Determine When a Quadratic Has Finite Minimum
Company: Citadel
Role: Data Scientist
Category: Machine Learning
Difficulty: medium
Interview Round: Technical Screen
Consider the unconstrained real-valued optimization problem
\[
\min_{x \in \mathbb{R}^n} f(x) = x^\top Qx + c^\top x,
\]
where \(Q \in \mathbb{R}^{n \times n}\) and \(c \in \mathbb{R}^n\). The matrix \(Q\) is not necessarily symmetric.
Discuss the conditions under which this problem has a finite minimum value. If the minimum is finite, characterize the minimizers and the optimal value.
Overview: This question evaluates understanding of quadratic optimization, linear algebra, and convexity concepts—particularly properties of quadratic forms, matrix symmetry, and conditions for boundedness in real-valued unconstrained minimization—and why these properties matter for existence of minima.