3,969 research outputs found
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
S-Lemma with Equality and Its Applications
Let and be two quadratic functions
having symmetric matrices and . The S-lemma with equality asks when the
unsolvability of the system implies the existence of a real
number such that . The
problem is much harder than the inequality version which asserts that, under
Slater condition, is unsolvable if and only if for some . In this paper, we
show that the S-lemma with equality does not hold only when the matrix has
exactly one negative eigenvalue and is a non-constant linear function
(). As an application, we can globally solve as well as the two-sided generalized trust region subproblem
without any condition. Moreover, the
convexity of the joint numerical range where is a (possibly non-convex) quadratic
function and are affine functions can be characterized
using the newly developed S-lemma with equality.Comment: 34 page
A Revisit to Quadratic Programming with One Inequality Quadratic Constraint via Matrix Pencil
The quadratic programming over one inequality quadratic constraint (QP1QC) is
a very special case of quadratically constrained quadratic programming (QCQP)
and attracted much attention since early 1990's. It is now understood that,
under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP).
The optimal solution to (QP1QC), if exists, can be obtained by a matrix rank
one decomposition of the optimal matrix X? to (PSDP). In this paper, we pay a
revisit to (QP1QC) by analyzing the associated matrix pencil of two symmetric
real matrices A and B, the former matrix of which defines the quadratic term of
the objective function whereas the latter for the constraint. We focus on the
\undesired" (QP1QC) problems which are often ignored in typical literature:
either there exists no Slater point, or (QP1QC) is unbounded below, or (QP1QC)
is bounded below but unattainable. Our analysis is conducted with the help of
the matrix pencil, not only for checking whether the undesired cases do happen,
but also for an alternative way to compute the optimal solution in comparison
with the usual SDP/rank-one-decomposition procedure.Comment: 22 pages, 0 figure
Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems
Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input
multiple-output (MIMO) relaying belongs to the class of difference-of-convex
functions (DC) programming problems. DC programming problems occur as well in
other signal processing applications and are typically solved using different
modifications of the branch-and-bound method. This method, however, does not
have any polynomial time complexity guarantees. In this paper, we show that a
class of DC programming problems, to which the sum-rate maximization in two-way
MIMO relaying belongs, can be solved very efficiently in polynomial time, and
develop two algorithms. The objective function of the problem is represented as
a product of quadratic ratios and parameterized so that its convex part (versus
the concave part) contains only one (or two) optimization variables. One of the
algorithms is called POlynomial-Time DC (POTDC) and is based on semi-definite
programming (SDP) relaxation, linearization, and an iterative search over a
single parameter. The other algorithm is called RAte-maximization via
Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors
method and an iterative search over two (or one, in its approximate version)
optimization variables. We also derive an upper-bound for the optimal values of
the corresponding optimization problem and show by simulations that this
upper-bound can be achieved by both algorithms. The proposed methods for
maximizing the sum-rate in the two-way AF MIMO relaying system are shown to be
superior to other state-of-the-art algorithms.Comment: 35 pages, 10 figures, Submitted to the IEEE Trans. Signal Processing
in Nov. 201
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