This looks clean on paper but there's a lot packed in.
Start by stating the necessary and sufficient conditions for a finite minimum: Q must be positive semidefinite and c must lie in the range of Q (i.e., the linear term must be orthogonal to the null space of Q). Then explain why these conditions ensure the quadratic form is bounded below and the linear term does not cause unbounded decrease along directions in the null space.
Pro tip: Mention that if Q is positive definite, the minimum is unique and finite for any c; if Q is only positive semidefinite, the minimum may be finite but non-unique, and you must check the range condition on c. This shows depth and avoids oversimplification.
Determine whether Q is positive definite, positive semidefinite, or indefinite. This dictates whether the quadratic form is strictly convex, convex, or unbounded below.
If Q is positive semidefinite but not positive definite, the linear term c must be orthogonal to the null space of Q (i.e., c ∈ range(Q)). Otherwise, the objective can be driven to -∞ along null directions.
Combine the findings: Q must be positive semidefinite, and c must lie in the range of Q. These conditions are both necessary and sufficient for a finite minimum.
If Q is positive definite, the minimizer is unique. If Q is only positive semidefinite, the set of minimizers is an affine subspace, and any solution can be found by solving the linear system Qx = -c/2 (or Qx = -c depending on convention).
Relate to optimization algorithms: positive definiteness ensures convergence of gradient descent, while semidefiniteness requires careful handling of null space components.
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