Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufﬁcient conditions for global max or

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


Critical point: sufficient conditions for local max.


(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


Critical point: sufficient conditions for local min.


(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


Critical point: sufficient conditions for local min.


(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


We have partial converse statement


(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




(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




(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




(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


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








Philippe Bich








Philippe Bich








Philippe Bich








Philippe Bich


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


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


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


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


Hessian and convexity: Questions

if f (x, y) = 2xy− 2x2 − y2 − 8x + 6y + 4 convex ? concave ? strictlyconvex ? strictly concave ?

Philippe Bich


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




(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




(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




(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


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


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

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Chapter 6: Convexity, Concavity and optimization without ...Chapter 6: Convexity, Concavity and optimization without constraints What about sufﬁcient conditions for global max or