  This paper generalizes a previously-conceived, continuation-based
optimization technique for scalar objective functions on constraint manifolds
to cases of periodic and quasiperiodic solutions of delay-differential
equations. A Lagrange formalism is used to construct adjoint conditions that
are linear and homogenous in the unknown Lagrange multipliers. As a
consequence, it is shown how critical points on the constraint manifold can be
found through several stages of continuation along a sequence of connected
one-dimensional manifolds of solutions to increasing subsets of the necessary
optimality conditions. Due to the presence of delayed and advanced arguments in
the original and adjoint differential equations, care must be taken to
determine the degree of smoothness of the Lagrange multipliers with respect to
time. Such considerations naturally lead to a formulation in terms of
multi-segment boundary-value problems (BVPs), including the possibility that
the number of segments may change, or that their order may permute, during
continuation. The methodology is illustrated using the software package coco on
periodic orbits of both linear and nonlinear delay-differential equations,
keeping in mind that closed-form solutions are not typically available even in
the linear case. Finally, we demonstrate optimization on a family of
quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with
delay and parametric forcing. The quasiperiodic case is a further original
contribution to the literature on optimization constrained by partial
differential BVPs.
