1,170 research outputs found
A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure
Yuan's theorem of the alternative is an important theoretical tool in
optimization, which provides a checkable certificate for the infeasibility of a
strict inequality system involving two homogeneous quadratic functions. In this
paper, we provide a tractable extension of Yuan's theorem of the alternative to
the symmetric tensor setting. As an application, we establish that the optimal
value of a class of nonconvex polynomial optimization problems with suitable
sign structure (or more explicitly, with essentially non-positive coefficients)
can be computed by a related convex conic programming problem, and the optimal
solution of these nonconvex polynomial optimization problems can be recovered
from the corresponding solution of the convex conic programming problem.
Moreover, we obtain that this class of nonconvex polynomial optimization
problems enjoy exact sum-of-squares relaxation, and so, can be solved via a
single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW
preprint, 22 page
A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs
We consider the multilinear polytope defined as the convex hull of the set of
binary points satisfying a collection of multilinear equations. The complexity
of the facial structure of the multilinear polytope is closely related to the
acyclicity degree of the underlying hypergraph. We obtain a polynomial-size
extended formulation for the multilinear polytope of beta-acyclic hypergraphs,
hence characterizing the acyclic hypergraphs for which such a formulation can
be constructed
Approximate matrix and tensor diagonalization by unitary transformations: convergence of Jacobi-type algorithms
We propose a gradient-based Jacobi algorithm for a class of maximization
problems on the unitary group, with a focus on approximate diagonalization of
complex matrices and tensors by unitary transformations. We provide weak
convergence results, and prove local linear convergence of this algorithm.The
convergence results also apply to the case of real-valued tensors
Multilinear Superhedging of Lookback Options
In a pathbreaking paper, Cover and Ordentlich (1998) solved a max-min
portfolio game between a trader (who picks an entire trading algorithm,
) and "nature," who picks the matrix of gross-returns of all
stocks in all periods. Their (zero-sum) game has the payoff kernel
, where is the trader's final wealth and
is the final wealth that would have accrued to a deposit into the best
constant-rebalanced portfolio (or fixed-fraction betting scheme) determined in
hindsight. The resulting "universal portfolio" compounds its money at the same
asymptotic rate as the best rebalancing rule in hindsight, thereby beating the
market asymptotically under extremely general conditions. Smitten with this
(1998) result, the present paper solves the most general tractable version of
Cover and Ordentlich's (1998) max-min game. This obtains for performance
benchmarks (read: derivatives) that are separately convex and homogeneous in
each period's gross-return vector. For completely arbitrary (even
non-measurable) performance benchmarks, we show how the axiom of choice can be
used to "find" an exact maximin strategy for the trader.Comment: 41 pages, 3 figure
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