Paris.

Optimization. Philippe Bich (Paris 1 Panthéon-Sorbonne and PSE) Paris, 2016.

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Last time Last time we have seen: exercises about topology, adherence, interior, continuity, intermediary value theorem, ... left-continuity, right continuity, upper semicontinuity, lower semicontinuity. Bolzano theorem (generalization of intermediary theorem). Hyperplane, half-plane Level set, interpretation of maximization or minimization problem with Level sets. Differentiable function from R to R, Taylor development. Differential of a function from Rn to R

Chapter 4: Differential calculus: reminders

Section 2: partial derivative, C k . Differential Dx f of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative Dx f (u) (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives denoted Dx f

∂k f (x) ∂xik

of f : C ⊂ Rn → R at x ∈ int(C),

f : C ⊂ Rn → R is C 0 , C 1 , ...C k if and only if....

Chapter 4: Differential calculus: reminders

Section 2: partial derivative, C k . Differential Dx f of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative Dx f (u) (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives denoted Dx f

∂k f (x) ∂xik

of f : C ⊂ Rn → R at x ∈ int(C),

f : C ⊂ Rn → R is C 0 , C 1 , ...C k if and only if....

Chapter 4: Differential calculus: reminders

Section 2: partial derivative, C k . Differential Dx f of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative Dx f (u) (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives denoted Dx f

∂k f (x) ∂xik

of f : C ⊂ Rn → R at x ∈ int(C),

f : C ⊂ Rn → R is C 0 , C 1 , ...C k if and only if....

Chapter 4: Differential calculus: reminders

Section 2: partial derivative, C k . Differential Dx f of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative Dx f (u) (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives denoted Dx f

∂k f (x) ∂xik

of f : C ⊂ Rn → R at x ∈ int(C),

f : C ⊂ Rn → R is C 0 , C 1 , ...C k if and only if....

Chapter 4: Differential calculus: reminders

Section 3: Gradient, hessian, link with differential. Gradient ∇fx of f : C ⊂ Rn → R at x ∈ int(C). First order approximation, geometric interpretation with tangent approximation. Schwarz’s Theorem (if f is C 2 ) and Hessian of f : C ⊂ Rn → R at x ∈ int(C), denoted Hess(f )(x) Second order approximation of a C 2 function f : C ⊂ Rn → R around x ∈ int(C).

Chapter 4: Differential calculus: reminders

Section 3: Gradient, hessian, link with differential. Gradient ∇fx of f : C ⊂ Rn → R at x ∈ int(C). First order approximation, geometric interpretation with tangent approximation. Schwarz’s Theorem (if f is C 2 ) and Hessian of f : C ⊂ Rn → R at x ∈ int(C), denoted Hess(f )(x) Second order approximation of a C 2 function f : C ⊂ Rn → R around x ∈ int(C).

Chapter 4: Differential calculus: reminders

Section 3: Gradient, hessian, link with differential. Gradient ∇fx of f : C ⊂ Rn → R at x ∈ int(C). First order approximation, geometric interpretation with tangent approximation. Schwarz’s Theorem (if f is C 2 ) and Hessian of f : C ⊂ Rn → R at x ∈ int(C), denoted Hess(f )(x) Second order approximation of a C 2 function f : C ⊂ Rn → R around x ∈ int(C).

Chapter 4: Differential calculus: reminders

Section 3: Notations for derivatives for vector-valued functions f = (f1 , ...fp , fp+1 , ..., fn ) where each fi is real-valued on IRN . - For example, we can compute, at x ∈ IRN , the derivative of the p ≤ n first components of f with respect to the k ≤ N-first variables. It is denoted...and it can be represented by the matrix... - If p = k = N = n, this is the jacobian of f = (f1 , ..., fn ) at x = (x1 , ..., xn ), denoted Jx (f )

Chapter 4: Differential calculus: reminders

Section 4: Chain rule: Example of Computation. Chain rule: how computing the differential of f (u1 (x1 , ..., xn ), u2 (x1 , ..., xn ), ..., un (x1 , ..., xn )).

Chapter 4: Differential calculus: reminders

Section 4: Implicit function theorem. Implicit equation: f (x, y ) = 0 Explicit equation: y = f (x) Find condition that guarantees you can pass from implicit form to explicit form!

