I have the following function in $p$ variables $y=\begin{bmatrix}y_1 & \dots & y_p \end{bmatrix}$:
$$
f(y) = \sum_{i=1}^p b_i \exp \left(\sum_{j \leq i} y_j \right) – \sum_{i=1} c_i y_i,
$$
where the $b_i$ and $c_i$ are positive constants. I am trying to determine if this function is convex. In my attempt to do so I tried to see if the Hessian matrix is positive definite.
I found the $p \times p$-dimensional Hessian matrix as follows:
$$
H=
\begin{bmatrix}
a_1 + \dots + a_p & a_2 + \dots + a_p & \cdots & a_p \\
a_2 + \dots + a_p & a_2 + \dots + a_p & & \\
\vdots & & \ddots & \vdots \\
a_p & a_p & \dots & a_p
\end{bmatrix},
$$
with $a_i=b_i \exp \left(\sum_{j \leq i} y_j \right)$, such that $a_p \geq a_{p-1} \geq \dots \geq a_1 > 0$. I feel like this matrix is positive definite, but am not sure how to show it. What I have tried so far is to see if $x^T H x > 0$. I have gotten to here so far:
$$
x^T A x = \sum_{i=1}^p \left(x_i^2 \sum_{j \geq i} a_j + x_i \sum_{j > i} x_j \sum_{k \geq j} a_k + x_i \sum_{j < i} x_j \sum_{k \geq i} a_k \right),
$$
but am unsure how to continue from here.
Best Answer
Let $v\in\mathbb{R}^p$ be a fixed vector. For $y\in\mathbb{R}^p$, let $\langle y,v\rangle$ denote the standard inner product. The function $h_v:\mathbb{R}^p\to\mathbb{R}$ defined by
$$h_v(y)=\exp(\langle y,v\rangle )$$ is convex, because $e^t$ is convex. Indeed, if $0\leq\lambda\leq 1$, then for every $y_1,y_2\in\mathbb{R}^p$:
$$ h_v(\lambda y_1+(1-\lambda)y_2)=e^{\lambda\langle y_1,v\rangle+(1-\lambda)\langle y_2,v\rangle}\leq \lambda e^{\langle y_1,v\rangle}+(1-\lambda)e^{\langle y_2,v\rangle}=\lambda h_v(y_1)+(1-\lambda)h_v(y_2) $$ by the convexity of $e^t$. For $1\leq k\leq p$, let $e_k$ denote the standard unit vectors in $\mathbb{R}^p$. Put $$v_k=\sum_{i=1}^ke_i$$ Given positive numbers $b_i$, the functions $b_i h_{v_i}$ are convex for each $i$, and so also their sum. Note that $$F(y)=\sum_{i=1}^pb_i\exp(\sum_{j\leq i}y_j)=\sum_{i=1}^p b_i h_{v_i}(y)$$ So $F$ is a convex function. Observe that the Hessian of $F$ coincides with with the Hessian of the given function $f$. Therefore, the function $f$ is also convex.
It can be shown that if $H_y$ is the Hessian of $f$ (or of $F$) evaluated at some point $y\in\mathbb{R}^p$ then
$$\langle H_y v,v\rangle=\exp(-\sum_{i=1}^py_i)\sum_{j=1}^pb_j\exp(\sum_{2\leq i\leq j}x_i)\left(\sum_{i=1}^jv_i\right)^2$$ which is seen to be positive for $v\neq 0$, but it is much easier to prove that $F$ is convex as above.