Question 4 only. Thank you! (1) Suppose that vec(x)inR^(n) and vec(x)=0. Suppose that vec(a),vec(b)inR^(n) and vec(a)=vec(b). Show that vec(x)*vec(a)=vec(x)*vec(b) (2) Explain why vec(x)*(A^(T)(vec(y)))=(Avec(x))*vec(y). Hint: Use that that vec(x)*vec(y)=vec(x)^(T)vec(y) (where the far right quantity is matrix multiplication) and the solutionspile.com

Question

Question 4 only. Thank you! (1) Suppose that

vec(x)inR^(n)

and

vec(x)>=0

. Suppose that

vec(a),vec(b)inR^(n)

and

vec(a)>=vec(b)

. Show that

vec(x)*vec(a)>=vec(x)*vec(b)

(2) Explain why

vec(x)*(A^(T)(vec(y)))=(Avec(x))*vec(y)

. Hint: Use that that

vec(x)*vec(y)=vec(x)^(T)vec(y)

(where the far right quantity is matrix multiplication) and the fact that

(AB)^(T)=B^(T)A^(T)

. (3) Let

A

be an

m\times n

matrix,

vec(c)inR^(n)

and

vec(b)inR^(m)

. Suppose that

Avec(x)<=vec(b)

and

vec(x)>=0

and

A^(T)vec(y)>=vec(c)

and

vec(y)>=0

. Show that

vec(x)*vec(c)<=vec(y)*vec(b)

WITHOUT using the duality theorem. Hint: Write

A^(T)vec(y)>=vec(c)

and use part (1) and part (2) above. (4) Let

vec(x_(0)),vec(y_(0))

satisfy the hypothesis of part 3 . Suppose that

vec(x_(0))*vec(c)=vec(y_(0))*vec(b)

Show that

vec(x_(0))

and

vec(y_(0))

are optimal solutions to the following problems

 maximize vec(x)*vec(c) subject to Avec(x)<=vec(b),vec(x)>=0
 minimize vec(y)*vec(b) subject to A^(T)vec(y)>=vec(c),vec(y)>=0

Accepted Answer

Expert Answer to -  Question 4 only. Thank you!  (1) Suppose that vec(x)inR^(n) and vec(x)>=0. Suppose that vec(a),v

Answer

Solution for -  Question 4 only. Thank you!  (1) Suppose that vec(x)inR^(n) and vec(x)>=0. Suppose that vec(a),v

Answer

This an additional answer to -  Question 4 only. Thank you!  (1) Suppose that vec(x)inR^(n) and vec(x)>=0. Suppose that vec(a),v

(Solved): Question 4 only. Thank you! (1) Suppose that vec(x)inR^(n) and vec(x)>=0. Suppose that vec(a),v ...

View Expert Answer

Expert Answer

Buy This Answer $5

Place Order

We Provide Services Across The Globe