Take the following optimisation problem $$ (*) \hspace{1cm} h(u_1,u_2,u_3)\equiv \min_{p_1,p_2,p_3} \sum_{j=1}^3(f(u_j,p_j))^2\\ \text{s.t. } g(u_j,p_j)\leq b_j \text{ }\forall j \in \{1,2,3\}\\ $$ where
- $u_j$, is an $m$-dimensional vector of real numbers, known by the researcher, for each $j\in \{1,2,3\}$.
- $p_j$ is an $n$-dimensional vector of real numbers, unknown by the researcher, for each $j\in \{1,2,3\}$.
- $b_j$ is a $k$-dimensional vector of real numbers, known by the researcher, for each $j\in \{1,2,3\}$.
- $f:(u_j, p_j)\in \mathbb{R}^m\times \mathbb{R}^n\mapsto f(u_j, p_j)\in \mathbb{R}$ is a function, known by the researcher, for each $j\in \{1,2,3\}$.
- $g:(u_j, p_j)\in \mathbb{R}^m\times \mathbb{R}^n\mapsto g(u_j, p_j)\in \mathbb{R}^k$ is a function, known by the researcher, for each $j\in \{1,2,3\}$.
Suppose we decompose problem $(*)$ in three optimisation problems indexed by $j\in \{1,2,3\}$: $$ h_j(u_j)\equiv \min_{p_j} (f(u_j,p_j))^2\\ \text{s.t. } g(u_j,p_j)\leq b_j\\ $$ Question: Assume that problem $(*)$ is feasible. Is it true that $$ h(u_1,u_2,u_3)=\sum_{j=1}^3 h(u_j) $$ ?
