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* An optimization problem with discrete variables is known as a ''[[discrete optimization]]'', in which an [[Mathematical object|object]] such as an [[integer]], [[permutation]] or [[Graph (discrete mathematics)|graph]] must be found from a [[countable set]].
* A problem with continuous variables is known as a ''[[continuous optimization]]'', in which an optimal value from a [[continuous function]] must be found. They can include [[Constrained optimization|constrained problem]]s and multimodal problems.
== Search space ==
In the context of an optimization problem, the '''search space''' refers to the set of all possible points or solutions that satisfy the problem's constraints, targets, or goals.<ref>{{Cite web |title=Search Space |url=https://courses.cs.washington.edu/courses/cse473/06sp/GeneticAlgDemo/searchs.html |access-date=2025-05-10 |website=courses.cs.washington.edu}}</ref> These points represent the feasible solutions that can be evaluated to find the optimal solution according to the objective function. The search space is often defined by the ___domain of the function being optimized, encompassing all valid inputs that meet the problem's requirements.<ref>{{Cite web |date=2020-09-22 |title=Search Space - LessWrong |url=https://www.lesswrong.com/w/search-space |access-date=2025-05-10 |website=www.lesswrong.com |language=en}}</ref>
The search space can vary significantly in size and complexity depending on the problem. For example, in a continuous optimization problem, the search space might be a multidimensional real-valued ___domain defined by bounds or constraints. In a discrete optimization problem, such as combinatorial optimization, the search space could consist of a finite set of permutations, combinations, or configurations.
In some contexts, the term ''search space'' may also refer to the optimization of the ___domain itself, such as determining the most appropriate set of variables or parameters to define the problem. Understanding and effectively navigating the search space is crucial for designing efficient algorithms, as it directly influences the computational complexity and the likelihood of finding an optimal solution.
==Continuous optimization problem==
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\end{align}</math>
where
* {{math|''f'' : [[Euclidean space|ℝ<sup>''n''</sup>]] → [[Real numbers|ℝ]]}} is the
* {{math|''g<sub>i</sub>''(''x'') ≤ 0}} are called
* {{math|''h<sub>j</sub>''(''x'') {{=}} 0}} are called
* {{math|''m'' ≥ 0}} and {{math|''p'' ≥ 0}}.
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