Chapter 4: Differential calculus: reminders

Section 4: Implicit function theorem. Implicit equation: f (x, y ) = 0 Explicit equation: y = f (x) Find condition that guarantees you can pass from implicit form to explicit form!

Chapter 4: Differential calculus: reminders

Section 4: Implicit function theorem. Implicit equation: f (x, y ) = 0 Explicit equation: y = f (x) Find condition that guarantees you can pass from implicit form to explicit form!

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Example1: f (x, y ) = ax + by + c = 0 Possible to write y = f (x) in a unique way if and only if.... Sometimes, possible to write y = f (x) only for x on some neighborhood. Example2: f (x, y ) = x 2 + y 2 − 1 = 0

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Example1: f (x, y ) = ax + by + c = 0 Possible to write y = f (x) in a unique way if and only if.... Sometimes, possible to write y = f (x) only for x on some neighborhood. Example2: f (x, y ) = x 2 + y 2 − 1 = 0

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Example1: f (x, y ) = ax + by + c = 0 Possible to write y = f (x) in a unique way if and only if.... Sometimes, possible to write y = f (x) only for x on some neighborhood. Example2: f (x, y ) = x 2 + y 2 − 1 = 0

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Example1: f (x, y ) = ax + by + c = 0 Possible to write y = f (x) in a unique way if and only if.... Sometimes, possible to write y = f (x) only for x on some neighborhood. Example2: f (x, y ) = x 2 + y 2 − 1 = 0

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Implicit function theorem (one variable): Let U and V two open subsets of R. Let f : U × V → R which is C 1 . ∂f ¯ ¯ (x , y ) 6= 0. Let (x¯ , y¯ ) ∈ U × V such that f (x¯ , y¯ ) = 0 and ∂y Then: (1) there exists Ux¯ and Vy¯ open neighborhood of x¯ and y¯ . (2) There exists a C 1 function g : Ux¯ → V such that For every (x, y ) ∈ Ux¯ × Vy¯ , f (x, y ) = 0 is equivalent to y = g(x). ∂f

(x¯ ,y¯ ) . (x¯ ,y¯ ) ∂y

Moreover, we have g 0 (x¯ ) = − ∂x ∂f

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Implicit function theorem (one variable): Let U and V two open subsets of R. Let f : U × V → R which is C 1 . ∂f ¯ ¯ (x , y ) 6= 0. Let (x¯ , y¯ ) ∈ U × V such that f (x¯ , y¯ ) = 0 and ∂y Then: (1) there exists Ux¯ and Vy¯ open neighborhood of x¯ and y¯ . (2) There exists a C 1 function g : Ux¯ → V such that For every (x, y ) ∈ Ux¯ × Vy¯ , f (x, y ) = 0 is equivalent to y = g(x). ∂f

(x¯ ,y¯ ) . (x¯ ,y¯ ) ∂y

Moreover, we have g 0 (x¯ ) = − ∂x ∂f

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Implicit function theorem (one variable): Let U and V two open subsets of R. Let f : U × V → R which is C 1 . ∂f ¯ ¯ (x , y ) 6= 0. Let (x¯ , y¯ ) ∈ U × V such that f (x¯ , y¯ ) = 0 and ∂y Then: (1) there exists Ux¯ and Vy¯ open neighborhood of x¯ and y¯ . (2) There exists a C 1 function g : Ux¯ → V such that For every (x, y ) ∈ Ux¯ × Vy¯ , f (x, y ) = 0 is equivalent to y = g(x). ∂f

(x¯ ,y¯ ) . (x¯ ,y¯ ) ∂y

Moreover, we have g 0 (x¯ ) = − ∂x ∂f

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Implicit function theorem (one variable): Let U and V two open subsets of R. Let f : U × V → R which is C 1 . ∂f ¯ ¯ (x , y ) 6= 0. Let (x¯ , y¯ ) ∈ U × V such that f (x¯ , y¯ ) = 0 and ∂y Then: (1) there exists Ux¯ and Vy¯ open neighborhood of x¯ and y¯ . (2) There exists a C 1 function g : Ux¯ → V such that For every (x, y ) ∈ Ux¯ × Vy¯ , f (x, y ) = 0 is equivalent to y = g(x). ∂f

