# Optimization problem

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Optimization_problem
> Markdown URL: https://mediated.wiki/source/Optimization_problem.md
> Source: https://en.wikipedia.org/wiki/Optimization_problem
> Source revision: 1355376974
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Problem of finding the best feasible solution

For broader coverage of this topic, see [Mathematical optimization](/source/Mathematical_optimization).

In [mathematics](/source/Mathematics), [engineering](/source/Engineering), [computer science](/source/Computer_science) and [economics](/source/Economics), an **optimization problem** is the [problem](/source/Computational_problem) of finding the *best* solution from all [feasible solutions](/source/Feasible_solution).

Optimization problems can be divided into two categories, depending on whether the [variables](/source/Variable_(mathematics)) are [continuous](/source/Continuous_variable) or [discrete](/source/Discrete_variable):

- An optimization problem with discrete variables is known as a *[discrete optimization](/source/Discrete_optimization)*, in which an [object](/source/Mathematical_object) such as an [integer](/source/Integer), [permutation](/source/Permutation) or [graph](/source/Graph_(discrete_mathematics)) must be found from a [countable set](/source/Countable_set).

- A problem with continuous variables is known as a *[continuous optimization](/source/Continuous_optimization)*, in which an optimal value from a [continuous function](/source/Continuous_function) must be found. They can include [constrained problems](/source/Constrained_optimization) 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.[1] 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.[2]

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

The *[standard form](/source/Canonical_form)* of a [continuous](/source/Continuity_(mathematics)) optimization problem is[3] minimize x f ( x ) s u b j e c t t o g i ( x ) ≤ 0 , i = 1 , … , m h j ( x ) = 0 , j = 1 , … , p {\displaystyle {\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\\&&&h_{j}(x)=0,\quad j=1,\dots ,p\end{aligned}}} where

- *f* : [ℝ*n*](/source/Euclidean_space) → [ℝ](/source/Real_numbers) is the *[objective function](/source/Objective_function)* to be minimized over the n-variable vector x,

- *gi*(*x*) ≤ 0 are called *inequality [constraints](/source/Constraint_(mathematics))*

- *hj*(*x*) = 0 are called *equality constraints*, and

- *m* ≥ 0 and *p* ≥ 0.

If *m* = *p* = 0, the problem is an unconstrained optimization problem. By convention, the standard form defines a **minimization problem**. A **maximization problem** can be treated by [negating](/source/Additive_inverse) the objective function.

## Combinatorial optimization problem

Main article: [Combinatorial optimization](/source/Combinatorial_optimization)

Formally, a [combinatorial optimization](/source/Combinatorial_optimization) problem A is a quadruple[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] (*I*, *f*, *m*, *g*), where

- I is a [set](/source/Set_(mathematics)) of instances;

- given an instance *x* ∈ *I*, *f*(*x*) is the set of feasible solutions;

- given an instance x and a feasible solution y of x, *m*(*x*, *y*) denotes the [measure](/source/Measure_(mathematics)) of y, which is usually a [positive](/source/Positive_(mathematics)) [real](/source/Real_number).

- g is the goal function, and is either [min](/source/Minimum_(mathematics)) or [max](/source/Maximum_(mathematics)).

The goal is then to find for some instance x an *optimal solution*, that is, a feasible solution y with m ( x , y ) = g { m ( x , y ′ ) : y ′ ∈ f ( x ) } . {\displaystyle m(x,y)=g\left\{m(x,y'):y'\in f(x)\right\}.}

For each combinatorial optimization problem, there is a corresponding [decision problem](/source/Decision_problem) that asks whether there is a feasible solution for some particular measure *m*0. For example, if there is a [graph](/source/Graph_(discrete_mathematics)) G which contains vertices u and v, an optimization problem might be "find a path from u to v that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from u to v that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

In the field of [approximation algorithms](/source/Approximation_algorithm), algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.[4]

## See also

- [Counting problem (complexity)](/source/Counting_problem_(complexity)) – Type of computational problem

- [Design Optimization](/source/Design_Optimization)

- [Ekeland's variational principle](/source/Ekeland's_variational_principle)

- [Function problem](/source/Function_problem) – Type of computational problem

- [Glove problem](/source/Glove_problem) – Optimization problem in operations research

- [Operations research](/source/Operations_research) – Discipline concerning the application of advanced analytical methods

- [Satisficing](/source/Satisficing) – Cognitive heuristic of searching for an acceptable decision − the optimum need not be found, just a "good enough" solution.

- [Search problem](/source/Search_problem) – Class of computational problems

- [Semi-infinite programming](/source/Semi-infinite_programming)

## References

1. **[^](#cite_ref-1)** ["Search Space"](https://courses.cs.washington.edu/courses/cse473/06sp/GeneticAlgDemo/searchs.html). *courses.cs.washington.edu*. Retrieved 2025-05-10.

1. **[^](#cite_ref-2)** ["Search Space - LessWrong"](https://www.lesswrong.com/w/search-space). *www.lesswrong.com*. 2020-09-22. Retrieved 2025-05-10.

1. **[^](#cite_ref-3)** Boyd, Stephen P.; Vandenberghe, Lieven (2004). [*Convex Optimization*](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=143) (pdf). Cambridge University Press. p. 129. [ISBN](/source/ISBN_(identifier)) [978-0-521-83378-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-83378-3).

1. **[^](#cite_ref-Ausiello03_4-0)** Ausiello, Giorgio; et al. (2003), *Complexity and Approximation* (Corrected ed.), Springer, [ISBN](/source/ISBN_(identifier)) [978-3-540-65431-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-65431-5)

## External links

- ["How Traffic Shaping Optimizes Network Bandwidth"](https://www.ipctech.com/how-traffic-shaping-optimizes-network-bandwidth). *IPC*. 12 July 2016. Retrieved 13 February 2017.

v t e Convex analysis and variational analysis Basic concepts Convex combination Convex function Convex set Topics (list) Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex Maps Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative Main results (list) Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu Sets Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope Series Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Duality Dual system Duality gap Strong duality Weak duality Applications and related Convexity in economics

Authority control databases GND

---
Adapted from the Wikipedia article [Optimization problem](https://en.wikipedia.org/wiki/Optimization_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Optimization_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.