I am facing an optimization problem stated below:

Find the values of $x_i$ where $i = 1, ..., 20$ and $0 \leq x_i \leq 1 $,

to maximize $y$:

$$y = \sum_{i = 1}^{20}\theta(1 - \theta)^{20 - i}(\alpha x_i + \beta(1 - x_i))$$

where $\theta \in (0, 1)$ and $\alpha, \beta \in R$

while making sure that

$$\sum_{i = 1}^{t}(1 - x_i) \leq a(\sum_{i = 1}^tx_i - b)_+$$

for any $t \in [1, 20]$, where $a, b \in R$

Could you advice what optimization algorithm/technique should I look into in order to solve this problem? I don't need a complete solution.

  • $\begingroup$ Have to tried goal seek / solver in Excel? $\endgroup$ – unseen_rider Sep 8 '17 at 15:20
  • $\begingroup$ What does $(\cdot)_+$ mean? $\endgroup$ – Rodrigo de Azevedo Sep 8 '17 at 18:00
  • $\begingroup$ @RodrigodeAzevedo I think $(x)_+$ means $max(x, 0)$. Related post $\endgroup$ – hklel Sep 8 '17 at 18:45

It's a completely standard linear program in the form $\max c^T x$ subject to $Ax \leq b$.

EDIT: I did not see the $()_{+}$ operator. With that, it is not a linear program but a nonconvex problem which at least requires mixed-integer linear programming.

  • $\begingroup$ Thanks! I was intimidated by the complex form and didn't realize it's a LP problem :/ $\endgroup$ – hklel Sep 8 '17 at 18:54
  • $\begingroup$ Well, unfortunately not, as the new edit now says. $\endgroup$ – Johan Löfberg Sep 8 '17 at 18:58

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