Minimizing difference and individual variables in convex problem

Question

Let's say I have the following optimization problem:

$$ \begin{align*} \min_{\mathbf{x},\mathbf{y}} & \sum_i x_i-y_i \\ \mathrm{s.t.} & \{\mathbf{x},\mathbf{y}\} \in C(\mathbf{x},\mathbf{y}), \end{align*} $$ where $\mathbf{x}$ and $\mathbf{y}$ are vectors, $x_i$ and $y_i$ are the $i^{\mathrm{th}}$ elements of these vectors, and $C(\mathbf{x},\mathbf{y})$ is a well-defined convex set in these variables.

I want to minimize this difference $\sum_i \left(x_i - y_i\right)$, but also want to make sure that $\mathbf{x}$ is also the smallest it can be element-wise. In other words, if there are many $\{\mathbf{x},\mathbf{y}\}$ that minimize the difference and that are feasible, I'd like to obtain with the element-wise smallest $\mathbf{x}$.

I thought about this, and it would seem to me that this is naturally true, but I'm not positive. Any insights on this? If not, how can I guarantee this property?

I do not understand the notation ${x,y} \in C(x,y)$. What are $a,b$? And what does $x-y$ mean? (Are x,y reals?) — copper.hat, Feb 25 '14 at 04:00
I'm sorry, the $a$ and $b$ were typos from the first time I typed the problem. They should be $x$ and $y$, as they appear in the current question. By $C(x,y)$, I mean a general set, such as $C(x,y) := {x,y \ |\ x\le 4, y\le 3}$, for instance. — user1496984, Feb 25 '14 at 04:43
What do you mean by minimising the vector $x-y$? Is it minimising some norm like Euclidean distance? In that case the minimum seems zero and you pick the "smallest" $x$ in whatever way you define small for the vector. — Macavity, Feb 25 '14 at 12:57
@Macavity Oops, sorry again. I tried to simplify the notation as much as possible, but screwed up in the process. The current one should make more sense. Any thoughts now? — user1496984, Feb 25 '14 at 23:09
@AnonSubmitter85 A solution that minimizes $\sum_i x_i$ would do, I suppose. Effectively, I need each of the elements of $\mathbf{x}$ to be as small as possible, so Pareto optimality would be best, but that may be an overkill at this point. — user1496984, Feb 26 '14 at 01:28
Perhaps an e.g might help. In 2D, let C be the unit circle. Then your current formulation suggests for optimum we should choose $x =(-\frac1{\sqrt2}, -\frac1{\sqrt2})$ and $y=(\frac1{\sqrt2}, \frac1{\sqrt2})$ so that the objective is minimised at $-2\sqrt2$ and $x$ is "smallest". Is this what you intended? — Macavity, Feb 26 '14 at 01:36

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

If you know that $\sigma = \min f(\mathbf{x})$ subject to $\mathbf{x}\in C$and you want to want to minimize $g(\mathbf{x})$ for all such $\mathbf{x}$, I think all you have to do is

$$ \min_{\mathbf{x}} g(\mathbf{x}) \;\;\; \mathrm{s.t.} \;\;\; f(\mathbf{x})=\sigma \wedge \mathbf{x} \in C. $$

So, if you can solve the original problem and get the minimum value of the function, you can then use that to minimize $g(\mathbf{x})$ over all possible solutions by solving the second optimization problem above.

There is also a somewhat related post here.

Minimizing difference and individual variables in convex problem

1 Answers1