Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and...
Transcript of Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and...
![Page 1: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/1.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Critical point: sufficient conditions for local max.
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) supx∈C
f (x)
Assume that x̄ is interior to C, that f is C2 at x̄, that x̄ is a critical point of f .If Hessx̄f is negative definite, then x̄ is a local solution of (P).
Philippe Bich
![Page 2: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/2.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Critical point: sufficient conditions for local max.
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) supx∈C
f (x)
Assume that x̄ is interior to C, that f is C2 at x̄, that x̄ is a critical point of f .If Hessx̄f is negative definite, then x̄ is a local solution of (P).
Philippe Bich
![Page 3: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/3.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Critical point: sufficient conditions for local min.
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) infx∈C
f (x)
Assume f is C2 at x̄, that x̄ is a critical point of f , and x̄ is interior to C. IfHessx̄f is positive definite, then x̄ is a local solution of (P).
see: Further MATHEMATICS FOR Economic Analysis, section 3.2.
Philippe Bich
![Page 4: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/4.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Critical point: sufficient conditions for local min.
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) infx∈C
f (x)
Assume f is C2 at x̄, that x̄ is a critical point of f , and x̄ is interior to C. IfHessx̄f is positive definite, then x̄ is a local solution of (P).
see: Further MATHEMATICS FOR Economic Analysis, section 3.2.
Philippe Bich
![Page 5: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/5.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
We have partial converse statement
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) supx∈C
f (x)
Assume f is C2 at x̄, that x̄ is a local solution of (P) interior to C. Then x̄ isa critical point of f , and Hessx̄f is negative semidefinite.
Philippe Bich
![Page 6: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/6.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
We have partial converse statement
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) supx∈C
f (x)
Assume f is C2 at x̄, that x̄ is a local solution of (P) interior to C. Then x̄ isa critical point of f , and Hessx̄f is negative semidefinite.
Philippe Bich
![Page 7: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/7.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
We have partial converse statement
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) infx∈C
f (x)
Assume f is C2 at x̄, that x̄ is a local solution of (P) interior to C. Then x̄ isa critical point of f , and Hessx̄f is positive semidefinite.
Philippe Bich
![Page 8: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/8.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
We have partial converse statement
TheoremConsider C a subset of Rn, a function f : Rn → R and consider theproblem
(P) infx∈C
f (x)
Assume f is C2 at x̄, that x̄ is a local solution of (P) interior to C. Then x̄ isa critical point of f , and Hessx̄f is positive semidefinite.
Philippe Bich
![Page 9: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/9.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
What about sufficient conditions for global max or global min ?
the function x2 has one critical point which is a global minimum.
This function is convex...
Is it a hazard ? No!
We will see that convexity and concavity play an important role toguarantee a critical point is a global solution.
But les us try to find criteria to say a function is convex, concave, ...
Philippe Bich
![Page 10: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/10.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
What about sufficient conditions for global max or global min ?
the function x2 has one critical point which is a global minimum.
This function is convex...
Is it a hazard ? No!
We will see that convexity and concavity play an important role toguarantee a critical point is a global solution.
But les us try to find criteria to say a function is convex, concave, ...
Philippe Bich
![Page 11: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/11.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
What about sufficient conditions for global max or global min ?
the function x2 has one critical point which is a global minimum.
This function is convex...
Is it a hazard ? No!
We will see that convexity and concavity play an important role toguarantee a critical point is a global solution.
But les us try to find criteria to say a function is convex, concave, ...
Philippe Bich
![Page 12: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/12.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
What about sufficient conditions for global max or global min ?
the function x2 has one critical point which is a global minimum.
This function is convex...
Is it a hazard ? No!
We will see that convexity and concavity play an important role toguarantee a critical point is a global solution.
But les us try to find criteria to say a function is convex, concave, ...
Philippe Bich
![Page 13: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/13.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
What about sufficient conditions for global max or global min ?
the function x2 has one critical point which is a global minimum.
This function is convex...
Is it a hazard ? No!
We will see that convexity and concavity play an important role toguarantee a critical point is a global solution.
But les us try to find criteria to say a function is convex, concave, ...
Philippe Bich
![Page 14: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/14.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
The sign of Hessian is a possible criterium for convexity
Equivalent condition for a C2 function f : U ⊂ Rn → R to be convex(U convex open): Hessx(f ) is positive semidefinite at every x ∈ U ifand only if f is convex.
