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I am working on a problem relating to what is known as the "Good Deal risk measure" for production valuation in incomplete markets. I have created the following primal optimization problem, and I am wondering how it can be converted to its dual formulation. I am mainly curios since this is a quadratic conic program, and so I am unsure how the quadratic constraint is handled.

Note that $N_{(t-1)}$ can be viewed as the nodes in a scenario tree at time step $t-1$, whilst $N_{(t-1)m}$ are the nodes at time $t$ which are descendant of node $m$ at time $t-1$. The $\delta_{tmn}$ term is the probability of reaching node $n$ at time $t$ from node m at time $t-1$.

$$ \begin{aligned} \min _{\beta_{t n}} & \sum_{m \in \mathcal{N}_{t-1}} \sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n} \cdot\left(P_{t n}-C\right) \cdot \mathbf{x}_{t n} \\ \text { s.t. } & \sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n} \cdot f_{t n}^i=f_{(t-1) m}^i \quad i \in \mathcal{I}, m \in \mathcal{N}_{t-1} \\ & \sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n}^2 \leq A_{(t-1) m}^2 \quad m \in \mathcal{N}_{t-1} \\ & \beta_{t n} \geq 0 \quad n \in \mathcal{N}_t \end{aligned} $$

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    $\begingroup$ The Mosek modeling cookbook has a chapter on duality for conic problems. docs.mosek.com/modeling-cookbook/duality.html $\endgroup$ Feb 23 at 18:28
  • $\begingroup$ I have tried to read the literature online, but am struggling to see how the general theory links to exactly my problem. Could you possibly indicate which section in particular that applies to my quadratic constraint? Do you have any suggestions for how it should look? $\endgroup$ Feb 23 at 19:34
  • $\begingroup$ If I am not mistaken, your only nonlinear constraint is $$\sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n}^2 \leq A_{(t-1) m}^2 \quad m \in \mathcal{N}_{t-1}$$ Define the set $K$ of all nonnegative vectors $\beta_t$ satisfying this constraint. This will be a cone and section 3.2.3 of the Mosek modeling cookbook linked above covers how to express this cone. Then your feasible region is in the form $M\beta_t = f, \, \beta_t \in K$ and you can apply the theory of conic duality described there. $\endgroup$ Feb 29 at 19:34

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