I have a question regarding the equivalence between the solutions of two mathematical problems.
Problem 1
Let $A\equiv \{1,...,N\}$ and let $P: A\rightarrow [0,1]$ such that $P(i)\geq 0$ $\forall i \in A$ and $\sum_{i=1}^N P(i)=1$. Let $v\equiv (v_1,...,v_N)\in V\subset \mathbb{R}^N$, where $V$ is finite. Let $\lambda\geq 0$. Let $\mathcal{P}$ be the function space collecting all possible functions $P$.
Consider the following problem:
$$ \text{Find $P\in \mathcal{P}$ such that $\forall i\in A$}\\ \begin{cases} \text{If $P(i)>0$ then $\sum_{v\in V} \frac{ \exp(v_i/\lambda)}{\sum_{j=1}^N P(j) \exp(v_j/\lambda)}=1$}\\ \text{If $P(i)=0$ then $\sum_{v\in V} \frac{ \exp(v_i/\lambda)}{\sum_{j=1}^N P(j) \exp(v_j/\lambda)}\leq 1$}\\ \end{cases} $$
Assume that we have imposed conditions on $V$ and $\lambda$ such that this problem has a unique solution $P^*$.
Problem 2
Consider the solution $P^*$ of problem 1 and suppose there exists $i\in A$ such that $P^*(i)=0$. Let $\tilde{A}\equiv A\setminus \{i\}$.
Let $\tilde{P}: \tilde{A}\rightarrow [0,1]$ such that $\tilde{P}(j)\geq 0$ $\forall j \in \tilde{A}$ and $\sum_{j\neq i}^N \tilde{P}(j)=1$. Let $\tilde{\mathcal{P}}$ be the function space collecting all possible functions $\tilde{P}$.
Consider the following problem:
$$ \text{Find $\tilde{P}\in \tilde{\mathcal{P}}$ such that $\forall j\in \tilde{A}$}\\ \begin{cases} \text{If $\tilde{P}(j)>0$ then $\sum_{v\in V} \frac{ \exp(v_j/\lambda)}{\sum_{k\neq i}^N \tilde{P}(k) \exp(v_k/\lambda)}=1$}\\ \text{If $\tilde{P}(j)=0$ then $\sum_{v\in V} \frac{ \exp(v_j/\lambda)}{\sum_{k\neq i}^N \tilde{P}(k) \exp(v_k/\lambda)}\leq 1$}\\ \end{cases} $$
Again, assume that we have imposed conditions on $V$ and $\lambda$ such that this problem has a unique solution $\tilde{P}^*$.
Question: Is it true that $\tilde{P}^*(j)=P^*(j)$ $\forall j \in \tilde{A}$? Why?
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