Formal definition of optimisation problems

When looking at these problems, we can formalise them as such:

• The solution space, $S=\{\text{set of all possible solutions}\}$
• A cost function, $h:S\to \mathbb{Q}$ where $h(\text{solution})=\text{value of solution}$
  • We define a global maximum, $h(s_{\text{global}})\ge h(s)\text{ for all }s\in S$
  • Likewise a local maximum, $h(s_{\text{local}})\ge h(s)\text{ for all ’neighbouring’ }s\in S$