256 research outputs found
On simultaneous diagonalization via congruence of real symmetric matrices
Simultaneous diagonalization via congruence (SDC) for more than two symmetric
matrices has been a long standing problem. So far, the best attempt either
relies on the existence of a semidefinite matrix pencil or casts on the complex
field. The problem now is resolved without any assumption. We first propose
necessary and sufficient conditions for SDC in case that at least one of the
matrices is nonsingular. Otherwise, we show that the singular matrices can be
decomposed into diagonal blocks such that the SDC of given matrices becomes
equivalently the SDC of the sub-matrices. Most importantly, the sub-matrices
now contain at least one nonsingular matrix. Applications to simplify some
difficult optimization problems with the presence of SDC are mentioned
An SDP Approach For Solving Quadratic Fractional Programming Problems
This paper considers a fractional programming problem (P) which minimizes a
ratio of quadratic functions subject to a two-sided quadratic constraint. As is
well-known, the fractional objective function can be replaced by a parametric
family of quadratic functions, which makes (P) highly related to, but more
difficult than a single quadratic programming problem subject to a similar
constraint set. The task is to find the optimal parameter and then
look for the optimal solution if is attained. Contrasted with the
classical Dinkelbach method that iterates over the parameter, we propose a
suitable constraint qualification under which a new version of the S-lemma with
an equality can be proved so as to compute directly via an exact
SDP relaxation. When the constraint set of (P) is degenerated to become an
one-sided inequality, the same SDP approach can be applied to solve (P) {\it
without any condition}. We observe that the difference between a two-sided
problem and an one-sided problem lies in the fact that the S-lemma with an
equality does not have a natural Slater point to hold, which makes the former
essentially more difficult than the latter. This work does not, either, assume
the existence of a positive-definite linear combination of the quadratic terms
(also known as the dual Slater condition, or a positive-definite matrix
pencil), our result thus provides a novel extension to the so-called "hard
case" of the generalized trust region subproblem subject to the upper and the
lower level set of a quadratic function.Comment: 26 page
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