convex duality principle can then be stated as follows: if the hard constraint and
the objective functionals are all convex, then we have pri = dual. (Here we do
not need the technical condition, provided we interpret the min and max as an
in mum and supremum, respectively, and use the convention that the minimum of
an infeasible problem is .)
1
For the case when there are no convexity assumptions on the hard constraint
or the objectives, then we still have pri
dual, but strict inequality can hold, in
which case the di erence is called the duality gap for the problem.
NOTES AND REFERENCES
143
Notes and References
See the Notes and References for chapters 13 and 14 for general books covering the notion
of convexity.
Closed-Loop Convex Specifications
The idea of a closed-loop formulation of controller design specications has a long history
see section 16.3. In the early work, however, convexity is not mentioned explicitly.
The explicit observation that many design specications are closed-loop convex can be
found in Salcudean's thesis
], Boyd et al.
], Polak and Salcudean
],
Sal86
BBB88
PS89
and Boyd, Barratt, and Norman
].
BBN90
Optimization
Many of the Notes and References from chapter 3 consider in detail the special case
of convex specications and functionals, and so are relevant. See also the Notes and
References from chapters 13 and 14.
Duality
The fact that the nonnegatively weighted-sum objectives yield all Pareto optimal specica-
tions is shown in detail in Da Cunha and Polak
], and in chapter 6 of Clarke
].
CP67
Cla83
The results on convex duality are standard, and can be found in complete detail in, e.g.,
Barbu and Precupanu
] or Luc
].
BP78
Luc89
A Stochastic Interpretation of Closed-Loop Convex Functionals
Jensen's inequality states that if is a convex functional,
, and stoch is any
H
2
H
H
zero-mean -valued random variable, then
H
( )
( + stoch)
(6.13)
H
E
H
H
i.e., zero-mean random uctuations in a transfer matrix increase, on average, the value of
a convex functional. In fact, Jensen's inequality characterizes convex functionals: if (6.13)
holds for all (deterministic) and all zero-mean stoch, then is convex.
H
H
144