(x¯ ,y¯ ) . (x¯ ,y¯ ) ∂y

Moreover, we have g 0 (x¯ ) = − ∂x ∂f

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: one variable Implicit function theorem (one variable): Let U and V two open subsets of R. Let f : U × V → R which is C 1 . ∂f ¯ ¯ (x , y ) 6= 0. Let (x¯ , y¯ ) ∈ U × V such that f (x¯ , y¯ ) = 0 and ∂y Then: (1) there exists Ux¯ and Vy¯ open neighborhood of x¯ and y¯ . (2) There exists a C 1 function g : Ux¯ → V such that For every (x, y ) ∈ Ux¯ × Vy¯ , f (x, y ) = 0 is equivalent to y = g(x). ∂f

(x¯ ,y¯ ) . (x¯ ,y¯ ) ∂y

Moreover, we have g 0 (x¯ ) = − ∂x ∂f

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: several variables Implicit function theorem (n + p variables, n equations): Let U and V two open subsets of Rp and Rn . Let f1 , ..., fn from U × V → R which are C 1 . Let (x¯ , y¯ ) = (x¯1 , ..., x¯p , y¯1 , ..., y¯n ) ∈ U × V such that ∂(f1 ,...,fn ) (x¯ , y¯ ) invertible (as a matrix). f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) Then: (1) there exists U(x¯1 ,...,x¯p ) and V(y¯1 ,...,y¯n ) open neighborhood of (x¯1 , ..., x¯p ) and (y¯1 , ..., y¯n ). (2) There exists a C 1 function g = (g1 , ..., gn ) : U(x¯1 ,...,x¯p ) → V(y¯1 ,...,y¯n ) such that For every (x, y ) ∈ U(x¯1 ,...,x¯p ) × V(y¯1 ,...,y¯n ) , f (x1 , ..., xp , y1 , ..., yn ) = 0 is equivalent to (y1 , ..., yn ) = g(x1 , ..., xp ). Moreover, we have ∂(f1 ,...,fn ) ∂g ¯ ¯ ¯ −1 ∂(f1 ,...,fn ) ¯ ¯ ∂(x1 ,...,xp ) (x ) = −( ∂(y1 ,...,yn ) (x , y )) . ∂(x1 ,...,xp ) (x , y ).

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: several variables Implicit function theorem (n + p variables, n equations): Let U and V two open subsets of Rp and Rn . Let f1 , ..., fn from U × V → R which are C 1 . Let (x¯ , y¯ ) = (x¯1 , ..., x¯p , y¯1 , ..., y¯n ) ∈ U × V such that ∂(f1 ,...,fn ) (x¯ , y¯ ) invertible (as a matrix). f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) Then: (1) there exists U(x¯1 ,...,x¯p ) and V(y¯1 ,...,y¯n ) open neighborhood of (x¯1 , ..., x¯p ) and (y¯1 , ..., y¯n ). (2) There exists a C 1 function g = (g1 , ..., gn ) : U(x¯1 ,...,x¯p ) → V(y¯1 ,...,y¯n ) such that For every (x, y ) ∈ U(x¯1 ,...,x¯p ) × V(y¯1 ,...,y¯n ) , f (x1 , ..., xp , y1 , ..., yn ) = 0 is equivalent to (y1 , ..., yn ) = g(x1 , ..., xp ). Moreover, we have ∂(f1 ,...,fn ) ∂g ¯ ¯ ¯ −1 ∂(f1 ,...,fn ) ¯ ¯ ∂(x1 ,...,xp ) (x ) = −( ∂(y1 ,...,yn ) (x , y )) . ∂(x1 ,...,xp ) (x , y ).

