MATH.9:5.2 - Restrict a unique optimum by exchange symmetry

Let x1,x2 be nonnegative real numbers with x1+x2=10, and minimize J=x1²+x2². Exchanging the two allocations preserves feasibility and cost.

If the minimizer is known to exist uniquely, the exchange must fix it. Hence x1=x2, and the constraint gives the candidate (5,5). For this example, existence, uniqueness and its optimality can also be established directly: every feasible point has the form (5+h,5-h), and

J(5+h,5-h)=50+2h².

The minimum is attained only at h=0. The direct calculation can finish this small problem; the symmetry deduction is reusable when uniqueness has another available justification.

If the same J is maximized on the segment, its maxima are (10,0) and (0,10). The exchange moves one to the other. Its fixed midpoint is the minimum, so the fixed-point equation alone would solve the wrong optimization question.

Now minimize x1²+2x2² with the same resource constraint. Exchange changes the criterion. Substituting x2=10-x1 gives

J=3*(x1-20/3)²+200/3,

so the unique optimum is (20/3,10/3). The old equality x1=x2 no longer follows. MATH.10 supplies the more general admissible-variation construction when the remaining optimization is the difficulty.