Paris.

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

Last time

Last time we have seen: value, solution, maximizing or minimizing sequence of an optimization problem. The value of an optimization problem can be finite or infinite. If the value of an optimization problem is finite, it can have an exact solution, or approximate solution (formalized by maximizing or minimizing sequences). We have begun topology (study of open, closed, compact sets, continuity, ...) which is important to get sufficient conditions for an optimization problem to have a solution.

Last time

Last time we have seen: value, solution, maximizing or minimizing sequence of an optimization problem. The value of an optimization problem can be finite or infinite. If the value of an optimization problem is finite, it can have an exact solution, or approximate solution (formalized by maximizing or minimizing sequences). We have begun topology (study of open, closed, compact sets, continuity, ...) which is important to get sufficient conditions for an optimization problem to have a solution.

Last time

Last time we have seen: value, solution, maximizing or minimizing sequence of an optimization problem. The value of an optimization problem can be finite or infinite. If the value of an optimization problem is finite, it can have an exact solution, or approximate solution (formalized by maximizing or minimizing sequences). We have begun topology (study of open, closed, compact sets, continuity, ...) which is important to get sufficient conditions for an optimization problem to have a solution.

Last time

Last time we have seen: value, solution, maximizing or minimizing sequence of an optimization problem. The value of an optimization problem can be finite or infinite. If the value of an optimization problem is finite, it can have an exact solution, or approximate solution (formalized by maximizing or minimizing sequences). We have begun topology (study of open, closed, compact sets, continuity, ...) which is important to get sufficient conditions for an optimization problem to have a solution.

Question of the week (DIFFICULT: NOT FOR THE EXAM!) Correction of the Question of the week If A ⊂ R, call A0 the set of accumulation points of A. Is it possible to find A such that A0 6= A00 and A00 6= A000 ? If yes, find one, otherwise prove it is impossible.

Question of the week (DIFFICULT: NOT FOR THE EXAM!) Correction of the Question of the week If A ⊂ R, call A0 the set of accumulation points of A. Is it possible to find A such that A0 6= A00 and A00 6= A000 ? If yes, find one, otherwise prove it is impossible.

Question of the week Correction of the Question of the week A={

1 1 + n 2 : n > 0, p > 0}. p 2 p

A0 = {

1 : p > 0} ∪ {0}. p

A00 = {0}.

A000 = ∅.

Question of the week Correction of the Question of the week A={

1 1 + n 2 : n > 0, p > 0}. p 2 p

A0 = {

1 : p > 0} ∪ {0}. p

A00 = {0}.

A000 = ∅.

Question of the week Correction of the Question of the week A={

1 1 + n 2 : n > 0, p > 0}. p 2 p

A0 = {

1 : p > 0} ∪ {0}. p

A00 = {0}.

A000 = ∅.

Question of the week Correction of the Question of the week A={

1 1 + n 2 : n > 0, p > 0}. p 2 p

A0 = {

1 : p > 0} ∪ {0}. p

A00 = {0}.

A000 = ∅.

Chapter 3: Topology

Correction of the exercises Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. It is compact ?

Chapter 3: Topology

Correction of the exercises Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. It is compact ?

Chapter 3: Topology

Correction of the exercises Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. It is compact ?

Chapter 3: Topology

Correction of the exercises Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. It is compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. It is compact ?

Chapter 3: Topology

Corrections of the exercises Find A and B such that A ∩ B 6= A ∩ B. Solution: A =Q, l B = IR −Q. l Find A such that ∂(int(A)) 6= ∂(A). Solution: A =Q. l Find A and B subsets of R such that A + B is not closed. Solution: A = ZZ and B = πZZ (DIFFICULT: NOT FOR THE EXAM!)

