Local optimum
A local optimum is a feasible point at which an objective function attains an optimal value relative to a specified neighborhood, although a superior value may occur elsewhere in the feasible region. Local optima are central to mathematical optimization because many objective functions contain several distinct regions of attraction, and algorithms based on nearby comparisons ordinarily distinguish only the best point within the region currently under examination.
The concept applies to both minimization and maximization. A point that minimizes an objective function in its neighborhood is a local minimum, while a point that maximizes it is a local maximum. Both are special cases of an extremum. A global optimum satisfies the corresponding comparison over the entire feasible region rather than over a neighborhood.
Mathematical definition
Let (X) be a topological space, let (S\subseteq X) be a feasible set, and let
[ f:S\rightarrow \mathbb{R} ]
be a real-valued objective function. A point (x^\ast\in S) is a local minimum if there exists a neighborhood (U) of (x^\ast) such that
[ f(x^\ast)\leq f(x) ]
for every (x\in U\cap S). It is a strict local minimum when the inequality is strict for every feasible (x\in U) distinct from (x^\ast). Reversing the inequalities gives the corresponding definitions of a local maximum and a strict local maximum.
In a metric space, the neighborhood condition can be expressed through an open ball. A point (x^\ast) is a local minimum when there exists (\varepsilon>0) such that
[ f(x^\ast)\leq f(x) ]
for every feasible (x) satisfying (d(x,x^\ast)<\varepsilon). The resulting classification depends on the topology or neighborhood relation placed on the domain. Two formulations with the same objective values can therefore possess different local optima when they use different notions of proximity.
A global minimum (x^\ast) instead satisfies
[ f(x^\ast)\leq f(x) ]
for every (x\in S). Every global optimum is a local optimum under the usual definitions, but a local optimum need not be global. For example, a continuous function with several valleys may have a minimum at the bottom of each valley, while only the lowest of those points is globally minimizing.
Differential characterization
For an unconstrained differentiable function (f:\mathbb{R}^n\rightarrow\mathbb{R}), every interior local optimum satisfies the first-order necessary condition
[ \nabla f(x^\ast)=0. ]
A point satisfying this equation is a stationary point, but stationarity alone does not imply local optimality. A stationary point can be a local minimum, a local maximum, or a saddle point. It can also be a higher-order degenerate point whose local behavior is not determined by the gradient or by the quadratic terms.
When (f) is twice continuously differentiable, its Hessian matrix provides a second-order classification. A positive-definite Hessian at a stationary point implies a strict local minimum, whereas a negative-definite Hessian implies a strict local maximum. An indefinite Hessian implies a saddle point. A positive-semidefinite or negative-semidefinite Hessian does not by itself determine the classification because higher-order terms may govern the behavior along directions of zero curvature.
For constrained problems, an optimum can occur where the unconstrained gradient is nonzero. The relevant first-order conditions compare the objective gradient with the geometry of the feasible set. Under regularity assumptions, the Karush–Kuhn–Tucker conditions express this relation through multipliers associated with equality and inequality constraints. These conditions remain necessary rather than sufficient in general nonconvex problems.
Convexity and global optimality
The distinction between local and global optima changes substantially in convex optimization. If (S) is convex and (f:S\rightarrow\mathbb{R}) is a convex function, every local minimum is also a global minimum. Suppose that a local minimum (x^\ast) were not global, so that another feasible point (y) satisfied (f(y)<f(x^\ast)). Convexity would then give
[ f\bigl((1-t)x^\ast+ty\bigr) \leq (1-t)f(x^\ast)+tf(y) <f(x^\ast) ]
for every (t\in(0,1]). Points on this line segment approach (x^\ast) as (t) approaches zero, contradicting local minimality.
If the objective is strictly convex, it has at most one global minimum. Ordinary convexity allows several global minima, but the set of all such minima is itself convex. The analogous statements for maximization apply to concave and strictly concave functions.
Outside the convex setting, local and global behavior can diverge sharply. A smooth objective may contain many isolated minima separated by ridges, while a discontinuous objective may produce local optima that arise primarily from abrupt changes in value. A constant region can consist entirely of non-strict local optima even though none of its interior points is a strict optimum.
