Distance Optimization with Equality Constraints

Introduction

A nearest-point problem asks which point on a specified curve or surface is closest to a given point. The constraint describes the allowed locations; the objective measures the distance from the given point.

These problems are natural applications of Lagrange multipliers, but setting them up carefully matters. In particular, minimizing squared distance is simpler than minimizing distance itself, and solving the multiplier equations produces candidates that still need to be checked.

Formulating the Problem

Let \(\mathbf{a}\in\mathbb{R}^n\) be the fixed point, and let \(\mathbf{x}\in\mathbb{R}^n\) be a point on the curve or surface. For a single equality constraint, write the feasible set as \(g(\mathbf{x})=0\). The distance from \(\mathbf{x}\) to \(\mathbf{a}\) is \(\|\mathbf{x}-\mathbf{a}\|\), so the squared distance is

Because the square-root function is strictly increasing on nonnegative numbers, minimizing \(D(\mathbf{x})\) gives exactly the same nearest points as minimizing \(\|\mathbf{x}-\mathbf{a}\|\). Squared distance is also differentiable everywhere, including at \(\mathbf{x}=\mathbf{a}\).

For one equality constraint, the Lagrange equations for an extremum of \(D\) are

Since \(\nabla D(\mathbf{x})=2(\mathbf{x}-\mathbf{a})\), the equations become

These equations apply at a constrained local extremum where the constraint is regular, meaning \(\nabla g(\mathbf{x})\ne\mathbf{0}\). A singular feasible point, where \(\nabla g(\mathbf{x})=\mathbf{0}\), must be checked separately: the regular Lagrange condition does not guarantee that it will be found. Even at regular points, the equations give stationary candidates, not an automatic proof that a candidate is the global nearest point.

Why the Lagrange Condition Appears

At a regular point of the curve or surface, its tangent directions are perpendicular to \(\nabla g\). To see why a nearest point must satisfy the Lagrange condition, take any differentiable feasible path \(\boldsymbol{\gamma}(t)\) through a constrained local minimum \(\mathbf{x}_*=\boldsymbol{\gamma}(0)\). Since the path stays on the constraint, \(g(\boldsymbol{\gamma}(t))=0\), and differentiation at \(t=0\) gives

The squared distance also has a local minimum along this path. The chain rule therefore gives

Thus \(\nabla D(\mathbf{x}_*)\) is perpendicular to every tangent direction. At a regular point of a single constraint, the normal directions are the multiples of \(\nabla g(\mathbf{x}_*)\). Therefore \(\nabla D(\mathbf{x}_*)=\lambda\nabla g(\mathbf{x}_*)\) for some scalar \(\lambda\), as required. Geometrically, at a nearest point, the displacement from the fixed point to the constraint is normal to the constraint.

Worked Example: Nearest Point on a Line

Worked Example: Nearest Point on a Circle

Worked Example: Nearest Point on a Plane

Checking Candidates and Avoiding a Sign Error

The circle example illustrates why solving the Lagrange equations is only part of a nearest-point problem: there can be multiple stationary candidates, and their objective values must be compared. If the feasible set is compact and the objective is continuous, a minimum exists; checking all relevant candidates then identifies it. For a noncompact set, verify that a nearest point exists or provide a direct global argument, as in the line and plane examples.

A frequent error in a line projection is using the wrong sign for the normal displacement. For the line and point in the first example, the residual at \((1,1)\) is \(3(1)+4(1)-10=-3\). Adding \(\frac{-3}{25}(3,4)\) gives \(\left(\frac{16}{25},\frac{13}{25}\right)\), but this point is not on the line:

The correction is to subtract the residual divided by the squared length of the normal, times the normal. Here that gives \((1,1)-\frac{-3}{25}(3,4)=\left(\frac{34}{25},\frac{37}{25}\right)\), which does satisfy the constraint. Substitution into the constraint is a simple but essential check.

Why These Problems Matter

Nearest-point calculations arise whenever a point must be adjusted to satisfy an exact condition while changing it as little as possible. The equality constraint might represent a geometric object or a required relationship among coordinates. The squared-distance objective expresses the size of the adjustment without introducing an unnecessary square root.

The reusable procedure is to define the squared distance, state the equality constraint, solve the Lagrange system at regular feasible points, and compare or otherwise verify the candidates. Keep track of singular feasible points as well, because they are not covered by the regular Lagrange condition.