Chapter 3: Topology

Corrections of the exercises Find A and B such that A ∩ B 6= A ∩ B. Solution: A =Q, l B = IR −Q. l Find A such that ∂(int(A)) 6= ∂(A). Solution: A =Q. l Find A and B subsets of R such that A + B is not closed. Solution: A = ZZ and B = πZZ (DIFFICULT: NOT FOR THE EXAM!)

Chapter 3: Topology

Corrections of the exercises Find A and B such that A ∩ B 6= A ∩ B. Solution: A =Q, l B = IR −Q. l Find A such that ∂(int(A)) 6= ∂(A). Solution: A =Q. l Find A and B subsets of R such that A + B is not closed. Solution: A = ZZ and B = πZZ (DIFFICULT: NOT FOR THE EXAM!)

Chapter 3: Topology

Corrections of the exercises Find A and B such that A ∩ B 6= A ∩ B. Solution: A =Q, l B = IR −Q. l Find A such that ∂(int(A)) 6= ∂(A). Solution: A =Q. l Find A and B subsets of R such that A + B is not closed. Solution: A = ZZ and B = πZZ (DIFFICULT: NOT FOR THE EXAM!)

Chapter 3: Topology

Corrections of the exercises Find A and B such that A ∩ B 6= A ∩ B. Solution: A =Q, l B = IR −Q. l Find A such that ∂(int(A)) 6= ∂(A). Solution: A =Q. l Find A and B subsets of R such that A + B is not closed. Solution: A = ZZ and B = πZZ (DIFFICULT: NOT FOR THE EXAM!)

Chapter 3: Topology

Corrections of the exercises Find A and B such that A ∩ B 6= A ∩ B. Solution: A =Q, l B = IR −Q. l Find A such that ∂(int(A)) 6= ∂(A). Solution: A =Q. l Find A and B subsets of R such that A + B is not closed. Solution: A = ZZ and B = πZZ (DIFFICULT: NOT FOR THE EXAM!)

Chapter 3: Topology

We continue the lecture: Section 2: Continuity C ⊂ Rn is pathwise connected if .... Intermediate value theorem for a continuous mapping f : R → R. Bolzano theorem: If a real-valued continuous function defined on pathwise connected subset C ⊂ Rn is sometimes positive and sometimes negative, it must be 0 at some point.

Chapter 3: Topology

We continue the lecture: Section 2: Continuity C ⊂ Rn is pathwise connected if .... Intermediate value theorem for a continuous mapping f : R → R. Bolzano theorem: If a real-valued continuous function defined on pathwise connected subset C ⊂ Rn is sometimes positive and sometimes negative, it must be 0 at some point.

Chapter 3: Topology

We continue the lecture: Section 2: Continuity C ⊂ Rn is pathwise connected if .... Intermediate value theorem for a continuous mapping f : R → R. Bolzano theorem: If a real-valued continuous function defined on pathwise connected subset C ⊂ Rn is sometimes positive and sometimes negative, it must be 0 at some point.

Chapter 3: Topology Exercise: a cyclist covers 10km in exactly one hour (but his speed may not be constant). Prove that there exists an interval of time of half an hour for which he covers exactly 5km.

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity for function f from a C ⊂ R to R. upper semicontinuity; lower semicontinuity for function f from a C ⊂ Rn to R. Link with epigraph and hypograph.

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity for function f from a C ⊂ R to R. upper semicontinuity; lower semicontinuity for function f from a C ⊂ Rn to R. Link with epigraph and hypograph.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of S = {x : f (x) ≤ c} when f : Rn → R in practice... Level sets. Interpretation of maximization problem with level sets: Example: Maxx 2 +y 2 ≤1 x + y

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of S = {x : f (x) ≤ c} when f : Rn → R in practice... Level sets. Interpretation of maximization problem with level sets: Example: Maxx 2 +y 2 ≤1 x + y

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of S = {x : f (x) ≤ c} when f : Rn → R in practice... Level sets. Interpretation of maximization problem with level sets: Example: Maxx 2 +y 2 ≤1 x + y

