I am trying to find the dual of a LP. So far I am stuck, can anyone help me please ?.
$$ \max_{x} \sum_{i=0}^{K-1} a_ix_i\\ \text{subject to} \sum_{i=0}^{t-1} x_i \leq t\;, t = 1\dots K\\ x_i \geq 0 $$
So far, I do not find a way to represent the first constraint in a vector / matrix form. If someone can point me a way to represent the first constraint in vectot/matrix form, it will be helpful.
Let $a$ denote the column-vector $a = (a_0,a_1,\dots,a_{K-1})$, and let $x$ denote the column-vector $x = (x_1,x_2,\dots,x_{K-1})$. Take $$ A = \pmatrix{ 1&0&\cdots & 0\\ 1&1 & \ddots & \vdots\\ \vdots & & \ddots & 0\\ 1 & & \cdots & 1 }, \qquad b = \pmatrix{1 \\ 2 \\ \vdots \\ K} $$ We can then restate the LP as $$ \text{maximize (over $x$) } \quad a^Tx\\ \text{subject to } \quad Ax \leq b, \quad x \geq 0 $$