65,942 research outputs found
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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
On the relationship between bilevel decomposition algorithms and direct interior-point methods
Engineers have been using bilevel decomposition algorithms to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upperlevel problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods
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Developments in linear and integer programming
In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
Interior Point Decoding for Linear Vector Channels
In this paper, a novel decoding algorithm for low-density parity-check (LDPC)
codes based on convex optimization is presented. The decoding algorithm, called
interior point decoding, is designed for linear vector channels. The linear
vector channels include many practically important channels such as inter
symbol interference channels and partial response channels. It is shown that
the maximum likelihood decoding (MLD) rule for a linear vector channel can be
relaxed to a convex optimization problem, which is called a relaxed MLD
problem. The proposed decoding algorithm is based on a numerical optimization
technique so called interior point method with barrier function. Approximate
variations of the gradient descent and the Newton methods are used to solve the
convex optimization problem. In a decoding process of the proposed algorithm, a
search point always lies in the fundamental polytope defined based on a
low-density parity-check matrix. Compared with a convectional joint message
passing decoder, the proposed decoding algorithm achieves better BER
performance with less complexity in the case of partial response channels in
many cases.Comment: 18 pages, 17 figures, The paper has been submitted to IEEE
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