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 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?

How to write it in CVX acceptable/equivalent form?

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…

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…