Citadel·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Citadel Data Scientist interview with a pretty heavy quant optimization problem. One question, no fluff, straight into the math.

Questions Asked (1)

Given an unconstrained quadratic optimization problem min f(x) = x^T Q x + c^T x over real-valued vectors, where Q is not necessarily symmetric, under what conditions does a finite minimum exist? If it does, characterize the set of minimizers and the optimal value.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

This one took me a second to settle into because Q being non-symmetric felt like a trick at first.

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AI HintsAI Generated

Suggested Approach

First, symmetrize Q by noting that x^T Q x = x^T (Q+Q^T)/2 x, so only the symmetric part matters for the quadratic term. Then analyze convexity and coercivity: a finite minimum exists iff the symmetric part is positive semidefinite and the linear term lies in its range. Characterize the minimizer set as an affine subspace and compute the optimal value using the pseudoinverse.

Pro tip: Emphasize that Q's asymmetry is irrelevant for the quadratic form; always symmetrize first. Also, mention that if the symmetric part is positive semidefinite but singular, the problem may still have a finite minimum if the linear term is in the range, and the solution set is affine.

1. Symmetrize the quadratic form

Replace Q with its symmetric part Q_s = (Q + Q^T)/2, since x^T Q x = x^T Q_s x for all x. This simplifies analysis without changing the problem.

2. Check convexity and coercivity

Examine the eigenvalues of Q_s. If Q_s is positive definite, f is strictly convex and coercive, so a unique finite minimum exists. If Q_s is positive semidefinite, f is convex but not strictly; a finite minimum exists iff the linear term c is in the range of Q_s (i.e., orthogonal to the null space of Q_s). If Q_s has a negative eigenvalue, f is unbounded below.

3. Characterize the set of minimizers

When a finite minimum exists, the minimizers are the solutions to the linear system Q_s x = -c/2 (or equivalently, the gradient condition). The solution set is an affine subspace: x* + null(Q_s), where x* is any particular solution (e.g., obtained via pseudoinverse).

4. Compute the optimal value

The optimal value is f(x*) = - (1/4) c^T Q_s^+ c, where Q_s^+ is the Moore-Penrose pseudoinverse of Q_s. This formula holds when c is in the range of Q_s.

Key Points to Mention

Only the symmetric part of Q matters for the quadratic form; asymmetry does not affect the optimization.
Conditions for finite minimum: Q_s positive semidefinite and c in range(Q_s).
If Q_s positive definite, unique minimizer; if positive semidefinite, minimizer set is affine (x* + null(Q_s)).
Optimal value formula: -1/4 c^T Q_s^+ c.
Connection to linear least squares and normal equations.
Unboundedness if Q_s has a negative eigenvalue or if c has a component outside range(Q_s) when Q_s is PSD.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.