Equivalent condition for a C2 function f : U ⊂ Rn → R to beconcave (U convex open): Hessx(f ) is negative semidefinite at everyx ∈ U if and only if f is concave.
See Section 2.3. in Further MATHEMATICS FOR Economic Analysis.
Philippe Bich
![Page 15: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/15.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
The sign of Hessian is a possible criterium for convexity
Equivalent condition for a C2 function f : U ⊂ Rn → R to be convex(U convex open): Hessx(f ) is positive semidefinite at every x ∈ U ifand only if f is convex.
Equivalent condition for a C2 function f : U ⊂ Rn → R to beconcave (U convex open): Hessx(f ) is negative semidefinite at everyx ∈ U if and only if f is concave.
See Section 2.3. in Further MATHEMATICS FOR Economic Analysis.
Philippe Bich
![Page 16: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/16.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Hessian and convexity
Sufficient condition for a C2 function f : U ⊂ Rn → R to be convex(U convex open): if Hessx(f ) is positive definite at every x ∈ U, thenf is strictly convex on U.
Sufficient condition for a C2 function f : U ⊂ Rn → R to be convex(U convex open): if Hessx(f ) is negative definite at every x ∈ U, thenf is strictly concave on U.
Philippe Bich
![Page 17: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/17.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Hessian and convexity
Sufficient condition for a C2 function f : U ⊂ Rn → R to be convex(U convex open): if Hessx(f ) is positive definite at every x ∈ U, thenf is strictly convex on U.
Sufficient condition for a C2 function f : U ⊂ Rn → R to be convex(U convex open): if Hessx(f ) is negative definite at every x ∈ U, thenf is strictly concave on U.
Philippe Bich
![Page 18: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/18.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Hessian and convexity: Questions
if f (x, y) = 2xy− 2x2 − y2 − 8x + 6y + 4 convex ? concave ? strictlyconvex ? strictly concave ?
Philippe Bich
![Page 19: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/19.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Now, we are ready for:Critical point: sufficient conditions for global max.
TheoremConsider U an open convex subset of Rn, a function f : Rn → R andconsider the problem
(P) infx∈U
f (x)
Assume f is convex on U. Then if x̄ is a critical point of f , it is a globalsolution of (P).
Philippe Bich
![Page 20: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/20.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Now, we are ready for:Critical point: sufficient conditions for global max.
TheoremConsider U an open convex subset of Rn, a function f : Rn → R andconsider the problem
(P) infx∈U
f (x)
Assume f is convex on U. Then if x̄ is a critical point of f , it is a globalsolution of (P).
Philippe Bich
![Page 21: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/21.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Now, we are ready for:Critical point: sufficient conditions for global max.
TheoremConsider U an open convex subset of Rn, a function f : Rn → R andconsider the problem
(P) supx∈U
f (x)
Assume f is concave on U. Then if x̄ is a critical point of f , it is a globalsolution of (P).
Philippe Bich
![Page 22: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/22.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
Now, we are ready for:Critical point: sufficient conditions for global max.
TheoremConsider U an open convex subset of Rn, a function f : Rn → R andconsider the problem
(P) supx∈U
f (x)
Assume f is concave on U. Then if x̄ is a critical point of f , it is a globalsolution of (P).
Philippe Bich
![Page 23: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/23.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
How to prove it ? Rests on the following theoremTheorem (Inequality of convexity) Assume f is C1 on some convex setU ⊂ Rn. Then the function f is convex if and only if for all(x, x′) ∈ U × U,
f (x′) ≥ f (x)+ < ∇f (x), x′ − x >
Philippe Bich
![Page 24: Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufficient conditions for global max or](https://reader030.fdocuments.in/reader030/viewer/2022040213/5e97d78db43ea4385a78a0e3/html5/thumbnails/24.jpg)
Chapter 6: Convexity, Concavity and optimization withoutconstraints
How to prove it ? Rests on the following theoremTheorem (Inequality of convexity) Assume f is C1 on some convex setU ⊂ Rn. Then the function f is convex if and only if for all(x, x′) ∈ U × U,
f (x′) ≥ f (x)+ < ∇f (x), x′ − x >
Philippe Bich