# How to express this objective function in CVX

The objective function is:

maximize \prod_{t=1}^T(1+p_th_t)^{c_t}

Here 0\le p \le 3 are continuos variables and c_t\in \{0,1\} are binary integer variables.

How can I express this in CVX?

Can I use geo_mean function?

The FAQ) applies here. First of all, products are rarely convex. You might be able to reformulate this using geo_mean instead.

That said, I have a sneaking suspicion that a reformulation to true MIDCP form is possible if you replace that objective function with a geo_mean. I’ll think about it.

@mcg Thank you very much. In fact, I obtained this objective reformulating my original objection function which is povided in the edited version of my post.

Your original objective is closer to convex. The right way to reformulate sum_log is with geo_mean. But there’s still the matter of the c_t's. But note that if c_t\in\{0,1\},

$$(1+p_th_t)^{c_t} = 1+c_tp_th_t$$

This isn’t true for all c_t, of course, just for binary c_t. So let’s take advantage of this. For each t, create a new variable, and add the constraints

$$q_t \leq 1 + h_t \min{p_t,3 c_t}$$

So I think what you can do is this:

\begin{array}{ll}
\text{maximize} & \left( \textstyle \prod_t q_t \right)^{1/T} \
\text{subject to} & q_t \leq 1 + h_t \min{ p_t, 3 c_t } \
& 0 \leq p_t \leq 3, ~ t=1,2,\dots,T \
& c_t\in{0,1} ~ t=1,2,\dots T \
& \text{other constraints}
\end{array}

In CVX you would have cvx_begin variables p(T) q(T) variable c(T) binary maximize(geo_mean(q)) q <= 1 + h .* min( p, 3 * c ) ... cvx_end You'll have to convince yourself this is equivalent.

Yes, that is not accepted. You will have to do a similar transformation the one I offer below.

Using MCG’s approach the constraints become

q_t\le1+h_t{\rm{min}}(p_t,3c_t), \forall t

\prod\limits_{t=1}^Nq_t\le Z which is equivalent to

 geo_mean(q)<=(Z)^{1/N}


Am I doing something wrong?

@mcg, please comment on my answer… Also, which solver I should be using for MIDCP, both Sedumi and SDTP3 do not support integer variables, I do not have access to MOSEK…

Then you cannot solve this problem with CVX.

Yes. READ THE FAQ.) The fact that you’re still trying to use prod instead of geo_mean tells me you’re just not understanding how CVX is used, and I am reluctant to help further until you do.

@mcg Thanks, ofcourse, I need to express the product in terms of geo-metric mean. I just wanted you to confirm the transformation…

Your edit looks good. But of course, without MOSEK or Gurobi you won’t be able to solve it…

@mcg, Thanks for your comments. I tried with MOSEK, it just fails to solve my problem. When I use Sedumi instead with variable c(T) not binary, it can solve the problem but that I do not want…