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: several variables Implicit function theorem (n + p variables, n equations): Let U and V two open subsets of Rp and Rn . Let f1 , ..., fn from U × V → R which are C 1 . Let (x¯ , y¯ ) = (x¯1 , ..., x¯p , y¯1 , ..., y¯n ) ∈ U × V such that ∂(f1 ,...,fn ) (x¯ , y¯ ) invertible (as a matrix). f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) Then: (1) there exists U(x¯1 ,...,x¯p ) and V(y¯1 ,...,y¯n ) open neighborhood of (x¯1 , ..., x¯p ) and (y¯1 , ..., y¯n ). (2) There exists a C 1 function g = (g1 , ..., gn ) : U(x¯1 ,...,x¯p ) → V(y¯1 ,...,y¯n ) such that For every (x, y ) ∈ U(x¯1 ,...,x¯p ) × V(y¯1 ,...,y¯n ) , f (x1 , ..., xp , y1 , ..., yn ) = 0 is equivalent to (y1 , ..., yn ) = g(x1 , ..., xp ). Moreover, we have ∂(f1 ,...,fn ) ∂g ¯ ¯ ¯ −1 ∂(f1 ,...,fn ) ¯ ¯ ∂(x1 ,...,xp ) (x ) = −( ∂(y1 ,...,yn ) (x , y )) . ∂(x1 ,...,xp ) (x , y ).

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: several variables Implicit function theorem (n + p variables, n equations): Let U and V two open subsets of Rp and Rn . Let f1 , ..., fn from U × V → R which are C 1 . Let (x¯ , y¯ ) = (x¯1 , ..., x¯p , y¯1 , ..., y¯n ) ∈ U × V such that ∂(f1 ,...,fn ) (x¯ , y¯ ) invertible (as a matrix). f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) Then: (1) there exists U(x¯1 ,...,x¯p ) and V(y¯1 ,...,y¯n ) open neighborhood of (x¯1 , ..., x¯p ) and (y¯1 , ..., y¯n ). (2) There exists a C 1 function g = (g1 , ..., gn ) : U(x¯1 ,...,x¯p ) → V(y¯1 ,...,y¯n ) such that For every (x, y ) ∈ U(x¯1 ,...,x¯p ) × V(y¯1 ,...,y¯n ) , f (x1 , ..., xp , y1 , ..., yn ) = 0 is equivalent to (y1 , ..., yn ) = g(x1 , ..., xp ). Moreover, we have ∂(f1 ,...,fn ) ∂g ¯ ¯ ¯ −1 ∂(f1 ,...,fn ) ¯ ¯ ∂(x1 ,...,xp ) (x ) = −( ∂(y1 ,...,yn ) (x , y )) . ∂(x1 ,...,xp ) (x , y ).

Chapter 4: Differential calculus: reminders Section 4: Implicit function theorem: several variables Implicit function theorem (n + p variables, n equations): Let U and V two open subsets of Rp and Rn . Let f1 , ..., fn from U × V → R which are C 1 . Let (x¯ , y¯ ) = (x¯1 , ..., x¯p , y¯1 , ..., y¯n ) ∈ U × V such that ∂(f1 ,...,fn ) (x¯ , y¯ ) invertible (as a matrix). f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) Then: (1) there exists U(x¯1 ,...,x¯p ) and V(y¯1 ,...,y¯n ) open neighborhood of (x¯1 , ..., x¯p ) and (y¯1 , ..., y¯n ). (2) There exists a C 1 function g = (g1 , ..., gn ) : U(x¯1 ,...,x¯p ) → V(y¯1 ,...,y¯n ) such that For every (x, y ) ∈ U(x¯1 ,...,x¯p ) × V(y¯1 ,...,y¯n ) , f (x1 , ..., xp , y1 , ..., yn ) = 0 is equivalent to (y1 , ..., yn ) = g(x1 , ..., xp ). Moreover, we have ∂(f1 ,...,fn ) ∂g ¯ ¯ ¯ −1 ∂(f1 ,...,fn ) ¯ ¯ ∂(x1 ,...,xp ) (x ) = −( ∂(y1 ,...,yn ) (x , y )) . ∂(x1 ,...,xp ) (x , y ).

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Homogenous functions, returns to scale, Euler theorem. Inverse function theorem .

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Homogenous functions, returns to scale, Euler theorem. Inverse function theorem .

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Homogenous functions, returns to scale, Euler theorem. Inverse function theorem .

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders

Section 5: Some applications macro model of income determination. Y =national income. T =taxes I =investment. C =consumption, function of Y − T . G =Government expediture. Y = C(Y − T ) + I + G. Possible to express Y as a function of I, G, T ? Variation of Y as a function of I, G, T ?

