Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.

For a function of three variables, each constraint forms a surface in three-dimensional space. Feasible points are those that satisfy both constraints simultaneously. These points usually lie along the curve where the two constraint surfaces intersect. Since optimization is limited to this curve, the goal is not to find the maximum or minimum over all space, but only among the points that meet both conditions.

At an optimal point, the objective function cannot increase or decrease along the feasible curve. The gradient of the objective function is therefore perpendicular to the tangent direction of the curve. The gradients of the two constraint functions are also perpendicular to their respective constraint surfaces. Together, these two constraint gradients define a plane of normal directions. For this reason, the gradient of the objective function can be expressed as a linear combination of the two constraint gradients.

To apply the method, two multipliers, commonly denoted by lambda and mu, are introduced. Partial derivatives are then taken with respect to each variable. These derivative conditions are combined with the two original constraint equations to form a complete system. Solving this system gives the candidate points for the constrained optimum. These points are then checked in the context of the problem to identify the required maximum or minimum.