6,759 research outputs found
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
Iterative Regularization for Learning with Convex Loss Functions
We consider the problem of supervised learning with convex loss functions and
propose a new form of iterative regularization based on the subgradient method.
Unlike other regularization approaches, in iterative regularization no
constraint or penalization is considered, and generalization is achieved by
(early) stopping an empirical iteration. We consider a nonparametric setting,
in the framework of reproducing kernel Hilbert spaces, and prove finite sample
bounds on the excess risk under general regularity conditions. Our study
provides a new class of efficient regularized learning algorithms and gives
insights on the interplay between statistics and optimization in machine
learning
The Necessary And Sufficient Condition for Generalized Demixing
Demixing is the problem of identifying multiple structured signals from a
superimposed observation. This work analyzes a general framework, based on
convex optimization, for solving demixing problems. We present a new solution
to determine whether or not a specific convex optimization problem built for
generalized demixing is successful. This solution will also bring about the
possibility to estimate the probability of success by the approximate kinematic
formula
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