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Chapter 7 / Missile Trajectory Optimisation
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7.9 Univariate Search and Termination
7.9.1 Fibonacci
7.9.2 Armijo
For locating local minima, directed searches and polynomial fitting
techniques are generally superior to methods based on function evaluations
that have a linear rate of convergence. Univariate minimisation for this
application is performed in the presence of constraining hyper-planes
bounding the feasible region. Methods that requiring data beyond a
constraining plane must limit the step length to stay in the feasible region
even if the minimum lies beyond it. The cost function is thus a complex
curve close to a boundary, particularly if penalty functions are used.
FIBONA_LS performs a Fibionacci univariate line search minimising a
convex cost function along a conjugate gradient direction, a combination
that ensures stability of the optimisation process. It uses a fixed number of
function evaluations to locate the minimum given a specified uncertainty
bound - note apriori knowledge of the number of function evaluations is
required. This method gives the optimal interval reduction for a given
number of function evaluations.
ARMIJO_LS performs an Armijo inexact line search guaranteeing a
minimum reduction in the function and hence convergence,
X
k + 1
=
X
k
7-18
−
m −1
( 0.
5 ) ⋅ [ B ] ⋅ g
Equation 7.9-1
(m) must satisfy Wolfe Condition 3, whilst (m-1) does not, starting with a
Newton step when (m) is zero. Bazaraa [B.4] provides a proof of convergence
when this technique is used to search in the steepest-descent direction.
7.10 Algorithm Implementation
Figure 7-2 encapsulates the major elements of the optimisation algorithm.
The cycle is packed into the duration of the next control step before it is
used by the missile autopilot. Any “spare” time is used for cost reduction
and control updates. During line searches the boundary conditions remain
unchanged. The steepest-descent direction, selected for its simplicity with
an Armijo search to determine step length (α), results in the control update,
U
k+
1
: =
U
k
− α ⋅
∂ H
∂ U
k
Equation 7.10-1
As the steepest-decent cost reduction slows, a conjugate gradient search
direction is used to explore cost reductions normal to the steepest-decent