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of S = {x : f (x) ≤ c} when f : Rn → R in practice... Level sets. Interpretation of maximization problem with level sets: Example: Maxx 2 +y 2 ≤1 x + y

Chapter 4: Differential calculus: reminders

Section 1: Reminders for real-valued function. Differentiable function. A real-valued function is C 0 , C 1 , C 2 if ... Rolle theorem for f : [a, b] → R continuous, differentiable on (a, b), with f (a) = f (b) = 0. Then there exists... Mean-Value theorem: f : [a, b] → R continuous, differentiable on (a, b). Then there exists... Taylor development at c: f : [a, b] → R assumed to be C k , on (a, b). Let c ∈ (a, c).

Chapter 4: Differential calculus: reminders

Section 1: Reminders for real-valued function. Differentiable function. A real-valued function is C 0 , C 1 , C 2 if ... Rolle theorem for f : [a, b] → R continuous, differentiable on (a, b), with f (a) = f (b) = 0. Then there exists... Mean-Value theorem: f : [a, b] → R continuous, differentiable on (a, b). Then there exists... Taylor development at c: f : [a, b] → R assumed to be C k , on (a, b). Let c ∈ (a, c).

Chapter 4: Differential calculus: reminders

Section 1: Reminders for real-valued function. Differentiable function. A real-valued function is C 0 , C 1 , C 2 if ... Rolle theorem for f : [a, b] → R continuous, differentiable on (a, b), with f (a) = f (b) = 0. Then there exists... Mean-Value theorem: f : [a, b] → R continuous, differentiable on (a, b). Then there exists... Taylor development at c: f : [a, b] → R assumed to be C k , on (a, b). Let c ∈ (a, c).

Chapter 4: Differential calculus: reminders

Section 1: Reminders for real-valued function. Differentiable function. A real-valued function is C 0 , C 1 , C 2 if ... Rolle theorem for f : [a, b] → R continuous, differentiable on (a, b), with f (a) = f (b) = 0. Then there exists... Mean-Value theorem: f : [a, b] → R continuous, differentiable on (a, b). Then there exists... Taylor development at c: f : [a, b] → R assumed to be C k , on (a, b). Let c ∈ (a, c).

Chapter 4: Differential calculus: reminders

Section 2: partial derivative, C k . Differential of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives for f : C ⊂ Rn → R at x ∈ int(C), denoted Dx f 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 of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives for f : C ⊂ Rn → R at x ∈ int(C), denoted Dx f 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 of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives for f : C ⊂ Rn → R at x ∈ int(C), denoted Dx f 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 of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative (in the direction u ∈ Rn ) of f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives for f : C ⊂ Rn → R at x ∈ int(C), denoted Dx f 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 of f : C ⊂ Rn → R at x ∈ int(C). First order approximation, geometric interpretation with tangent approximation. Schwarz’s theorem 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 of f : C ⊂ Rn → R at x ∈ int(C). First order approximation, geometric interpretation with tangent approximation. Schwarz’s theorem 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 of f : C ⊂ Rn → R at x ∈ int(C). First order approximation, geometric interpretation with tangent approximation. Schwarz’s theorem 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 N = n, the jacobian of f = (f1 , ..., fn ) at x = (x1 , ..., xn ) is the matrix 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¯ ) invertible (as a matrix). Then: f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) (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

∂g ∂(x1 ,...,xp )

∂(f1 ,...,fn ) −1 ∂(f1 ,...,fn ) = −( ∂(y ) . ∂(x1 ,...,xp ) . 1 ,...,yn

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¯ ) invertible (as a matrix). Then: f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) (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

∂g ∂(x1 ,...,xp )

∂(f1 ,...,fn ) −1 ∂(f1 ,...,fn ) = −( ∂(y ) . ∂(x1 ,...,xp ) . 1 ,...,yn

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¯ ) invertible (as a matrix). Then: f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) (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