Chapter 4: Differential calculus: reminders Section 5: Some applications Homogenous function. Definition: f (x1 , ..., xn ) homogenous of degree k on a domain D ⊂ IRn if f (tx1 , .., txn ) = t k f (x1 , ..., xn ). Example: f (x, y ) = x a .y b homogenous of degree a + b (depending on a + b, constant, decreasing or increasing return to scale). Theorem (Euler): Let D ⊂ IRn an open domain such that for some t > 0, (x1 , ..., xn ) ∈ D ⇔ (tx1 , ..., txn ). ∈ D. Assume f (x1 , ..., xn ) is C 1 on D. Then f is homogenous of degree k if and only n X ∂f (x1 , ..., xn ) i=1

∂xi

= kf (x1 , ..., xn ).

Chapter 4: Differential calculus: reminders Section 5: Some applications Homogenous function. Definition: f (x1 , ..., xn ) homogenous of degree k on a domain D ⊂ IRn if f (tx1 , .., txn ) = t k f (x1 , ..., xn ). Example: f (x, y ) = x a .y b homogenous of degree a + b (depending on a + b, constant, decreasing or increasing return to scale). Theorem (Euler): Let D ⊂ IRn an open domain such that for some t > 0, (x1 , ..., xn ) ∈ D ⇔ (tx1 , ..., txn ). ∈ D. Assume f (x1 , ..., xn ) is C 1 on D. Then f is homogenous of degree k if and only n X ∂f (x1 , ..., xn ) i=1

∂xi

= kf (x1 , ..., xn ).

Chapter 4: Differential calculus: reminders Section 5: Some applications Homogenous function. Definition: f (x1 , ..., xn ) homogenous of degree k on a domain D ⊂ IRn if f (tx1 , .., txn ) = t k f (x1 , ..., xn ). Example: f (x, y ) = x a .y b homogenous of degree a + b (depending on a + b, constant, decreasing or increasing return to scale). Theorem (Euler): Let D ⊂ IRn an open domain such that for some t > 0, (x1 , ..., xn ) ∈ D ⇔ (tx1 , ..., txn ). ∈ D. Assume f (x1 , ..., xn ) is C 1 on D. Then f is homogenous of degree k if and only n X ∂f (x1 , ..., xn ) i=1

∂xi

= kf (x1 , ..., xn ).

Chapter 4: Differential calculus: reminders Section 5: Some applications Homogenous function. Definition: f (x1 , ..., xn ) homogenous of degree k on a domain D ⊂ IRn if f (tx1 , .., txn ) = t k f (x1 , ..., xn ). Example: f (x, y ) = x a .y b homogenous of degree a + b (depending on a + b, constant, decreasing or increasing return to scale). Theorem (Euler): Let D ⊂ IRn an open domain such that for some t > 0, (x1 , ..., xn ) ∈ D ⇔ (tx1 , ..., txn ). ∈ D. Assume f (x1 , ..., xn ) is C 1 on D. Then f is homogenous of degree k if and only n X ∂f (x1 , ..., xn ) i=1

∂xi

= kf (x1 , ..., xn ).

Chapter 4: Differential calculus: reminders Section 5: Some applications Homogenous function. Definition: f (x1 , ..., xn ) homogenous of degree k on a domain D ⊂ IRn if f (tx1 , .., txn ) = t k f (x1 , ..., xn ). Example: f (x, y ) = x a .y b homogenous of degree a + b (depending on a + b, constant, decreasing or increasing return to scale). Theorem (Euler): Let D ⊂ IRn an open domain such that for some t > 0, (x1 , ..., xn ) ∈ D ⇔ (tx1 , ..., txn ). ∈ D. Assume f (x1 , ..., xn ) is C 1 on D. Then f is homogenous of degree k if and only n X ∂f (x1 , ..., xn ) i=1

∂xi

= kf (x1 , ..., xn ).

Chapter 4: Differential calculus: reminders

Section 5: Some applications Inverse function theorem. Let f : Rn → Rn be C 1 . Assume the Jacobian J(x¯1 ,...,x¯n ) (f ) at some (x¯1 , ..., x¯n ) is invertible. Then: There exists: i) some open neighborhood Vx¯ and an open neighborhood Vf (x¯) . ii) A function f −1 which is C 1 from Vf (x¯) to Vx¯ such that: f ◦ f −1 = id on Vf (x¯) and f −1 ◦ f = id on Vx¯ . Moreover, we have Jf (x¯) (f −1 ) = Jx (f )−1 .

