(Unconstrained Optimization-Two Variables) Consider the function: f(x1, x2) = 4x1x2 − (x1)2x2 − x1(x2)2 Find a...
(Unconstrained Optimization-Two Variables)
Consider the function: f(x1, x2) = 4x1x2 − (x1)2x2 − x1(x2)2
Find a local maximum. Note that you should find 4 points that satisfy First Order Condition for maximization, but only one of them satisfies Second Order Condition for maximization.
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(Unconstrained Optimization-Two Variables)
Consider the function: f(x1, x2) =
4x1x2 −
(x1)2x2 −
x1(x2)2
Find a...
Question 3: Math Review-(Unconstrained) Optimization To find the value of x that maximizes f(x), first find...
Question 3: Math Review-(Unconstrained) Optimization To find the value of x that maximizes f(x), first find the values of z that satisfies f,(z) 0 and f"(x) <0. If there is only one such value, that is the value of r that maximizes f(x). If there are many values of z, say x1,T2, ,ZA, that satisfy those conditions, compare the values f(x), f(r2),..../(xN) and find out which r among i, r2,... .xv gives the global maximum ชา 1. Find out the...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2 - 6x1x2 – 18x1 X1 X2 Subject to 4x1 – 3x2 s 20 X1 + 2x2 < 10 -X1 < 0, - x2 < 0 a) Is this problem convex? Justify your answer. (5 Points) b) Form the Lagrange function. (5 Points) c) Formulate KKT conditions. (10 Points) d) Recall that one technique for finding roots of KKT condition is to check all permutations...
U = 8x10.5+ 2x2, where x1 is the quantity of good 1 consumed, and x2 is...
U = 8x10.5+ 2x2, where x1 is the quantity of good 1 consumed, and x2 is the quantity of good 2 consumed. (Yes the x is raised) 8x1.5 Suppose that the consumer has a budget of M = $400 to spend and that good 1 has a price of p1= 2, and good 2 has a price of p2= 8. Answer the following questions, and write your answers in the Answer Sheet. Write the person’s budget constraint as an equation,...
1. Consider the problem minimize f (x1, x2) = x} + 2x3 – 21 – 4x2...
1. Consider the problem minimize f (x1, x2) = x} + 2x3 – 21 – 4x2 + 2. (a) (4 points) Find all of the points (21, x2)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (c) (2 points) Is there a global minimizer?
Consider the following. Xi' = 3x1 - 2x2 x1(0) = 3 xz' = 2x1 – 2x2,...
Consider the following. Xi' = 3x1 - 2x2 x1(0) = 3 xz' = 2x1 – 2x2, *2(0) = (a) Transform the given system into a single equation of second order by solving the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for X1. (Use xp1 for xı' and xpP1 for x1".) xpP1 – xP1 – 2x1 = 0 (b) Find X1 and x2 that also satisfy the initial conditions. *2(t) =
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12...
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12 + 2. (a) (4 points) Find all of the points (21,22)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (C) (2 points) Is there a global minimizer?
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12...
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12 + 2. (a) (4 points) Find all of the points (21,22)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (C) (2 points) Is there a global minimizer?
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12...
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12 + 2. (a) (4 points) Find all of the points (21,22)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (C) (2 points) Is there a global minimizer?
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x =...
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
8. [10 points) a. Consider the function f (x1, x2) = x1 - xż. Investigate convexity...
8. [10 points) a. Consider the function f (x1, x2) = x1 - xż. Investigate convexity of this function in R2. What can you say about minimum and maximum values of the function and its behavior at the origin. b. Consider the function f(x1, x2) = x1+xz over the domain C = = {x € R2 : || xt||1 < 1}. Find the maximum of the function f over C.
Question 3: Math Review-(Unconstrained) Optimization To find the value of x that maximizes f(x), first find the values of z that satisfies f,(z) 0 and f"(x) <0. If there is only one such value, that is the value of r that maximizes f(x). If there are many values of z, say x1,T2, ,ZA, that satisfy those conditions, compare the values f(x), f(r2),..../(xN) and find out which r among i, r2,... .xv gives the global maximum ชา 1. Find out the...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2 - 6x1x2 – 18x1 X1 X2 Subject to 4x1 – 3x2 s 20 X1 + 2x2 < 10 -X1 < 0, - x2 < 0 a) Is this problem convex? Justify your answer. (5 Points) b) Form the Lagrange function. (5 Points) c) Formulate KKT conditions. (10 Points) d) Recall that one technique for finding roots of KKT condition is to check all permutations...
1. Consider the problem minimize f (x1, x2) = x} + 2x3 – 21 – 4x2 + 2. (a) (4 points) Find all of the points (21, x2)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (c) (2 points) Is there a global minimizer?
Consider the following. Xi' = 3x1 - 2x2 x1(0) = 3 xz' = 2x1 – 2x2, *2(0) = (a) Transform the given system into a single equation of second order by solving the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for X1. (Use xp1 for xı' and xpP1 for x1".) xpP1 – xP1 – 2x1 = 0 (b) Find X1 and x2 that also satisfy the initial conditions. *2(t) =
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12 + 2. (a) (4 points) Find all of the points (21,22)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (C) (2 points) Is there a global minimizer?
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12 + 2. (a) (4 points) Find all of the points (21,22)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (C) (2 points) Is there a global minimizer?
+ 1. Consider the problem minimize f (x1, x2) = x;} +233 – -21 - 4.12 + 2. (a) (4 points) Find all of the points (21,22)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (C) (2 points) Is there a global minimizer?
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
8. [10 points) a. Consider the function f (x1, x2) = x1 - xż. Investigate convexity of this function in R2. What can you say about minimum and maximum values of the function and its behavior at the origin. b. Consider the function f(x1, x2) = x1+xz over the domain C = = {x € R2 : || xt||1 < 1}. Find the maximum of the function f over C.
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