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Determine When a Quadratic Has Finite Minimum

CitadelData Scientist · Technical Screen
MediumData ScientistTechnical ScreenMachine Learning

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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.

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.