Preliminary notation: Consider a finite set $\mathcal{Y}\equiv \{1,...,L\}$ and a function $u:\mathcal{Y}\rightarrow \mathbb{R}$. Let $\Delta(\mathcal{Y})$ be the set of probability distributions over $\mathcal{Y}$, that is $$ \Delta(\mathcal{Y})\equiv \{P\in \mathbb{R}^L: P_y\geq 0 \forall y\in \mathcal{Y}, \sum_{y\in \mathcal{Y}}P_y=1\} $$

Consider the set $\mathcal{Y}^*\equiv argmax_{y\in \mathcal{Y}}u(y)$. For each $y\in \mathcal{Y}^*$, take $P\in \Delta(\mathcal{Y})$ such that $P_y=1$. Collect all such $|\mathcal{Y}^*|$ vectors into the set $\mathcal{B}$. For example. let $\mathcal{Y}^*=\{1,2\}$ and $L=3$. Then, $$ \mathcal{B}\equiv \{(1,0,0),(0,1,0)\} $$ Let $co\{\mathcal{B}\}$ denote the convex hull of $\mathcal{B}$.

The linear optimisation problem: I have to solve with Matlab the following linear minimisation problem $$ \min_{P\in co\{\mathcal{B}\}}f'P $$ where $f$ is a known $L\times 1$ vector of parameters.

Question: I want to rewrite this linear minimisation problem in the canonical form requested by solvers (e.g., Matlab). That is, $$ \min_P f'P\\ \text{s.t. } A_{eq}P=b_{eq}\\ A_{ineq}P \leq b_{ineq}\\ $$ Could you help? I can easily accommodate $P_y\geq 0$ $\forall y\in \mathcal{Y}$ and $\sum_{y\in \mathcal{Y}}P_y=1$ into the canonical contraints above, but I'm struggling to accommodate the part defining $\mathcal{B}$.


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