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Choquet integral
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<p>A Choquet integral is a <a href="/facts/Subadditive/jfRLRWfX">subadditive</a> or <a href="/facts/Superadditive/hc4kIlu3">superadditive</a> integral created by the French mathematician <a href="/facts/Gustave_Choquet/anvF1pPY">Gustave Choquet</a> in 1953. It was initially used in <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a> and <a href="/facts/Potential_theory/3sXGIcDA">potential theory</a>, but found its way into <a href="/facts/Decision_theory/TbXcQX0v">decision theory</a> in the 1980s, where it is used as a way of measuring the expected <a href="/facts/Utility_theory/wlMDG74L">utility</a> of an uncertain event. It is applied specifically to <a href="/facts/Membership_function_(mathematics)/NlIzCXrR">membership functions</a> and <a href="/facts/Capacity_of_a_set/37hACvFX">capacities</a>. In <a href="/facts/Imprecise_probability/unnG18zY">imprecise probability theory</a>, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone <a href="/facts/Upper_and_lower_probabilities/aVzUcjgR">lower probability</a>, or the upper expectation induced by a 2-alternating <a href="/facts/Upper_and_lower_probabilities/aVzUcjgR">upper probability</a>. </p><p>Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the <a href="/facts/Ellsberg_paradox/9J9oueMS">Ellsberg paradox</a> and the <a href="/facts/Allais_paradox/97shWpJg">Allais paradox</a>. </p><p><a href="/facts/Multiobjective_optimization/1Hb4gjGJ">Multiobjective optimization</a> problems seek <a href="/facts/Pareto_optimal/nMTdVPqy">Pareto optimal</a> solutions, but the Pareto set of such solutions can be extremely large, especially with multiple objectives. To manage this, optimization often focuses on a specific function, such as a <a href="/facts/Weighted_sum/YNJL7o99">weighted sum</a>, which typically results in solutions forming a <a href="/facts/Convex_envelope/D4Id3JDS">convex envelope</a> of the feasible set. However, to capture non-convex solutions, alternative <a href="/facts/Aggregation_problem/ASmljGAp">aggregation</a> operators like the Choquet integral can be used. </p>