Is is possible to formulate the maximization of the sum of gamma cumulative distributions as a disciplined quasiconvex problem (DQCP) in CVXPY?
I'm trying to solve the following problem:
Given $F(x) = \int_0^x t^{\alpha-1}exp(-t) dt$ (proportional to the CDF of the Gamma distribution with parameters $\alpha \in \mathbb{N}$ and $\beta = 1$), and a set of non-negative parameters $w \in \mathbb{R_+}^{n \times m}$
$$ \max_{x \in \mathbb{R}^n, y \in \mathbb{R}^{n \times m}} \sum_{i=1}^{n} F(x_i) $$
s.t. $$ \sum_{j=1}^m w_{ij} y_{ij} = x_i \quad \forall i \in 1 \dots n $$ $$ \sum_{i=1}^n y_{ij} = 1 \quad \forall j \in 1 \dots m $$ $$ x \ge 0 $$ $$ y_{ij} \ge 0 \quad \forall i \in 1 \dots n, j \in 1 \dots m $$
As the sum of the integrals of positive functions, $\sum F$ is increasing over its domain and therefore quasiconcave. Is there a way to express this within the DQCP ruleset for a given $\alpha$?
For example, if $\alpha = 2$, we have $\sum F(x_i) = 2 - \sum e^{-x_i} (x_i^2 + 2x_i+2) $.
Using cp.multiply does not return an expression that is known to be quasiconcave.
import cvxpy as cp
import numpy as np
# Problem data.
np.random.seed(1)
n = 10
m = 5
w = np.random.rand(m,n)
# Variables
x = cp.Variable(n, nonneg=True)
y = cp.Variable((m,n), nonneg=True)
# Constraints
constraints = [x == cp.sum(cp.multiply(w,y), axis=0)]
constraints += [cp.sum(y, axis=1) == 1]
# Test
poly = cp.square(x) + 2*x + 2
cp.multiply(cp.exp(-x), p).is_quasiconcave() # Returns false
```