Equality constraints

Here, we consider a quadratic optimization problemquadratic optimization problem with equality constraints, where is symmetric and , and . We also assume that has multiple solutions, since the other cases are trivial.

We define as being a matrix whose columns form a basis of .

is convex since it is an affine subspace.

Let . is equivalent to the following unconstrained problem: Proof:
All vectors in are reachable as for some . Define by . Then we get

is convex if and only if is positive semidefinite.

Proof:
The function from the previous proof translates lines from to lines in and applies to them, so is convex on iff is convex, which by the previous theorem is true iff is positive semidefinite.

Let . A point is an optimal solution to if and only if for some such that .

Proof:
Follows directly by also applying this theoremthis theorem.