chapter 6: convexity, concavity and optimization without ...chapter 6: convexity, concavity and...
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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