I have following two optimization problems. The first problem is $$(P1) \quad \max \sum_{i=1}^{K-1} \delta_i \log(1+y_ib_i)\\ \sum_{i=1}^{K-1}\delta_i y_ib_i \leq L \\ 0\leq \delta_i\leq 1 \\y_i \geq 0$$ where $L$ is some constant and $b_i$ are also constant but they are non-increasing with $i$. The second optimization problem is $$(P2) \quad \max \sum_{i=1}^{K-1} \log(1+x_ib_i)\\\sum_{i=1}^{N-1}x_ib_i\leq H\\x_i\geq0$$ where $b_i$'s has same properties as of problem $(P1)$ and $H>L$ and $H$ and $L$ are both positive constants. So my question is can we show that the maximum value achieved in $(P1)$ is smaller than the maximum value achieved in $(P2)$? Any help in this regard will be much appreciated. Thanks in advance.
Can we even solve the first problem?