1,953 research outputs found
A New Approach to Yakubovich's s-Lemma
Subject to regularity assumptions, Yakubovich's s-Lemma characterizes the quadratic functions f(x) defined on a finite-dimensional space which are copositive with a given quadratic function q(x). This result has far-reaching consequences in optimization and control theory. Several approaches to its proof are known, some of which generalize to Hilbert spaces. In this paper we explore a new geometric approach to the proof of this classical result
The S-Procedure via dual cone calculus
Given a quadratic function h that satisfies a Slater condition, Yakubovich’s S-Procedure (or S-Lemma) gives a characterization of all other quadratic functions that are copositive with in a form that is amenable to numerical computations. In this paper we present a deep-rooted connection between the S-Procedure and the dual cone calculus formula , which holds for closed convex cones in . To establish the link with the S-Procedure, we generalize the dual cone calculus formula to a situation where is nonclosed, nonconvex and nonconic but exhibits sufficient mathematical resemblance to a closed convex one. As a result, we obtain a new proof of the S-Lemma and an extension to Hilbert space kernels
The Nesterov-Todd Direction and its Relation to Weighted Analytic Centers
The subject of this report concerns differential-geometric properties of the Nesterov-Todd search direction for linear optimization over symmetric cones. In particular, we investigate the rescaled asymptotics of the associated flow near the central path. Our results imply that the Nesterov-Todd direction arises as the solution of a Newton system defined in terms of a certain transformation of the primal-dual feasible domain. This transformation has especially appealing properties which generalize the notion of weighted analytic centers for linear programming
Algebraic Tail Decay of Condition Numbers for Random Conic Systems under a General Family of Input Distributions
We consider the conic feasibility problem associated with linear homogeneous systems of inequalities. The complexity of iterative algorithms for solving this problem depends on a condition number. When studying the typical behaviour of algorithms under stochastic input one is therefore naturally led to investigate the fatness of the distribution tails of the random condition number that ensues. We study an unprecedently general class of probability models for the random input matrix and show that the tails decay at algebraic rates with an exponent that naturally emerges when applying a theory of uniform absolute continuity which is also developed in this paper.\ud
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Raphael Hauser was supported through grant NAL/00720/G from the Nuffield Foundation and through grant GR/M30975 from the Engineering and Physical Sciences Research Council of the UK. Tobias Müller was partially supported by EPSRC, the Department of Statistics, Bekker-la-Bastide fonds, Dr Hendrik Muller's Vaderlandsch fonds, and Prins Bernhard Cultuurfonds
The continuous Newton-Raphson method can look ahead
This paper is about an intriguing property of the continuous Newton-Raphson method for the minimization of a continuous objective function f: if x is a point in the domain of attraction of a strict local minimizer x* then the flux line of the Newton-Raphson flow that starts in x approaches x* from a direction that depends only on the behavior of f in arbitrarily small neighborhoods around x and x*. In fact, if F is a sufficiently benign perturbation of f on an open region D not containing x, then the two flux lines through x defined by the Newton-Raphson vector fields that correspond to f and F differ from one another only within D.\ud
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The work was supported by EPSRC grant GR/S34472 (R. Hauser) and by the Clarendon Fund, Oxford University Press and ORS Award, Universities UK (J Nedic
Optimal execution strategy with an uncertain volume target
In the seminal paper on optimal execution of portfolio transactions, Almgren
and Chriss (2001) define the optimal trading strategy to liquidate a fixed
volume of a single security under price uncertainty. Yet there exist
situations, such as in the power market, in which the volume to be traded can
only be estimated and becomes more accurate when approaching a specified
delivery time. During the course of execution, a trader should then constantly
adapt their trading strategy to meet their fluctuating volume target. In this
paper, we develop a model that accounts for volume uncertainty and we show that
a risk-averse trader has benefit in delaying their trades. More precisely, we
argue that the optimal strategy is a trade-off between early and late trades in
order to balance risk associated with both price and volume. By incorporating a
risk term related to the volume to trade, the static optimal strategies
suggested by our model avoid the explosion in the algorithmic complexity
usually associated with dynamic programming solutions, all the while yielding
competitive performance
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
Distribution of Aligned Letter Pairs in Optimal Alignments of Random Sequences
Considering the optimal alignment of two i.i.d. random sequences of length
, we show that when the scoring function is chosen randomly, almost surely
the empirical distribution of aligned letter pairs in all optimal alignments
converges to a unique limiting distribution as tends to infinity. This
result is interesting because it helps understanding the microscopic path
structure of a special type of last passage percolation problem with correlated
weights, an area of long-standing open problems. Characterizing the microscopic
path structure yields furthermore a robust alternative to optimal alignment
scores for testing the relatedness of genetic sequences
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