Ⓑ Optimization

Let \(\Theta\) and \(X\) be two nonempty metric spaces, \(\Gamma\colon \Theta \rightrightarrows X\) a continuous and compact-valued correspondence, and \(f\colon X\times \Theta \to \mathbb{R}\) a real-valued function that is continuous with respect to the product metric \(d((x_1, \theta_1),(x_2, \theta_2)):= d_{X}(x_1, x_2) + d_{\Theta}(\theta_1, \theta_2)\).

Now, consider the optimization problem \(\max_{x\in \Gamma(\theta)} f(x, \theta).\) Define the set-valued solution function \(\sigma\colon \Theta \to 2^{X}\) and the value function \(F\colon \Theta \to \mathbb{R} \cup \{+\infty\}\) as follows:

Then, the following are true:

1. \(\sigma(\theta)\) is an upper hemicontinuous and compact-valued correspondence.

2. \(F(\theta)\) is a continuous real-valued function.