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load('optmiz);
/* This macsyma batch file illustrates how to use the function
STAP for determining the stationary points of a function of 1 or more
variables, either unconstrained or subject to equality &/or inequality
constraints. As a prerequisite, see the text file OPTMIZ USAGE. The
examples here are chosen to be instructive, geometrically visualizable,
and easy enough to be solved by hand. For a more thorough discussion
and results of some more demanding test cases, see the report
"Automatic analytical optimization using computer algebraic
manipulation", by David R. Stoutemyer, (ALOHA project technical report,
University of Hawaii, Honolulu, June 1974).
Here is an example with a maximum: */
stapoints(5*log(y) - 3*x**2*y - 4*y) ;
/* Here is an example with a saddle, which also illustrates
that we may use subscripted variables: */
/* this example is broken
stapoints(atan(x[1]) + atanh(x[2]) + x[2]/x[1]) ; */
/* Here is a famous example by Peano, for which the second-
order test reveals only that the stationary point is not a maximum.
It turns out that the stationary point is a saddle, unless A=B, in
which case it is a minimum, along with all other points on the curve
Y = C + (B*X)**2. See "Theory of Maxima and Minima" by H. Hancock,
(Dover Press) for a discussion of how to analytically determine the
nature of a stationary point when the 2nd-order test is inconclusive.
This example also illustrates how we may explicitly indicate the
decision variables. */
peano: (y - c - (a*x)**2)*(y - c - (b*x)**2) ;
stapoints(peano, [], [], [x,y]) ;
/* Note how the answer is independent of A and B, but not C.
Now, note how when A=B, giving a non-isolated stationary point, an
extra parameter is automatically introduced to describe the stationary
curve: */
stapoints(ev(peano,a=b), [], [], [x,y]) ;
/* The limitations of STAP are primarily dependent upon the
limitations of the macsyma SOLVE command for solving nonlinear
equations. SOLVE will attempt to find a closed-form solution to a
single equation that is irrational or involves elementary
transcendental functions; but as of June 1974, SOLVE is intended
to solve simultaneous equations only if they are polynomial
in the unknowns. However, it generally converts rational equations
to this form by multiplying each equation by the least common
denominator of its terms; so it may solve simultaneous rational
equations too. Thus, as in our first two examples, the objective may
contain any terms with rational gradients, such as ATAN(R), ATANH(R),
or LOG(R), where R is rational in the decision variables. The clearing
of the denominator may result in a "solution" which is a pole for some
gradient components while a zero for the others. Although not a true
solution to the equations, this is a bonus in our case, because we are
interested in all extrema, not just stationary points. In fact, we
are usually more interested in any poles of the proper sign than in
stationary points of the proper type with finite objective values.
Poles, if found, are usually revealed in an alarming way by one or
more large-magnitude components in GRADSUB or OBJSUB, or by an error
interrupt such as a zerodivide.
A surprising number of optimization problems can be converted to the
required form by a change of variable For example, fractional powers
of a decision variable, X, may be eliminated by letting X = Z**Q,
where Q is the least common denominator of the fractional powers of x.
For example: */
frac: x**(2/3) - 2*x*y + y ;
ev(frac, x=z**3);
stapoints(%) ;
/* A similar technique may be used to eliminate fractional powers
of more complicated subexpressions, provided we include appropriate
supplementary equality constraints. For example: */
sqrt(x+y)*y - x ;
stapoints(ev(%,sqrt(x+y)=z), [], z**2-x-y) ;
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