∂g ∂(x1 ,...,xp )

∂(f1 ,...,fn ) −1 ∂(f1 ,...,fn ) = −( ∂(y ) . ∂(x1 ,...,xp ) . 1 ,...,yn

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¯ ) invertible (as a matrix). Then: f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) (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

∂g ∂(x1 ,...,xp )

∂(f1 ,...,fn ) −1 ∂(f1 ,...,fn ) = −( ∂(y ) . ∂(x1 ,...,xp ) . 1 ,...,yn

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¯ ) invertible (as a matrix). Then: f (x¯ , y¯ ) = 0 and ∂(y 1 ,...,yn ) (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

∂g ∂(x1 ,...,xp )

∂(f1 ,...,fn ) −1 ∂(f1 ,...,fn ) = −( ∂(y ) . ∂(x1 ,...,xp ) . 1 ,...,yn

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 Definition of A ⊂ Rn convex. Examples. Convex hull, Unit simplex, simplex. Strict convexity.

Chapter 5: Convexity

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

Chapter 5: Convexity

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

Chapter 5: Convexity

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

Chapter 5: Convexity

Section 2: Convex functions convex (concave) C 2 function from R to R. Strictly convex or concave C 2 function General defintion of convex (concave), strict or not, f : A ⊂ Rn → R. Quasi-concavity, First Properties....Properties with Hypograph, Epigraph, Upper level sets. regularity of convex (concave) function. Convexity inequalities.

Chapter 5: Convexity

Section 2: Convex functions convex (concave) C 2 function from R to R. Strictly convex or concave C 2 function General defintion of convex (concave), strict or not, f : A ⊂ Rn → R. Quasi-concavity, First Properties....Properties with Hypograph, Epigraph, Upper level sets. regularity of convex (concave) function. Convexity inequalities.

Chapter 5: Convexity

Section 2: Convex functions convex (concave) C 2 function from R to R. Strictly convex or concave C 2 function General defintion of convex (concave), strict or not, f : A ⊂ Rn → R. Quasi-concavity, First Properties....Properties with Hypograph, Epigraph, Upper level sets. regularity of convex (concave) function. Convexity inequalities.

Chapter 5: Convexity

Section 2: Convex functions convex (concave) C 2 function from R to R. Strictly convex or concave C 2 function General defintion of convex (concave), strict or not, f : A ⊂ Rn → R. Quasi-concavity, First Properties....Properties with Hypograph, Epigraph, Upper level sets. regularity of convex (concave) function. Convexity inequalities.

Chapter 5: Convexity

Section 2: Convex functions convex (concave) C 2 function from R to R. Strictly convex or concave C 2 function General defintion of convex (concave), strict or not, f : A ⊂ Rn → R. Quasi-concavity, First Properties....Properties with Hypograph, Epigraph, Upper level sets. regularity of convex (concave) function. Convexity inequalities.

Chapter 5: Convexity

Section 3: Hessian and convexity symetric real matrix, reduction. negative (or positive) semidefinite matrix, negative (or positive) semidefinite matrix. 2 definitions. Application to Hessian when n = 2: trace and determinant. Application to Hessian when n = 3: criterium for strict concavity, concavity. Convexity inequalities.

Chapter 5: Convexity

Section 3: Hessian and convexity symetric real matrix, reduction. negative (or positive) semidefinite matrix, negative (or positive) semidefinite matrix. 2 definitions. Application to Hessian when n = 2: trace and determinant. Application to Hessian when n = 3: criterium for strict concavity, concavity. Convexity inequalities.

Chapter 5: Convexity

Section 3: Hessian and convexity symetric real matrix, reduction. negative (or positive) semidefinite matrix, negative (or positive) semidefinite matrix. 2 definitions. Application to Hessian when n = 2: trace and determinant. Application to Hessian when n = 3: criterium for strict concavity, concavity. Convexity inequalities.