On Squared-Variable Formulations

Understand

We revisit a formulation technique for inequality constrained optimization problems that has been known for decades: the substitution of squared variables for nonnegative variables.

• Using this technique, inequality constraints are converted to equality constraints via the introduction of a squared-slack variable.
• Such formulations have the superficial advantage that inequality constraints can be dispensed with altogether.
• But there are clear disadvantages, not least being that first-order optimal points for the squared-variable reformulation may not correspond to first-order optimal points for the original problem, because the Lagrange multipliers may have the wrong sign.