Identify if optimization problem is convex or non-convex?

Question

I have formulated optimization problem for building, where cost concerns with energy consumption and constraints are related to hardware limits and model of building. To solve this formulation, I need to know if problem is convex or non-convex, to select appropriate tool to solve the same.  (constraints are nonlinear in nature) Thanks a million in advance.

Answer 1

In my understanding (and in a very large generality), an optimization problem $$\min\{f(x), \;\;x\in C\}$$ is convex if $C$ is convex, and $f:C\to\mathbb{R}$ is convex.

As an example, we often give the following framework, where $C$ is described by equality and inequality constraints:

$$\min\{f(x), \;\;\forall 1\leq i \leq n, g_i(x)=0, \;\;\forall 1\leq j\leq m, h_j(x)\leq 0\}$$ (here $g_i:X\to\mathbb{R}$, $h_j:X\to\mathbb{R}$, where $X$ is a Banach space, say).\ To see if this problem satisfies the previous definition of a convex problem, we need to check if $$C:=\{\forall 1\leq i \leq n, g_i(x)=0, \;\;\forall 1\leq j\leq m, h_j(x)\leq 0\}$$ is convex, which is the case if $1\leq i \leq n, g_i$ is AFFINE, and $\forall 1\leq j\leq m, h_j$ is convex (because the lower level set of a convex function is convex, and the intersection of convex sets is convex).

The previous conditions are almost equivalent to the convexity of $C$ (almost because you could imagine that one of the constraint is irrelevant), so in this setting, this is what people usually call a convex problem.

Well, to conclude I just add that a non-convex problem, is just a problem which is not convex with respect to the previous definitions!