13.24
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, t…
Consider a rectangular shipping carton with dimensions x, y, and z. The goal is to minimize the surface area to reduce material costs. At the same time, the carton must meet two conditions: a fixed volume and a shipping size requirement.
In this problem, surface area is the quantity being minimized, while volume and shipping size are the constraints.
Geometrically, each constraint forms a surface in three-dimensional space. Any carton design that meets both constraints must lie on the curve where these two surfaces intersect.
To find the optimal point on this curve, the method of Lagrange multipliers with two constraints is used. This introduces two multipliers, lambda and mu. At this point, the gradient of the surface area is a linear combination of the gradients of the two constraints. This means the gradient of the surface area lies in the plane spanned by the two constraint gradients.
Next, partial derivatives with respect to each variable are computed. These derivative equations, together with the two constraint equations, form a system. Solving this system gives the values of x, y, and z that minimize the surface area while meeting both requirements.
View the full transcript and gain access to JoVE Core videos
Q1: What is the main difference between Lagrange multipliers with one constraint versus two constraints?
With one constraint, the gradient of the objective function is parallel to a single constraint gradient. With two constraints, the gradient of the objective function becomes a linear combination of two constraint gradients, forming a plane of normal directions. This allows optimization along the curve where both constraint surfaces intersect, rather than just on a single surface.
Q2: How do constraint surfaces relate to feasible points in a two-constraint optimization problem?
Each constraint forms a surface in three-dimensional space. Feasible points are those satisfying both constraints simultaneously, and they typically lie along the curve where the two constraint surfaces intersect. The optimization problem is restricted to finding the maximum or minimum only among points on this intersection curve, not across all space.
Q3: Why is the gradient of the objective function a linear combination of the constraint gradients at an optimal point?
At an optimal point, the objective function cannot increase or decrease along the feasible curve, so its gradient must be perpendicular to the curve's tangent direction. The two constraint gradients are perpendicular to their respective surfaces and together define a plane. The objective gradient lies in this plane, making it expressible as a linear combination of the two constraint gradients.
Q4: What role do the multipliers lambda and mu play in solving a two-constraint Lagrange multiplier problem?
Lambda and mu are scalar multipliers that express the objective function's gradient as a linear combination of the two constraint gradients. They appear in the partial derivative equations that form part of the complete system. Solving for these multipliers alongside the variables yields candidate points for the constrained optimum.
Q5: How is the system of equations formed when applying Lagrange multipliers with two constraints?
Partial derivatives are taken with respect to each variable, incorporating the two multipliers. These derivative conditions are combined with the two original constraint equations to form a complete system. Solving this system simultaneously gives the values of all variables and multipliers, identifying candidate points for the constrained maximum or minimum.
Q6: In the shipping carton example, what quantities represent the objective function and the constraints?
The surface area of the carton is the objective function being minimized to reduce material costs. The two constraints are the fixed volume requirement and the shipping size requirement. The optimal carton design must satisfy both constraints while achieving the minimum surface area.
Q7: What steps follow after solving the system of equations in a two-constraint Lagrange multiplier problem?
After solving the system, candidate points are identified. These points must then be checked in the context of the problem to determine which represents the required maximum or minimum. This verification ensures the solution is physically meaningful and satisfies all problem requirements.