-period
financial market, and we are interested in the computation of model-free upper
and lower bounds for their arbitrage-free prices. We work in a completely
realistic setting, in that we only assume the knowledge of traded prices for
other single- and multi-asset derivatives, and even allow for the presence of
bid-ask spread in these prices. We provide a fundamental theorem of asset
pricing for this market model, as well as a superhedging duality result, that
allows to transform the abstract maximization problem over probability measures
into a more tractable minimization problem over vectors, subject to certain
constraints. Then, we recast this problem into a linear semi-infinite
optimization problem, and provide two algorithms for its solution. These
algorithms provide upper and lower bounds for the prices that are
-optimal, as well as a characterization of the optimal pricing
measures. Moreover, these algorithms are efficient and allow the computation of
bounds in high-dimensional scenarios (e.g. when ). Numerical experiments
using synthetic data showcase the efficiency of these algorithms, while they
also allow to understand the reduction of model-risk by including additional
information, in the form of known derivative prices
Discrete element modelling of rock cutting processes interaction with evaluation of tool wear
The document presents a numerical model of rocks and soils using spherical Discrete Elements, also called Distinct Elements. The motion of spherical elements is described by means of equations of rigid body dynamics. Explicit integration in time yields high computational efficiency. Spherical elements interact among one another with contact forces, both in normal and tangential directions. Efficient contact search scheme based on the octree structures has been implemented. Special constitutive model of contact interface taking into account cohesion forces allows us to model fracture and decohesion of materials. Numerical simulation predicts wear of rock cutting tools. The developed numerical algorithm of wear evaluation allows us us to predict evolution of the shape of the tool caused by wear. Results of numerical simulation are validated by comparison with experimental data.Postprint (published version
Convex Optimization: Algorithms and Complexity
This monograph presents the main complexity theorems in convex optimization
and their corresponding algorithms. Starting from the fundamental theory of
black-box optimization, the material progresses towards recent advances in
structural optimization and stochastic optimization. Our presentation of
black-box optimization, strongly influenced by Nesterov's seminal book and
Nemirovski's lecture notes, includes the analysis of cutting plane methods, as
well as (accelerated) gradient descent schemes. We also pay special attention
to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror
descent, and dual averaging) and discuss their relevance in machine learning.
We provide a gentle introduction to structural optimization with FISTA (to
optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror
prox (Nemirovski's alternative to Nesterov's smoothing), and a concise
description of interior point methods. In stochastic optimization we discuss
stochastic gradient descent, mini-batches, random coordinate descent, and
sublinear algorithms. We also briefly touch upon convex relaxation of
combinatorial problems and the use of randomness to round solutions, as well as
random walks based methods.Comment: A previous version of the manuscript was titled "Theory of Convex
Optimization for Machine Learning
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