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. The theorem is primarily used in mathematical economics and optimal control.Let θ ∈ Θ {displaystyle heta in Theta } , and suppose G {displaystyle G} is an open set containing ( A ∩ B ) ( θ ) {displaystyle (Acap B)( heta )} . If A ( θ ) ⊆ G {displaystyle A( heta )subseteq G} , then the result follows immediately. Otherwise, observe that for each x ∈ A ( θ ) ∖ G {displaystyle xin A( heta )setminus G} we have x ∉ B ( θ ) {displaystyle x otin B( heta )} , and since B {displaystyle B} is closed there is a neighborhood U x × V x {displaystyle U_{x} imes V_{x}} of ( θ , x ) {displaystyle ( heta ,x)} in which x ′ ∉ B ( θ ′ ) {displaystyle x' otin B( heta ')} whenever ( θ ′ , x ′ ) ∈ U x × V x {displaystyle ( heta ',x')in U_{x} imes V_{x}} . The collection of sets { G } ∪ { V x : x ∈ A ( θ ) ∖ G } {displaystyle {G}cup {V_{x}:xin A( heta )setminus G}} forms an open cover of the compact set A ( θ ) {displaystyle A( heta )} , which allows us to extract a finite subcover G , V x 1 , … , V x n {displaystyle G,V_{x_{1}},dots ,V_{x_{n}}} . Then whenever θ ∈ U x 1 ∩ ⋯ ∩ U x n {displaystyle heta in U_{x_{1}}cap dots cap U_{x_{n}}} , we have A ( θ ) ⊆ G ∪ V x 1 ∪ ⋯ ∪ V x n {displaystyle A( heta )subseteq Gcup V_{x_{1}}cup dots cup V_{x_{n}}} , and so ( A ∩ B ) ( θ ) ⊆ G {displaystyle (Acap B)( heta )subseteq G} . This completes the proof. ◻ {displaystyle square } Fix θ ∈ Θ {displaystyle heta in Theta } , and let ε > 0 {displaystyle varepsilon >0} be arbitrary. For each x ∈ C ( θ ) {displaystyle xin C( heta )} , there exists a neighborhood U x × V x {displaystyle U_{x} imes V_{x}} of ( θ , x ) {displaystyle ( heta ,x)} such that whenever ( θ ′ , x ′ ) ∈ U x × V x {displaystyle ( heta ',x')in U_{x} imes V_{x}} , we have f ( x ′ , θ ′ ) < f ( x , θ ) + ε {displaystyle f(x', heta ')<f(x, heta )+varepsilon } . The set of neighborhoods { V x : x ∈ C ( θ ) } {displaystyle {V_{x}:xin C( heta )}} covers C ( θ ) {displaystyle C( heta )} , which is compact, so V x 1 , … , V x n {displaystyle V_{x_{1}},dots ,V_{x_{n}}} suffice. Furthermore, since C {displaystyle C} is upper hemicontinuous, there exists a neighborhood U ′ {displaystyle U'} of θ {displaystyle heta } such that whenever θ ′ ∈ U ′ {displaystyle heta 'in U'} it follows that C ( θ ′ ) ⊆ ⋃ k = 1 n V x k {displaystyle C( heta ')subseteq igcup _{k=1}^{n}V_{x_{k}}} . Let U = U ′ ∩ U x 1 ∩ ⋯ ∩ U x n {displaystyle U=U'cap U_{x_{1}}cap dots cap U_{x_{n}}} . Then for all θ ′ ∈ U {displaystyle heta 'in U} , we have f ( x ′ , θ ′ ) < f ( x k , θ ) + ε {displaystyle f(x', heta ')<f(x_{k}, heta )+varepsilon } for each x ′ ∈ C ( θ ′ ) {displaystyle x'in C( heta ')} , as x ′ ∈ V x k {displaystyle x'in V_{x_{k}}} for some k {displaystyle k} . It follows thatFix θ ∈ Θ {displaystyle heta in Theta } , and let ε > 0 {displaystyle varepsilon >0} be arbitrary. By definition of f ∗ {displaystyle f^{*}} , there exists x ∈ C ( θ ) {displaystyle xin C( heta )} such that f ∗ ( θ ) < f ( x , θ ) + ε 2 {displaystyle f^{*}( heta )<f(x, heta )+{frac {varepsilon }{2}}} . Now, since f {displaystyle f} is lower semicontinuous, there exists a neighborhood U 1 × V {displaystyle U_{1} imes V} of ( θ , x ) {displaystyle ( heta ,x)} such that whenever ( θ ′ , x ′ ) ∈ U 1 × V {displaystyle ( heta ',x')in U_{1} imes V} we have f ( x , θ ) < f ( x ′ , θ ′ ) + ε 2 {displaystyle f(x, heta )<f(x', heta ')+{frac {varepsilon }{2}}} . Observe that C ( θ ) ∩ V ≠ ∅ {displaystyle C( heta )cap V eq emptyset } (in particular, x ∈ C ( θ ) ∩ V {displaystyle xin C( heta )cap V} ). Therefore, since C {displaystyle C} is lower hemicontinuous, there exists a neighborhood U 2 {displaystyle U_{2}} such that whenever θ ′ ∈ U 2 {displaystyle heta 'in U_{2}} there exists x ′ ∈ C ( θ ′ ) ∩ V {displaystyle x'in C( heta ')cap V} . Let U = U 1 ∩ U 2 {displaystyle U=U_{1}cap U_{2}} . Then whenever θ ′ ∈ U {displaystyle heta 'in U} there exists x ′ ∈ C ( θ ′ ) ∩ V {displaystyle x'in C( heta ')cap V} , which implies 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. The theorem is primarily used in mathematical economics and optimal control. Maximum Theorem. Let X {displaystyle X} and Θ {displaystyle Theta } be topological spaces, f : X × Θ → R {displaystyle f:X imes Theta o mathbb {R} } be a continuous function on the product X × Θ {displaystyle X imes Theta } , and C : Θ ⇉ X {displaystyle C:Theta ightrightarrows X} be a compact-valued correspondence such that C ( θ ) ≠ ∅ {displaystyle C( heta ) eq emptyset } for all θ ∈ Θ {displaystyle heta in Theta } . Define the marginal function (or value function) f ∗ : Θ → R {displaystyle f^{*}:Theta o mathbb {R} } by and the set of maximizers C ∗ : Θ ⇉ X {displaystyle C^{*}:Theta ightrightarrows X} by If C {displaystyle C} is continuous (i.e. both upper and lower hemicontinuous) at θ {displaystyle heta } , then f ∗ {displaystyle f^{*}} is continuous and C ∗ {displaystyle C^{*}} is upper hemicontinuous with nonempty and compact values. As a consequence, the sup {displaystyle sup } may be replaced by max {displaystyle max } and the a r g sup {displaystyle mathrm {arg} ,sup } by a r g max { extstyle mathrm {arg} ,max } . The theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter. In this case, Θ {displaystyle Theta } is the parameter space, f ( x , θ ) {displaystyle f(x, heta )} is the function to be maximized, and C ( θ ) {displaystyle C( heta )} gives the constraint set that f {displaystyle f} is maximized over. Then, f ∗ ( θ ) {displaystyle f^{*}( heta )} is the maximized value of the function and C ∗ {displaystyle C^{*}} is the set of points that maximize f {displaystyle f} .