sci.op-research

Hello,

I'm solving an nonlinear+nonconvex optimal control problem as an NLP
(I'm discretizing the system using backward Euler and posing each
equation as a constraint). My discrete time vector is k={0,1,2...,tf}.
I'm solving this system repeatedly with a shrinking horizon at each
stage (to mimic a controller known as MPC).

Problem P(j)
----------------
min z(tf)
s.t.
f(x'(k), x(k), u(k), k) = 0, x(0) = x0
g(x(k), u(k), k) = 0
k={0,1,2...,tf}.

I'm trying to show that Bellman's principle of optimality holds, i.e.
the solution that I obtain by solving a 1-stage problem from 0..tf
must correspond exactly to the solution I obtain by solving the
multistage problem. Unfortunately, my results seem to indicate
otherwise.

One of the implicit stipulations of Bellman's principle of optimality
is the fact that a global solution is obtained at every stage.

My question is: does Bellman's principle hold for:
a) nonconvex problems, where local optima are obtained at every stage.
b) convex problems with nonunique solutions (i.e. nonstrict convex)

Thanks!

David