Chapter 4: Differential calculus: reminders

Section 5: Some applications Inverse function theorem. Let f : Rn → Rn be C 1 . Assume the Jacobian J(x¯1 ,...,x¯n ) (f ) at some (x¯1 , ..., x¯n ) is invertible. Then: There exists: i) some open neighborhood Vx¯ and an open neighborhood Vf (x¯) . ii) A function f −1 which is C 1 from Vf (x¯) to Vx¯ such that: f ◦ f −1 = id on Vf (x¯) and f −1 ◦ f = id on Vx¯ . Moreover, we have Jf (x¯) (f −1 ) = Jx (f )−1 .

Chapter 4: Differential calculus: reminders

Section 5: Some applications Inverse function theorem. Let f : Rn → Rn be C 1 . Assume the Jacobian J(x¯1 ,...,x¯n ) (f ) at some (x¯1 , ..., x¯n ) is invertible. Then: There exists: i) some open neighborhood Vx¯ and an open neighborhood Vf (x¯) . ii) A function f −1 which is C 1 from Vf (x¯) to Vx¯ such that: f ◦ f −1 = id on Vf (x¯) and f −1 ◦ f = id on Vx¯ . Moreover, we have Jf (x¯) (f −1 ) = Jx (f )−1 .

Chapter 4: Differential calculus: reminders

Section 5: Some applications Inverse function theorem. Let f : Rn → Rn be C 1 . Assume the Jacobian J(x¯1 ,...,x¯n ) (f ) at some (x¯1 , ..., x¯n ) is invertible. Then: There exists: i) some open neighborhood Vx¯ and an open neighborhood Vf (x¯) . ii) A function f −1 which is C 1 from Vf (x¯) to Vx¯ such that: f ◦ f −1 = id on Vf (x¯) and f −1 ◦ f = id on Vx¯ . Moreover, we have Jf (x¯) (f −1 ) = Jx (f )−1 .

Chapter 5: Convexity

Section 1: convexity of set linear combination, affine combination, convex combination. 2 equivalent definitions of A ⊂ Rn convex. Examples. Convex hull, Unit simplex, simplex. Compactness. Strict convexity.

Chapter 5: Convexity

Section 1: convexity of set linear combination, affine combination, convex combination. 2 equivalent definitions of A ⊂ Rn convex. Examples. Convex hull, Unit simplex, simplex. Compactness. Strict convexity.

Chapter 5: Convexity

Section 1: convexity of set linear combination, affine combination, convex combination. 2 equivalent definitions of A ⊂ Rn convex. Examples. Convex hull, Unit simplex, simplex. Compactness. Strict convexity.

Chapter 5: Convexity

Section 1: convexity of set linear combination, affine combination, convex combination. 2 equivalent definitions of A ⊂ Rn convex. Examples. Convex hull, Unit simplex, simplex. Compactness. Strict convexity.

Chapter 5: Convexity

Section 2: Convex functions Case of C 2 functions: convex (concave) function from R to R. Strictly convex or strictly concave C 2 function General definition (without C 2 ) of convex, concave, Strictly convex or strictly concave, When f : A ⊂ Rn → R. Quasi-concavity, Quasi-concavity. First Properties....Properties with Hypograph, Epigraph, Upper level sets.

Chapter 5: Convexity

Section 2: Convex functions Case of C 2 functions: convex (concave) function from R to R. Strictly convex or strictly concave C 2 function General definition (without C 2 ) of convex, concave, Strictly convex or strictly concave, When f : A ⊂ Rn → R. Quasi-concavity, Quasi-concavity. First Properties....Properties with Hypograph, Epigraph, Upper level sets.

Chapter 5: Convexity

Section 2: Convex functions Case of C 2 functions: convex (concave) function from R to R. Strictly convex or strictly concave C 2 function General definition (without C 2 ) of convex, concave, Strictly convex or strictly concave, When f : A ⊂ Rn → R. Quasi-concavity, Quasi-concavity. First Properties....Properties with Hypograph, Epigraph, Upper level sets.

