Paris.

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

Last time

Last time we have seen: a lot of reminders. Some "basic" exercices. sup, inf, max, min of real valued functions and of subsets of R. Intuitively, max and min are used if there are exact solution, and sup and inf if there are only approximate solution!

Last time

Last time we have seen: a lot of reminders. Some "basic" exercices. sup, inf, max, min of real valued functions and of subsets of R. Intuitively, max and min are used if there are exact solution, and sup and inf if there are only approximate solution!

Last time

Last time we have seen: a lot of reminders. Some "basic" exercices. sup, inf, max, min of real valued functions and of subsets of R. Intuitively, max and min are used if there are exact solution, and sup and inf if there are only approximate solution!

Last time

Last time we have seen: a lot of reminders. Some "basic" exercices. sup, inf, max, min of real valued functions and of subsets of R. Intuitively, max and min are used if there are exact solution, and sup and inf if there are only approximate solution!

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints). The value of (P) (resp. Q) is ... (can be finite or infinite).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints). The value of (P) (resp. Q) is ... (can be finite or infinite).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints). The value of (P) (resp. Q) is ... (can be finite or infinite).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints). The value of (P) (resp. Q) is ... (can be finite or infinite).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a local solution of (P) if ... x ∈ E is a local solution of (Q) if...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a local solution of (P) if ... x ∈ E is a local solution of (Q) if...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a local solution of (P) if ... x ∈ E is a local solution of (Q) if...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a local solution of (P) if ... x ∈ E is a local solution of (Q) if...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a local solution of (P) if ... x ∈ E is a local solution of (Q) if...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a local solution of (P) if ... x ∈ E is a local solution of (Q) if...

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... Existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... Existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... Existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... Existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... Existence

Chapter 2: Exercise

Consider (P) max x∈C

ex . x

Find value, solution, maximizing sequence when C = [1, 2], when C = [0, 10], when C = [0, +∞[. Consider (Q) min x∈C

ex . x

Find value, solution, local solution, when C = (−∞, 0) ∪ (0, +∞).

Chapter 2: Exercise

Consider (P) max x∈C

ex . x

Find value, solution, maximizing sequence when C = [1, 2], when C = [0, 10], when C = [0, +∞[. Consider (Q) min x∈C

ex . x

Find value, solution, local solution, when C = (−∞, 0) ∪ (0, +∞).

Chapter 2: Exercise

Consider (P) max x∈C

ex . x

Find value, solution, maximizing sequence when C = [1, 2], when C = [0, 10], when C = [0, +∞[. Consider (Q) min x∈C

ex . x

Find value, solution, local solution, when C = (−∞, 0) ∪ (0, +∞).

Chapter 2: Exercise

Consider (P) max x∈C

ex . x

Find value, solution, maximizing sequence when C = [1, 2], when C = [0, 10], when C = [0, +∞[. Consider (Q) min x∈C

ex . x

Find value, solution, local solution, when C = (−∞, 0) ∪ (0, +∞).

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to now if (P) has a value, i.e. if there exists a solution of (P) and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to now if (P) has a value, i.e. if there exists a solution of (P) and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to now if (P) has a value, i.e. if there exists a solution of (P) and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to now if (P) has a value, i.e. if there exists a solution of (P) and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, ...see chapter 3. When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. See chapter 4. Thus it is often interesting to compute the set of critical points of f to have candidate to be solution of an optimization problem. Under some convexity assumption (of the domain, of the function), we can get the converse statement: In chapter 5, important properties of convexity.

Chapter 2: Optimization (vocabulary) Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, ...see chapter 3. When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. See chapter 4. Thus it is often interesting to compute the set of critical points of f to have candidate to be solution of an optimization problem. Under some convexity assumption (of the domain, of the function), we can get the converse statement: In chapter 5, important properties of convexity.

Chapter 2: Optimization (vocabulary) Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, ...see chapter 3. When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. See chapter 4. Thus it is often interesting to compute the set of critical points of f to have candidate to be solution of an optimization problem. Under some convexity assumption (of the domain, of the function), we can get the converse statement: In chapter 5, important properties of convexity.

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ...Characterization with sequences, subsequences... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology

Section 1: Open subset, closed subset, ... 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

Section 1: Open subset, closed subset, ... 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

Section 1: Open subset, closed subset, ... 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

Section 1: Open subset, closed subset, ... 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

Questions Find open subsets whose intersection is not open. Find A and B such that A ∩ B 6= A ∩ B. Find A such that ∂(int(A)) 6= ∂(A). Find A and B subsets of R such that A + B is not closed.

