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Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions. Heuristics and metaheuristics make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found.
The goal is then to find for some instance x an optimal solution, that is, a feasible solution y with (,) = {(, ′): ′ ()}. For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m 0 .
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
Local search is an anytime algorithm; it can return a valid solution even if it's interrupted at any time after finding the first valid solution. Local search is typically an approximation or incomplete algorithm because the search may stop even if the current best solution found is not optimal. This can happen even if termination happens ...
TSP is known to be NP-hard so an optimal solution for even a moderate size problem is difficult to solve. Instead, the greedy algorithm can be used to give a good but not optimal solution (it is an approximation to the optimal answer) in a reasonably short amount of time. The greedy algorithm heuristic says to pick whatever is currently the ...
the optimal set is convex; if the objective function is strictly convex, then the problem has at most one optimal point. These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.
In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, tune, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem or a machine learning problem, especially with incomplete or imperfect information or limited computation capacity.