Chapter 5: Convexity

Section 2: Convex functions Case of C 2 functions: convex (concave) function from R to R. Strictly convex or strictly concave C 2 function General definition (without C 2 ) of convex, concave, Strictly convex or strictly concave, When f : A ⊂ Rn → R. Quasi-concavity, Quasi-concavity. First Properties....Properties with Hypograph, Epigraph, Upper level sets.

Chapter 5: Convexity Section 2: Convex functions Convexity inequality 1: on the growth rate for f on an interval of R. Convexity inequality 2: for f differentiable on a convex subset C of Rn . Then f is convex if and only if ∀(x, y ) ∈ C × C, f (y ) ≥ f (x)+ < ∇f (x), y − x > . Convexity inequality 3: for f differentiable on a convex subset C of Rn . Then f is convex if and only if ∀(x, y ) ∈ C × C, < ∇f (y ) − ∇f (x), y − x >≥ 0.

Chapter 5: Convexity Section 2: Convex functions Convexity inequality 1: on the growth rate for f on an interval of R. Convexity inequality 2: for f differentiable on a convex subset C of Rn . Then f is convex if and only if ∀(x, y ) ∈ C × C, f (y ) ≥ f (x)+ < ∇f (x), y − x > . Convexity inequality 3: for f differentiable on a convex subset C of Rn . Then f is convex if and only if ∀(x, y ) ∈ C × C, < ∇f (y ) − ∇f (x), y − x >≥ 0.

Chapter 5: Convexity Section 2: Convex functions Convexity inequality 1: on the growth rate for f on an interval of R. Convexity inequality 2: for f differentiable on a convex subset C of Rn . Then f is convex if and only if ∀(x, y ) ∈ C × C, f (y ) ≥ f (x)+ < ∇f (x), y − x > . Convexity inequality 3: for f differentiable on a convex subset C of Rn . Then f is convex if and only if ∀(x, y ) ∈ C × C, < ∇f (y ) − ∇f (x), y − x >≥ 0.

Chapter 5: Convexity

Section 2: Convex functions: regularity A convex (resp. concave) function on a convex subset C of Rn is continuous on the interior of C.

Chapter 5: Convexity

Section 3: Hessian and convexity symetric real matrix, reduction. negative (or positive) semidefinite matrix, negative (or positive) semidefinite matrix. 2 definitions (with eigenvalues or not). Application to Hessian when n = 2: trace and determinant. Application to Hessian when n = 3: criterium for strict concavity, concavity.

Chapter 5: Convexity

Section 3: Hessian and convexity symetric real matrix, reduction. negative (or positive) semidefinite matrix, negative (or positive) semidefinite matrix. 2 definitions (with eigenvalues or not). Application to Hessian when n = 2: trace and determinant. Application to Hessian when n = 3: criterium for strict concavity, concavity.

Chapter 5: Convexity

Section 3: Hessian and convexity symetric real matrix, reduction. negative (or positive) semidefinite matrix, negative (or positive) semidefinite matrix. 2 definitions (with eigenvalues or not). Application to Hessian when n = 2: trace and determinant. Application to Hessian when n = 3: criterium for strict concavity, concavity.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions Section 1: Weirstrass theorem: Existence for continuous mappings on a compact subset Let f : K → R. i) Assume f is continuous. ii) Assume K ⊂ Rn is compact. Then (P) max f (x) x∈K

and (Q) min f (x) x∈K

have at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions Section 1: Weirstrass theorem: Existence for continuous mappings on a compact subset Let f : K → R. i) Assume f is continuous. ii) Assume K ⊂ Rn is compact. Then (P) max f (x) x∈K

and (Q) min f (x) x∈K

have at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions Section 1: Weirstrass theorem: Existence for continuous mappings on a compact subset Let f : K → R. i) Assume f is continuous. ii) Assume K ⊂ Rn is compact. Then (P) max f (x) x∈K

and (Q) min f (x) x∈K

have at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions Section 1: Weirstrass theorem: Existence for continuous mappings on a compact subset Let f : K → R. i) Assume f is continuous. ii) Assume K ⊂ Rn is compact. Then (P) max f (x) x∈K

and (Q) min f (x) x∈K

have at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = +∞ (f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) min f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = +∞ (f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) min f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = +∞ (f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) min f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = +∞ (f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) min f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = −∞ (-f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) max f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = −∞ (-f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) max f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = −∞ (-f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) max f (x) x∈C

has at least one solution.

