53 research outputs found
The necessary and sufficient conditions of copositive tensors
In this paper, it is proved that (strict) copositivity of a symmetric tensor
is equivalent to the fact that every principal sub-tensor of
has no a (non-positive) negative -eigenvalue. The
necessary and sufficient conditions are also given in terms of the
-eigenvalue of the principal sub-tensor of the given tensor. This
presents a method of testing (strict) copositivity of a symmetric tensor by
means of the lower dimensional tensors. Also the equivalent definition of
strictly copositive tensors is given on entire space .Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608
An Alternative Perspective on Copositive and Convex Relaxations of Nonconvex Quadratic Programs
We study convex relaxations of nonconvex quadratic programs. We identify a
family of so-called feasibility preserving convex relaxations, which includes
the well-known copositive and doubly nonnegative relaxations, with the property
that the convex relaxation is feasible if and only if the nonconvex quadratic
program is feasible. We observe that each convex relaxation in this family
implicitly induces a convex underestimator of the objective function on the
feasible region of the quadratic program. This alternative perspective on
convex relaxations enables us to establish several useful properties of the
corresponding convex underestimators. In particular, if the recession cone of
the feasible region of the quadratic program does not contain any directions of
negative curvature, we show that the convex underestimator arising from the
copositive relaxation is precisely the convex envelope of the objective
function of the quadratic program, providing another proof of Burer's
well-known result on the exactness of the copositive relaxation. We also
present an algorithmic recipe for constructing instances of quadratic programs
with a finite optimal value but an unbounded doubly nonnegative relaxation.Comment: 26 page
(Global) Optimization: Historical notes and recent developments
Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning
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