Maximum theorem

The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters.The statement was first proven by Claude Berge in 1959.[1] The theorem is primarily used in mathematical economics and optimal control.is continuous (i.e. both upper and lower hemicontinuous) atis upper-hemicontinuous with nonempty and compact values.The maximum theorem can be used for minimization by considering the functionThe theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter.The result is that if the elements of an optimization problem are sufficiently continuous, then some, but not all, of that continuity is preserved in the solutions.Throughout this proof we will use the term neighborhood to refer to an open set containing a particular point.We preface with a preliminary lemma, which is a general fact in the calculus of correspondences.is upper hemicontinuous and compact-valued, andforms an open cover of the compact set, which allows us to extract a finite subcoverBy upper hemicontinuity, there is a neighborhoodis upper hemicontinuous, nonempty and compact-valued, thenis upper hemicontinuous, there exists a neighborhoodis lower semicontinuous, there exists a neighborhoodis lower hemicontinuous, there exists a neighborhoodUnder the hypotheses of the Maximum theorem,is an upper hemicontinuous correspondence with compact values.is nonempty, observe that the functionThe Extreme Value theorem implies thata closed subset of the compact set, the preliminary Lemma implies thatA natural generalization from the above results gives sufficient local conditions forto be nonempty, compact-valued, and upper semi-continuous.is single-valued, and thus is a continuous function rather than a correspondence.[15] It is also possible to generalize Berge's theorem to non-compact correspondences if the objective function is K-inf-compact.[16] Consider a utility maximization problem where a consumer makes a choice from their budget set.Translating from the notation above to the standard consumer theory notation, Then, Proofs in general equilibrium theory often apply the Brouwer or Kakutani fixed-point theorems to the consumer's demand, which require compactness and continuity, and the maximum theorem provides the sufficient conditions to do so.
continuityoptimizedClaude Bergemathematical economicsoptimal controlproductcorrespondencehemicontinuousopen setExtreme Value theoremquasiconcaveconcaveconvexcorrespondencesutility maximization problemutility functionbudget setindirect utility functionMarshallian demandgeneral equilibrium theoryBrouwerKakutani fixed-point theoremsEnvelope theoremBrouwer fixed point theoremKakutani fixed point theorem for correspondencesMichael selection theoremKim C. BorderFeinberg, Eugene A.Convex analysisvariational analysisConvex combinationConvex functionConvex setTopics (list)Choquet theoryConvex geometryConvex metric spaceConvex optimizationDualityLagrange multiplierLegendre transformationLocally convex topological vector spaceSimplexConvex conjugateClosedLogarithmicallyProperPseudo-Quasi-Invex functionSemi-continuitySubderivativeCarathéodory's theoremEkeland's variational principleFenchel–Moreau theoremFenchel-Young inequalityJensen's inequalityHermite–Hadamard inequalityKrein–Milman theoremMazur's lemmaShapley–Folkman lemmaUrsescuConvex hullOrthogonallyEffective domainEpigraphHypographJohn ellipsoidRadial setAlgebraic interiorZonotopeDual systemDuality gapStrong dualityWeak dualityConvexity in economics