Chapter 6: Existence of solutions of optimization problem without differentiability assumptions

Section 2: Weakening of Weirstrass theorem with coercivity Let f : C → R. Assume that if limkxk→+∞ f (x) = −∞ (-f is said coercive). ii) Assume C ⊂ Rn is closed. Then (P) max f (x) x∈C

has at least one solution.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions Section 1:Optimization problem with no constraints on an open domain Necessary condition Let f : U → R, where U ⊂ Rn open. i) Assume x¯ is a local solution of (P) min f (x) x∈U

then: Then: i) If f is differentiable at x, we have ∇f (x¯ ) = 0, i.e. x¯ is a critical point of f . ii) If f is C 2 at x¯ , then Hessx (f ) is positive semidefinite.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions Section 1:Optimization problem with no constraints on an open domain Necessary condition Let f : U → R, where U ⊂ Rn open. i) Assume x¯ is a local solution of (P) min f (x) x∈U

then: Then: i) If f is differentiable at x, we have ∇f (x¯ ) = 0, i.e. x¯ is a critical point of f . ii) If f is C 2 at x¯ , then Hessx (f ) is positive semidefinite.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions Section 1:Optimization problem with no constraints on an open domain Necessary condition Let f : U → R, where U ⊂ Rn open. i) Assume x¯ is a local solution of (P) min f (x) x∈U

then: Then: i) If f is differentiable at x, we have ∇f (x¯ ) = 0, i.e. x¯ is a critical point of f . ii) If f is C 2 at x¯ , then Hessx (f ) is positive semidefinite.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions Section 1:Optimization problem with no constraints on an open domain Necessary condition Let f : U → R, where U ⊂ Rn open. i) Assume x¯ is a local solution of (P) max f (x) x∈U

then: Then: i) If f is differentiable at x, we have ∇f (x¯ ) = 0, i.e. x¯ is a critical point of f . ii) If f is C 2 at x¯ , then Hessx (f ) is positive seminegative.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions Section 1:Optimization problem with no constraints on an open domain Necessary condition Let f : U → R, where U ⊂ Rn open. i) Assume x¯ is a local solution of (P) max f (x) x∈U

then: Then: i) If f is differentiable at x, we have ∇f (x¯ ) = 0, i.e. x¯ is a critical point of f . ii) If f is C 2 at x¯ , then Hessx (f ) is positive seminegative.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions Section 1:Optimization problem with no constraints on an open domain Necessary condition Let f : U → R, where U ⊂ Rn open. i) Assume x¯ is a local solution of (P) max f (x) x∈U

then: Then: i) If f is differentiable at x, we have ∇f (x¯ ) = 0, i.e. x¯ is a critical point of f . ii) If f is C 2 at x¯ , then Hessx (f ) is positive seminegative.

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions

Section 2:Optimization problem with no constraints on an open domain Sufficient condition for local optimum Let f : U → R, where U ⊂ Rn open. If f is C 2 at x¯ a critical point of f and if Hessx (f ) is positive definite, then x¯ is a local solution of (P) min f (x) x∈U

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions

Section 2:Optimization problem with no constraints on an open domain Sufficient condition for local optimum Let f : U → R, where U ⊂ Rn open. If f is C 2 at x¯ a critical point of f and if Hessx (f ) is positive negative, then x¯ is a local solution of (P) max f (x) x∈U

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions

Section 3:Optimization problem with no constraints on an open domain Sufficient condition for global minimum Let f : U → R, where U ⊂ Rn open. If f is convex on U convex, then any critical point x¯ is a global solution of (P) min f (x) x∈U

Chapter 7: Existence of solutions of optimization problem with differentiability assumptions

Section 3:Optimization problem with no constraints on an open domain Sufficient condition for global minimum Let f : U → R, where U ⊂ Rn open. If f is concave on U convex, then any critical point x¯ is a global solution of (P) max f (x) x∈U