Is this sum of convex and concave functions a convex function?

Question

Samuel Hooper · Accepted Answer

Not in general, of course. For example, if

β

is the zero vector,

A

is the zero matrix, and

B

is the identity matrix, then the function is not convex. So the answer depends on the values of

β, A, B

. The value of

Y

is not relevant, because it appears only in the terms that are linear in

X

and therefore do not affect convexity. Only quadratic terms matter. And for a quadratic form, being convex is equivalent to being positive semidefinite.
Note that

(X β) (X β)^{T}

can be written as

Tr (X C X^{T})

where

V = β \otimes β^{T}

is the Kronecker product (or outer product), which is a positive semidefinite matrix of rank 1. So the matter reduces to the convexity of

X \mapsto Tr (X C X^{T}) + Tr (X^{T} (A - B) X)

Unfortunately these two terms cannot be combined easily: the position of

X^{T}

matters. In practical terms: if

A - B

is positive semidefinite, then you have convexity. If

A - B

is not positive definite, then it's unlikely that the sum is convex.

Is this sum of convex and concave functions a convex function?

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