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Let $y \in \mathbb{R}^n$ be an arbitrary vector.

What type of optimization question is

$$f(x) = \dfrac{1}{2}x^TAy$$

where we seek to minimize $x$ over $\mathbb{R}^n$. I know it is not a QP. Is it just a linear program? The fact we have an $A$ matrix makes me uncertain

What are the conditions for convexity?

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    Is $f$ not linear in $x$? – 2017-02-15
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    Yes, it's a linear program – 2017-02-15
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    Note that as any linear transformation can be represented just as multiplication with matrix, than $Ay$ is just some linear transformation of $y$. Or simplier its just another vector. So $f$ is just a scalar product of two vectors. So yes, it is linear programming. For convexity (if I understand you correctly) conditions, you should take $x$ from some subset, but you haven't mentioned anything about it. – 2017-02-15
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    This is a linear problem. Without any constraints, it's a "void" problem as it's infimums is $-\infty$. With some constraints, things can become a bit more interesting. – 2017-02-15

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This is a convex and linear optimization problem. It is solvable if and only if $Ay=0$.