294 research outputs found
Self-scaled barriers for irreducible symmetric cones
Self-scaled barrier functions are fundamental objects in the theory of
interior-point methods for linear optimization over symmetric cones, of which
linear and semidefinite programming are special cases. We are classifying all
self-scaled barriers over irreducible symmetric cones and show that these
functions are merely homothetic transformations of the universal barrier
function. Together with a decomposition theorem for self-scaled barriers this
concludes the algebraic classification theory of these functions. After
introducing the reader to the concepts relevant to the problem and tracing the
history of the subject, we start by deriving our result from first principles
in the important special case of semidefinite programming. We then generalise
these arguments to irreducible symmetric cones by invoking results from the
theory of Euclidean Jordan algebras.Comment: 12 page
Self-scaled barrier functions on symmetric cones and their classification
Self-scaled barrier functions on self-scaled cones were introduced through a
set of axioms in 1994 by Y.E. Nesterov and M.J. Todd as a tool for the
construction of long-step interior point algorithms. This paper provides firm
foundation for these objects by exhibiting their symmetry properties, their
intimate ties with the symmetry groups of their domains of definition, and
subsequently their decomposition into irreducible parts and algebraic
classification theory. In a first part we recall the characterisation of the
family of self-scaled cones as the set of symmetric cones and develop a
primal-dual symmetric viewpoint on self-scaled barriers, results that were
first discovered by the second author. We then show in a short, simple proof
that any pointed, convex cone decomposes into a direct sum of irreducible
components in a unique way, a result which can also be of independent interest.
We then show that any self-scaled barrier function decomposes in an essentially
unique way into a direct sum of self-scaled barriers defined on the irreducible
components of the underlying symmetric cone. Finally, we present a complete
algebraic classification of self-scaled barrier functions using the
correspondence between symmetric cones and Euclidean Jordan algebras.Comment: 17 page
Interior-point algorithms for convex optimization based on primal-dual metrics
We propose and analyse primal-dual interior-point algorithms for convex
optimization problems in conic form. The families of algorithms we analyse are
so-called short-step algorithms and they match the current best iteration
complexity bounds for primal-dual symmetric interior-point algorithm of
Nesterov and Todd, for symmetric cone programming problems with given
self-scaled barriers. Our results apply to any self-concordant barrier for any
convex cone. We also prove that certain specializations of our algorithms to
hyperbolic cone programming problems (which lie strictly between symmetric cone
programming and general convex optimization problems in terms of generality)
can take advantage of the favourable special structure of hyperbolic barriers.
We make new connections to Riemannian geometry, integrals over operator spaces,
Gaussian quadrature, and strengthen the connection of our algorithms to
quasi-Newton updates and hence first-order methods in general.Comment: 36 page
Solving symmetric matrix word equations via symmetric space machinery
It has recently been proved by Hillar and Johnson [C.J. Hillar, C.R. Johnson, Symmetric word equations in two positive definite letters, Proc. Amer. Math. Soc. 132 (2004) 945-953] that every symmetric word equation in positive definite matrices has a positive definite solution (existence theorem). In this paper we partially solve the uniqueness conjecture via the symmetric space and non-positive curvature machinery existing in the open convex cone of positive definite matrices. Unique positive definite solutions are obtained in terms of geometric and weighted means and as fixed points of explicit strict contractions on the cone of positive definite matrices. © 2005 Elsevier Inc. All rights reserved
An Algorithm for Nonsymmetric Conic Optimization Inspired by MOSEK
We analyze the scaling matrix, search direction, and neighborhood used in
MOSEK's algorithm for nonsymmetric conic optimization [Dahl and Andersen,
2019]. It is proven that these can be used to compute a near-optimal solution
to the homogeneous self-dual model in polynomial time.Comment: 29 page
Full Newton Step Interior Point Method for Linear Complementarity Problem Over Symmetric Cones
In this thesis, we present a new Feasible Interior-Point Method (IPM) for Linear Complementarity Problem (LPC) over Symmetric Cones. The advantage of this method lies in that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. By suitable choice of parameters we prove the global convergence of iterates which always stay in the the central path neighborhood. A global convergence of the method is proved and an upper bound for the number of iterations necessary to find ε-approximate solution of the problem is presented
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Real Algebraic Geometry With a View Toward Moment Problems and Optimization
Continuing the tradition initiated in MFO workshop held in 2014, the aim of this workshop was to foster the interaction between real algebraic geometry, operator theory, optimization, and algorithms for systems control. A particular emphasis was given to moment problems through an interesting dialogue between researchers working on these problems in finite and infinite dimensional settings, from which emerged new challenges and interdisciplinary applications
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