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.


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!

| cite | improve this answer | |
  • $\begingroup$ Thanks a ton for your response.. $\endgroup$ – Tejaswinee Darure Jan 26 '16 at 10:13
  • $\begingroup$ To be precise, it is sufficient that $g_i$ is affine. $\endgroup$ – gerw Jan 27 '16 at 10:20
  • $\begingroup$ Thanks @gerw, I made the correction. $\endgroup$ – John Steinbeck Jan 27 '16 at 10:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.