Given $ U = \{(p_1,q_1),(p_2,q_2),…., (p_N,q_N)\}$ , where $ 0<p_1,p_2,…, p_N \le 1$ and $ 0<q_1,q_2,…, q_N$ , how to solve the following problem:

$ $ \begin{align} \max_{S\subseteq U }\max_\sigma&\quad \sum_{i=1}^{|S|}p_{\sigma_i}q_{\sigma_i}\Pi_{j=1}^{i-1}(1-p_{\sigma_j})\ s.t. & \quad |S|=K\le N. \end{align}$ $

where $ \sigma$ specifies the order of elements in $ S$ .

Suppose that we know the set of $ S$ , then $ \sigma$ should order the elements by $ q$ . (Otherwise, show the contradiction by switching two elements)

My question is: Suppose that $ (p_n,q_n)$ lies in $ S^*$ . Then when $ q_n$ increases, it will still lie in $ S^*$ , but will $ (p_n,q_n)$ be moved up?