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I have often heard people say that the set of integers is automatically a non-convex set.

This idea often comes up in Optimization - suppose there is some function you are trying to optimize (e.g. the objective function in the Travelling Salesman Problem). Suppose we know that this function is "convex" because it can be written as a linear function, and all linear functions are convex.(Is linear function convex or concave?)

In the Travelling Salesman Problem, we add "integer constraints" to this problem function (https://en.wikipedia.org/wiki/Travelling_salesman_problem#Miller%E2%80%93Tucker%E2%80%93Zemlin_formulation[23]) . Since we are adding integer constraints to this problem, and the set of integers is said to be non-convex - we are now dealing with the optimization of a convex function over a non-convex set.

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I am told that any time we have integer constraints - this effectively turns the optimization problem into a non-convex problem: Similarly, a Convex Function over a Non-Convex Set is said to automatically become Non-Convex (I don't know if there is a mathematical for this fact, but it sounds logical).

My Question: I am still struggling to understand why any set of integers (this includes integer constraints) are automatically non-convex?

Again, I can understand why this might be on an informal level - in a set of integers, we can not go from the number 2 to any number between 3, except the number 3 itself. That is, we can not access the numbers 2.1, 2.11, 2.111, 2.21, 2.9999 etc. This creates a "break"/"gap" in the set which if we were to visualize would end up looking like a typical non-convex set:

enter image description here

Is my intuition correct about this? Or am I missing something?

Thanks!

stats_noob
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  • The definition of a convex set is that any straight line between two points in the set is also in the set. So your 'informal' understanding is a precise proof. – Calvin Khor Feb 17 '22 at 05:39

2 Answers2

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You should return to the definition of convex sets.

We say set $C$ is a convex set if and only if for every $\lambda \in [0,1]$, $x\in C$, and $y \in C$, we have $$ \lambda x + (1-\lambda) y \in C. $$

For your example we have $C=\mathbb{N}$, then consider $x=1$, $y=2$, and $\lambda=1/2$. In this case $\lambda x + (1-\lambda)y =3/2$ which obviously is not a member of $C$.

MMH
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    Interesting. I didn't known this definition. It's very intuitive and corresponds to geometric convexity – Yuriy S Feb 17 '22 at 05:44
  • @ Mahdi: Thank you so much for your answer! By this definition, does this mean that the "Travelling Salesman Problem" is Non-Convex? Since the classic Simplex Algorithm can only be used for LP problems with Convex Constraints - does this mean that the Simplex Algorithm CAN NOT be used to solve the Travelling Salesman Problem? Thanks! – stats_noob Mar 02 '22 at 19:55
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In discrete optimization, we use the convention that a discrete problem is called convex if the relaxations are convex. Otherwise, any MINLP would be non-convex. For example, a MIQP can be convex or not (determined by whether the Q matrix in the objective is positive-definite). I.e. we believe convex MINLPs, MIQPs and MIQCPs exist.

You can argue that any discrete variable makes the problem non-convex. But that is somewhat non-standard.

Erwin Kalvelagen
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  • @ Erwin: Thank you so much for your answer! By this definition, does this mean that the "Travelling Salesman Problem" is Non-Convex? Since the classic Simplex Algorithm can only be used for LP problems with Convex Constraints - does this mean that the Simplex Algorithm CAN NOT be used to solve the Travelling Salesman Problem? Thanks! – stats_noob Mar 02 '22 at 19:55
  • Convexity is not an issue for linear models. It is only interesting for nonlinear models. But no. TSP (when formulated as a linear MIP) has linear relaxations. – Erwin Kalvelagen Mar 02 '22 at 20:39
  • Thank you! So you mean that TSP can not be solved using simplex? – stats_noob Mar 02 '22 at 21:04
  • Can you please take a look at this question? https://math.stackexchange.com/questions/4392641/can-we-take-the-derivative-of-a-function-over-a-non-convex-set thank you! – stats_noob Mar 02 '22 at 21:04
  • Simplex is for pure continuous LPs. TSP when formulated as a MIP needs Branch & Bound on top of Simplex. (And to add to the confusion, there is also a Simplex method for unconstrained NLP models). – Erwin Kalvelagen Mar 02 '22 at 21:10