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load('optmiz);
/*
Now let's try the famous post-office parcel problem: maximize the
volume of a rectangular parallelopiped parcel, subject to the
constraints that the length plus the girth can't exceed 72,
and that the length, width, and depth cannot be negative or
exceed 42. */
vol: l*w*d ;
eqcon: l + 2*w + 2*d - 72 ;
/* However, we may simplify the problem further, saving
additional time, by making the change of variable L=42-S**2, which
automatically satisfies the upper bound on L; and we may solve the
equality constraint for either W or D, then eliminate it from the
objective These changes reduce the problem to two simultaneous
nonlinear equations: */
solve(eqcon,w);
volsub: ev(vol,%);
ev(%, l=42-s**2);
stapoints(%) ;
/* It is almost always worth solving the largest possible subset
of the equality constraints for variables that enter them linearly,
then using this solution to eliminate these variables from the
remaining constraints and from the objective. This is also worth
doing for variables that enter nonlinearly, provided it introduces no
fractional powers in the objective or remaining constraints.
For example, to find the stationary points of X + 3*Y - 6*Z**2,
subject to X**2 + Y**2 + Z**2 = 1, it is well worth eliminating Z.
The change of variable for eliminating the inequality constraint
L <= 42, is equivalent to converting the inequality to an equality by
introducing a squared slack variable, solving for L, then eliminating
L from VOLSUB. From this viewpoint, the "change of variable"
technique is seen to be applicable to a great variety of inequality
constraints, not merely upper and lower bounds as is implied in most
textbooks. Together with applicable equality constraints, it is
generally worth including in the elimination the largest possible
subset of inequality constraints for which the eliminated variables
enter linearly, up to the number of decision variables minus the number
of equality constraints.
Another technique for treating an inequality constraint is to
solve the problem with the constraint assumed active, and also with
the constraint ignored, checking any stationary points for violation
of the constraint in the latter case: */
stapoints(ev(volsub,l=42)) ;
stapoints(volsub) ;
/* The second-order test is inadequate when
constraints have been artificially activated. For example, the one
eigenvalue for the case D = 15/2 above is negative, but this
stationary point is actually a saddle. To see this, we must check the
unactivated gradient at this point to see if it points into or out of
the feasible region: */
ev(grad, d=15/2, l=42);
decslkmults;
/* GRAD is a global varible bound by STAP to the
symbolic expression for the gradient, and DECSLKMULTS is bound to the
variables that the gradient is taken with respect to, in the order
of their components in GRAD. Thus, -405/4 points in, making the
point a saddle.
The report referenced at the beginning of this demonstration
explains how to generalize this test to more than one constraint.
When there is more than one inequality constraint, each feasible
combination must be activated, unless some additional convexity
requirements are satisfied. Nevertheless, this combinatorial
activation technique is probably capable of treating
larger problems than either the change-of-variable or the squared-
slack-variable-with-multiplier techniques of treating inequality
constraints.
We have already seen how STAP finds poles of the objective or gradient
only by accident. The following examples are included as reminders
about other limitations of using calculus to find extrema:*/
stapoints(x, [], x**3-y**2) ;
/* Lagrange multipliers require the constraints to have con-
tinuous tangents, and this is violated for the above example at X=0,
Y=0, where the objective is a minimum. A direct use of elimination
would fail too, because it results in an undefined derivative at the
minimum. In such cases, each piece between discontinuities must be
considered a separate constraint.
The next example has a minimum at X=0, Y=0, where the two
active constraints are parallel, making them linearly dependent.
Such cusps or "wiskers" on the feasible region violate the so-called
"constraint-qualification" requirement; so the extremum is not found:*/
stapoints(x, [-y, y-x**3]) ;
/* The following example doesn't have a strictly feasible point;
so it violates Slater's condition; and the extremum at X=0 is not
found: */
stapoints(x, x**2) ;
/* Finally, it is important to remember that unbounded regions
may have nonstationary or asymptotically stationary points
at infinity, which STAP will not find. Such situations are usually
obvious from qualitative considerations, provided the objective and
constraints are not too complicated and numerous, but the macsyma LIMIT
function can be of help. However, it is important to remember that a
multivariate limit depends upon the way it is taken. This is
illustrated by the following example, where you will have to type
NONZERO; in response to two interrupts: */
peano: (y - c - (a*x)**2)*(y - c - (b*x)**2) ;
limit(peano, y, inf);
limit(peano, x, inf);
radcan(ev(peano, y=c+(a**2+b**2)*x**2/2));
limit(%, x, inf);
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