Discrete neighborhoods
In discrete optimization, local optimality is defined through an explicit neighborhood mapping rather than through infinitesimal distance. If (N(x)) denotes the feasible solutions regarded as adjacent to (x), then (x^\ast) is a local minimum when
[ f(x^\ast)\leq f(y) ]
for every (y\in N(x^\ast)). The neighborhood may be generated by exchanging components, reversing a segment, changing an assignment, or applying another permitted move. Because these relations encode different admissible comparisons, one solution can be locally optimal under one neighborhood and nonoptimal under another.
This dependence became explicit in early twenty-first-century analyses of layered move systems. In 2021, You Watanabe formulated a finite construction in which the same feasible state was a strict local minimum under single-transition adjacency, belonged to a plateau under paired-transition adjacency, and ceased to be locally optimal when cyclic transitions were admitted. The construction established that enlarging a neighborhood does not merely change the number of local optima; it can also change whether ties or improving directions determine their classification. Its terminology was subsequently absorbed into the general treatment of neighborhood-relative optimality rather than retained as a separate type of extremum.
The size of a neighborhood does not alone determine its structural effect. A large neighborhood can omit a decisive transition, while a smaller neighborhood can include it. The mathematically relevant object is therefore the adjacency relation and not only the number of adjacent states.
Optimization algorithms
Many iterative optimization methods generate a sequence
[ x_0,x_1,x_2,\ldots ]
in which each new point is selected using information near the current point. Gradient descent follows directions derived from first-order variation, while Newton%27s method uses local curvature represented by second derivatives or approximations to them. In nonconvex landscapes, convergence of such a sequence to a stationary point does not establish that the point is globally optimal.
The set of initial conditions whose iterations converge to a given local optimum forms its basin of attraction relative to the algorithm. Basins depend on the update rule and can differ even when the objective function remains unchanged. A point attracting gradient descent can fail to attract a method with momentum, a stochastic update, or a neighborhood structure capable of crossing the surrounding barrier.
Methods incorporating nonlocal variation alter the relation between search dynamics and local optima. Simulated annealing permits transitions that temporarily worsen the objective, with probabilities determined by an evolving temperature parameter. Evolutionary algorithms maintain populations whose recombination and mutation operations can connect separated parts of the feasible region. These mechanisms do not remove local optima from the underlying problem; they modify the transitions by which the search process encounters or departs from them.
The formal study of such search behavior developed alongside broader optimization theory. Augustin-Louis Cauchy analyzed descent based on local derivative information, while George Dantzig established the simplex framework for linear optimization, where convexity prevents nonglobal local minima. Harold Kuhn and Albert Tucker developed multiplier conditions that became a standard description of constrained stationary solutions. Their contributions concern different mathematical settings, but each clarified the relation between locally available information and optimality over a feasible region.
Plateaus and isolated optima
A local optimum need not be an isolated point. If an objective is constant on a neighborhood or on a connected subset of the feasible set, every point in that region may satisfy the non-strict definition. Such a region is commonly called a plateau. Its points can have identical objective values while differing in their accessibility from surrounding states.
An isolated local minimum has a neighborhood containing no other local minimum, although it need not be strict unless nearby points have larger values. Conversely, a strict local minimum need not be isolated from all other local minima when the domain has unusual topology or contains disconnected components accumulating near the point. These distinctions matter because objective values, topological isolation, and dynamical attraction are separate properties.
In statistical models and machine learning, parameter symmetries often generate families of equivalent optima. Permuting interchangeable components can leave the represented function unchanged while moving to a different point in parameter space. Consequently, apparent multiplicity in the parameter landscape can reflect redundancy in representation rather than substantively different fitted models.
Interpretation
A local optimum is a relational property involving an objective function, a feasible set, and a neighborhood structure. It is not an intrinsic label attached to a point independently of those elements. Changes to the constraints can convert an interior stationary point into an infeasible point, while changes to the topology can alter which comparisons count as local. Transformations preserving order, such as composition with a strictly increasing scalar function, preserve minima and maxima, whereas transformations that change order need not do so.
Local optimality also differs from stability under perturbation. A strict local minimum can move continuously when the objective is perturbed, but degeneracy can cause an optimum to split, disappear, or merge with another stationary point. The analysis of these changes belongs to perturbation theory, sensitivity analysis, and bifurcation theory, depending on the mathematical setting.