Nonnegativity Constraint¶

Definition¶

A nonnegativity constraint requires a decision variable to satisfy $x_j\ge0$ $0$ . It excludes negative quantities while permitting zero. For a vector of variables, $\boldsymbol{x}\ge0$ means $x_j\ge0$ for every component $j$ . In the course's augmented simplex models, all original and auxiliary variables are nonnegative.

Interpretation¶

A nonnegativity constraint says that a variable cannot go below zero. Produced or shipped amounts are naturally of this kind, but the model only knows it once $x_j\ge0$ is written down.

The first quadrant shaded, with the region below the line 2x1 + 2.4x2 = 7.2 marked as feasible. The coordinate axes are the nonnegativity constraints; points with a negative coordinate are excluded.

The two axes are the boundaries of $x_1\ge0$ and $x_2\ge0$ $0$ , which together keep the first quadrant. The line $2x_1+2.4x_2=7.2$ $7.2$ , meeting the axes at $(3.6,0)$ and $(0,3)$ , cuts off the darker triangle of feasible points; everything with a negative coordinate is excluded.

With two nonnegative variables $x,y\ge0$ the feasible region lies in the first quadrant. Drag the peak of the objective to the left of the $y$ -axis: the optimum moves onto the axis and the constraint $x\ge0$ binds.

Zero is allowed: corners of a feasible region often lie on an axis, such as $(0,80)$ in the Giapetto region. The constraints matter even when they look harmless: without them, maximizing $x_1$ over the half-plane $2x_1+2.4x_2\le7.2$ would have no optimum, since $x_2$ could be made as negative as wanted. A variable that can be a profit or a loss takes either sign; it is left unrestricted or handled by a transformation. A common slip is to read $x_j\ge0$ as $x_j>0$ $0$ , or as " $x_j$ is a whole number"; it means neither.

Key Results¶

Nonnegativity is a constraint, not an automatic consequence of a variable name.
With two nonnegative variables, feasible points lie in the first quadrant.
Nonnegative continuous variables may still take fractional values.
At an optimum a nonnegativity constraint is either slack ( $x_{j}^{*} >$ $x_j^\ast>0$ $0$ ) or binding ( $x_{j}^{*} =$ $x_j^\ast=0$ $0$ ); in nonlinear problems this is what the Kuhn-Tucker conditions with nonnegativity constraints handle.

Examples¶

$x_s\ge0$ prevents negative soldier production.
$s_1\ge0$ ensures that a slack variable represents unused rather than exceeded capacity.

Prerequisites¶

Sources¶

OR Modelling · Week 1
Math IV: Multivariate Calculus · Week 7

Aliases: non-negativity constraint, sign restriction, nonnegativity restriction, sign constraint

Last updated: 26 October 2026 00:09