Smoothing / Mixing Variables

Nudge variables toward equality without worsening the objective; the extremum sits at the balanced point.

The idea

Smoothing, also called mixing variables, proves a symmetric inequality by moving the variables toward equality one pair at a time.

Symmetric inequalities are usually tight where all the variables are equal, and the method aims directly at that point. Start from an arbitrary configuration and pick two variables that differ. Replace them with a more balanced pair — most often both set to their average — chosen so that the constraint still holds and the objective does not move in the wrong direction. The objective at the new configuration is at least as good as before, and the variables are strictly closer to being equal.

Repeating the move drives the configuration toward the all-equal point while the objective never worsens. If the extremum is reached this way, it is attained where the variables agree, and the inequality reduces to evaluating the balanced point — usually a one-variable computation. The figure below traces such a run for three variables $a$, $b$, $c$ with their sum held fixed.

The balancing move must be verified, not assumed: it has to preserve the constraint, and it has to move the objective in one consistent direction. A move that can sometimes worsen the objective proves nothing, because then the balanced point need not be extremal.

Ways to work on it

• Walkthrough. Why balancing a pair with the sum fixed moves a symmetric objective, and what each move must guarantee.
• Practice. Evaluate a symmetric objective at its smoothed all-equal point.
• Hardest. Maximize a product under a fixed sum when a cap on one variable blocks full equality.

Not sure where to start? Take the ten-question placement test.