 Feasible triangle with a squared-distance objective

Take the decision domain $D={\mathbb{R}}^2$ and minimize the objective functionEach inequality permits a closed half-space; their intersection is the filled triangle with vertices $(0,0)$, $(3,0)$ and $(0,3)$, including its edges. The feasible region is a compact set and a convex set. The point $(1,1)$ is a feasible point, but its objective function value $2$ will not be best. Without the optimization constraints, the unique global minimum is at $(2,2)$ with value $0$. That point is infeasible because $2+2>3$.

The objective function is the squared Euclidean distance from $(2,2)$. Its positive level sets are circles centered there. Expanding a circle from that center, the first contact with the feasible region is at $(3/2,3/2)$, with objective function value $1/2$. Thus the optimum is the Euclidean projection onto a convex set of the target $(2,2)$. Squaring a nonnegative Euclidean distance preserves which point is nearest.^[ref]

Here is a direct certificate, without relying on the picture. Put $s=x+y$ and $d=x-y$. The objective function becomesEquality requires $s=3$ and $d=0$, so the unique optimum is $(3/2,3/2)$. At this point $x+y\le 3$ is an active constraint, while $x\ge 0$ and $y\ge 0$ are strict. An active constraint identifies a tight restriction at a specified point; it alone does not establish optimality. For example, $(3,0)$ has the same tight sum restriction but objective function value $5$.

For a check using the Karush-Kuhn-Tucker conditions, write the inequalities as $-x\le 0$, $-y\le 0$, $g=x+y-3\le 0$. Their Lagrange multipliers at the optimum are $(0,0,1)$. The gradient equation isFeasibility, nonnegative Lagrange multipliers and complementary slackness also hold. With this strictly convex function and affine optimization constraints, the Karush-Kuhn-Tucker conditions certify the unique global minimum.^[ref]