Notations

Definitions

N

total number of variables

M

total amount of resource

i

variable index

j

resource index

X

decision variable vector $x=(x_1,\cdots ,x_N)$

$f_i\left(x_i\right)$

the objective function related to variable $x_i$

$g_i\left(x_i\right)$

the derivative function of $f_i\left(x_i\right)$ , $g_i\left(x_i\right)=\text{d}{f}_i\left(x_i\right)/\text{d}{x}_i$

$k_i\left(x_i\right)$

the derivative function of $g_i\left(x_i\right)$ , $k_i\left(x_i\right)=\text{d}{g}_i\left(x_i\right)/\text{d}{x}_i$

$h_i(\cdot )$

the inverse function of $g_i\left(x_i\right)$ , $h_i(\cdot )=g_i^{-1}(\cdot )$

$c_{i,j}$

coefficient of variable i of resource j

$C_j$

available amount of resource j

$\lambda$

the Lagrange multiplier vector for the resource constraints

$w$

the Lagrange multiplier vector for the variable constraints

$v$

the Lagrange multiplier vector for the variable constraints

$f(\cdot )$

The objective function vector