Chapter 3: Topology

Questions Find open subsets whose intersection is not open. Find A and B such that A ∩ B 6= A ∩ B. Find A such that ∂(int(A)) 6= ∂(A). Find A and B subsets of R such that A + B is not closed.

Chapter 3: Topology

Questions Find open subsets whose intersection is not open. Find A and B such that A ∩ B 6= A ∩ B. Find A such that ∂(int(A)) 6= ∂(A). Find A and B subsets of R such that A + B is not closed.

Chapter 3: Topology

Questions Find open subsets whose intersection is not open. Find A and B such that A ∩ B 6= A ∩ B. Find A such that ∂(int(A)) 6= ∂(A). Find A and B subsets of R such that A + B is not closed.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ Rn . A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ Rn . A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ Rn . A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ Rn . A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set.

Chapter 3: Topology

Section 2: Continuity C ⊂ Rn is pathwise connected if .... Bolzano theorem: If a continuous function defined on pathwise connected subset C ⊂ Rn is sometimes positive and sometimes negative, it must be 0 at some point. Particular case: Intermediate value theorem.

Chapter 3: Topology

Section 2: Continuity C ⊂ Rn is pathwise connected if .... Bolzano theorem: If a continuous function defined on pathwise connected subset C ⊂ Rn is sometimes positive and sometimes negative, it must be 0 at some point. Particular case: Intermediate value theorem.

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. upper semicontinuity; lower semicontinuity.

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity. upper semicontinuity; lower semicontinuity.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of {x : f (x) ≤ c}. Level sets. Interpretation of maximization problem with level sets.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of {x : f (x) ≤ c}. Level sets. Interpretation of maximization problem with level sets.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of {x : f (x) ≤ c}. Level sets. Interpretation of maximization problem with level sets.

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 , ... 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...

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 , ... 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...

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 , ... 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...

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 ) for f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives for 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 of f : C ⊂ Rn → R at x ∈ int(C). Directional derivative (in the direction u ∈ Rn ) for f : C ⊂ Rn → R at x ∈ int(C). Partial derivatives for 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 of f : C ⊂ Rn → R at x ∈ int(C). Hessian of f : C ⊂ Rn → R at x ∈ int(C). 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). Hessian of f : C ⊂ Rn → R at x ∈ int(C). 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). Hessian of f : C ⊂ Rn → R at x ∈ int(C). Second order approximation of a C 2 function f : C ⊂ Rn → R around x ∈ int(C).

Chapter 4: Differential calculus: reminders

Section 4: Example of Computation, 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: Example of Computation, 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, y ) 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, y ) 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, y ) 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, y ) in a unique way if and only if.... 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, y ) in a unique way if and only if.... 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, y ) in a unique way if and only if.... 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):

Chapter 4: Differential calculus: reminders

Section 4: Implicit function theorem: several variables Example2: f (x1 , ..., xp , xp+1 , ..., xn ) = 0 where f : Rn → Rp : a11 x1 + a12 x2 + ...a1p xp + ... + a1n xn = 0 a21 x1 + a22 x2 + ...a2p xp + ... + a2n xn = 0 ... ap1 x1 + ap2 x2 + ...app xp + ... + apn xn = 0 Possible to write (x1 , ..., xp ) = g(xp+1 , ..., xn ) in a unique way if and only if.... Implicit function theorem (several variables):

Chapter 4: Differential calculus: reminders

Section 4: Implicit function theorem: several variables Example2: f (x1 , ..., xp , xp+1 , ..., xn ) = 0 where f : Rn → Rp : a11 x1 + a12 x2 + ...a1p xp + ... + a1n xn = 0 a21 x1 + a22 x2 + ...a2p xp + ... + a2n xn = 0 ... ap1 x1 + ap2 x2 + ...app xp + ... + apn xn = 0 Possible to write (x1 , ..., xp ) = g(xp+1 , ..., xn ) in a unique way if and only if.... Implicit function theorem (several variables):

Chapter 4: Differential calculus: reminders

Section 4: Implicit function theorem: several variables Example2: f (x1 , ..., xp , xp+1 , ..., xn ) = 0 where f : Rn → Rp : a11 x1 + a12 x2 + ...a1p xp + ... + a1n xn = 0 a21 x1 + a22 x2 + ...a2p xp + ... + a2n xn = 0 ... ap1 x1 + ap2 x2 + ...app xp + ... + apn xn = 0 Possible to write (x1 , ..., xp ) = g(xp+1 , ..., xn ) in a unique way if and only if.... Implicit function theorem (several variables):

Question of the week 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 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.