eldo_user

Optimization
Eldo Optimizer SQP Algorithm
Role of Tolerances in the Eldo Optimizer SQP Method
This topic describes the role of the parameters TOL_GRAD, TOL_FEAS and TOL_OPT.
For more information on these parameters, see Eldo Optimizer/SQP Parameters in the Eldo
Reference Manual.
Suppose a value of x that is a local minimizer of f(x) in the interval a ≤ x ≤ b is to be obtained:
Minimize: f(x)
Subject to: a ≤ x ≤ b
Or using a SPICE formulation:
.OPTIMIZE
* Minimize Statement
.OBJECTIVE EXTRACT_INFO LABEL=F
+ {$MACRO|FUNCTION}
+ GOAL=MINIMIZE
* Design variable specification
.PARAMOPT X=(X0, A, B)
The optimality conditions OPTIM(x) in the definition of the TOL_GRAD parameter are listed
in Table 13-2:
Table 13-2. Optimality Conditions
OPTIM(x) x
f’(x)
a < x < b
min(f’(x), 0) x = a
max(f’(x),0) x = b
Consider the first case in Figure 13-3. The minimum is the point x = b. The blue vector
represents the derivative f’(x) at point b. The derivative f’(x) is negative, so max(f’(x), 0) = 0. A
zero or very small OPTIM(x) indicates that the point x is an optimum. The number TOL_GRAD
is used in the stopping test of the Eldo optimizer SQP method. The point x is optimal when the
absolute value of OPTIM(x) is less than TOL_GRAD.
Consider the second case in Figure 13-3. Point y is an unconstrained minimizer of f, since it lies
strictly in the interval [a, b]. The optimality condition is then OPTIM(y) = f’(y) which is the
slope of f at point y. Here OPTIM(y) equals zero.
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Eldo® User's Manual, 15.3