Appl Math Optim 39:121–142 (1999)
1999 Springer-Verlag New York Inc.
Some Randomized Algorithms for
Convex Quadratic Programming
Institut f¨ur Angewandte Mathematik und Statistik, Universit¨at W¨urzburg,
Am Hubland, D-97074 W¨urzburg, Germany
Abstract. We adapt some randomized algorithms of Clarkson  for linear pro-
gramming to the framework of so-called LP-type problems, which was introduced
by Sharir and Welzl . This framework is quite general and allows a uniﬁed
and elegant presentation and analysis. We also show that LP-type problems include
minimization of a convex quadratic function subject to convex quadratic constraints
as a special case, for which the algorithms can be implemented efﬁciently, if only
linear constraints are present. We show that the expected running times depend only
linearly on the number of constraints, and illustrate this by some numerical results.
Even though the framework of LP-type problems may appear rather abstract at ﬁrst,
application of the methods considered in this paper to a given problem of that type
is easy and efﬁcient. Moreover, our proofs are in fact rather simple, since many
technical details of more explicit problem representations are handled in a uniform
manner by our approach. In particular, we do not assume boundedness of the feasible
set as required in related methods.
Key Words. Randomized algorithms, Convex programming, Quadratic program-
ming, Linear programming.
AMS Classiﬁcation. Primary 90C20, Secondary 90C05, 90C25.