TITLE: Help understand equivalence of two optimization problems
QUESTION [5 upvotes]: The problem is the circled part. I don't fully understand why they are equivalent. In their notation, two vectors ${\bf x} \ge {\bf y}$ means every component of $\bf x$ is larger or equal to the corresponding component of vector $\bf y$.



My attempt and more explanation (might contain error): 
By my understanding, $\bf x$ here should be treated as fixed. The first optimization is varying ${\bf a}_i$, and the second optimization is varying ${\bf p}_i$. 
I think the two optimizations are equivalent in the sense that $\max_{{\bf a}_i} {\bf a}_i^T{\bf x} =\min_{{\bf p}_i} {\bf p}^T_i{\bf d}_i$.
It is easy to see ${\bf D}_i{\bf a}_i \le {\bf d}_i$ is equivalent to ${\bf p}_i^T{\bf D}_i{\bf a}_i \le {\bf p}_i^T{\bf d}_i, \forall {\bf p}_i \ge {\bf 0}$. Now if there exists ${\bf p}_i$ s.t. ${\bf p}_i^T{\bf D}_i = {\bf x}^T$, then we have
$${{\bf{x}}^T}{{\bf{a}}_i} = {\bf{p}}_i^T{{\bf{D}}_i}{{\bf{a}}_i} \leqslant {\bf{p}}_i^T{{\bf{d}}_i},\forall {{\bf{p}}_i} \geqslant {\bf{0}},{{\bf{p}_i}^T}{{\bf{D}}_i} = {\bf{x}}_i^T$$
Thus the maximum value of ${{\bf{x}}^T}{{\bf{a}}_i}$ is at most $\inf \{ {\bf{p}}_i^T{{\bf{d}}_i}:{{\bf{p}}_i} \geqslant {\bf{0}},{{\bf{p}}_i^T}{{\bf{d}}_i} = {\bf{x}}_i^T\} $. Here comes the question: First,

The equivalence in the circled part of the screen shot means 
$$\max \{ {{\bf{x}}^T}{{\bf{a}}_i}:{\bf{p}}_i^T{{\bf{D}}_i}{{\bf{a}}_i} \leqslant {\bf{p}}_i^T{{\bf{d}}_i},\forall {{\bf{p}}_i} \geqslant {\bf{0}}\}  = \inf \{ {\bf{p}}_i^T{{\bf{d}}_i}:{{\bf{p}}_i} \geqslant {\bf{0}},{{\bf{x}}^T}{{\bf{a}}_i} = {\bf{p}}_i^T\} $$
but I can only arrive at "$\le$" rather tahn $=$ from above discussion.
  $$\max \{ {{\bf{x}}^T}{{\bf{a}}_i}:{\bf{p}}_i^T{{\bf{D}}_i}{{\bf{a}}_i} \leqslant {\bf{p}}_i^T{{\bf{d}}_i},\forall {{\bf{p}}_i} \geqslant {\bf{0}}\}  \leqslant \inf \{ {\bf{p}}_i^T{{\bf{d}}_i}:{{\bf{p}}_i} \geqslant {\bf{0}},{{\bf{x}}^T}{{\bf{a}}_i} = {\bf{p}}_i^T\} $$
I am not able to see the full equivalence.

Secondly, 

What happens if there does not exist ${\bf p}_i$ s.t. ${\bf p}_i^T{\bf D}_i = {\bf x}^T$?


Paper: https://faculty.fuqua.duke.edu/~dbbrown/bio/papers/bertsimas_brown_caramanis_11.pdf

REPLY [3 votes]: I am very sure this example 4.2.1 is exactly what you ask for.
This is from the book "convex optimization theory" written by the author of the paper you link in the question.