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Chapter 7 / Missile Trajectory Optimisation
_ _
This condition ensures that the step length chosen results in a new gradient
that is > 0 but < (ε2 := 0.5) with respect to the initial gradient.
WOLFE CONDITION 3
( X ) H ( X ) ≥ − ε ⋅ α ⋅ ( d g )
H •
k
− k+
1
7-17
3
k
k
Equation 7.8-5
This condition ensures that the function value is less than that its initial
value using (ε3 := 0.0001) to prevent cycling. With this value the function
must be reduced by 0.01% of the linear prediction. This condition is
automatically satisfied in Steepest Descent if (ε1 := 1 and ε2 := 0).
7.8.1 Newton - Raphson Method
For Steepest Descent, [B] is replaced by the identity matrix [I]. This is the
simplest of the gradient-based algorithms and is used to initialise many
other more complex algorithms. Minimisation in function space using an
inexact line search is conceptually simple, computationally efficient, robust
away from an optimum solution. Improvement at each step is generally
assured and accommodating control constraints is simple. Detrimentally,
convergence close to an optimal solution is slow and prone to “zigzagging”,
although this should not be a problem given continually changing
boundary conditions.
7.8.2 Conjugate Gradient Method
Conjugate Gradient methods are often selected as an alternative to Newton-
Raphson methods if Hessians are unavailable. The method searches along
“n-1” conjugate directions following a steepest-decent step,
T
( ≠ j ) ∧ ( k , j ) ∈ [ 1(
1 ) n ] ⇒ d ⋅ [ G ] ⋅ d : = 0
k k
j
[ 1 ( 1 ) n − 1 ] ⇒ d : = g + βk
dk
-1
k ∈
⋅
k
k
Equation 7.8-6
Equation 7.8-7
The use of inexact line searches can lead to slow convergence. For nonquadratic
functions convergence takes more than (n) iterations and the
process must be restarted when (k := n), generally using a steepest-decent
step. CONJ_GRADIENT uses an implementation of Shanno’s Conjugate
Gradient method recommended by Vorley [V.4] , a memoryless quasi-Newton
algorithm modified for constraints in conjunction with a Fibonacci search.
This algorithm is used to examine alternative directions when the cost
reduction using the steepest-descent method begins to slow.