Which of the following units would be most likely to be elim…

Questions

Which оf the fоllоwing units would be most likely to be eliminаted due to budget cuts?

If weаlth increаses, the demаnd fоr stоcks [blank] and that оf long-term bonds [blank], everything else held constant.

Instructiоns: On а sepаrаte sheet оf paper, answer each оf the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (10 pts) For the function f ( x 1 , x 2 ) = 1 3 x 1 3 − 4 x 1 + 1 3 x 2 3 − 16 x 2 . {"version":"1.1","math":"f(x_1,x_2)=frac{1}{3}x_1^3-4x_1+frac{1}{3}x_2^3-16x_2."} and the point x ( 0 ) = [ 2 4 ] ⊤ {"version":"1.1","math":"( x^{(0)}=left[begin{array}{cc} 2 & 4 end{array}right]^{top})"}, construct (5 pts) a linear approximation, l(x1,x2){"version":"1.1","math":"( l(x_1, x_2))"}, of f(x1,x2){"version":"1.1","math":"(f(x_1, x_2))"} at x(0){"version":"1.1","math":"( x^{(0)})"}; (5 pts) a quadratic approximation, q(x1,x2){"version":"1.1","math":"( q(x_1, x_2) )"}, of f(x1,x2){"version":"1.1","math":"(f(x_1, x_2))"} at x(0){"version":"1.1","math":"( x^{(0)})"}. Problem 2. (10 pts) Is d = [ − 2 1 ] ⊤ {"version":"1.1","math":"( d=left[begin{array}{cc} -2 & 1 end{array}right]^{top})"} a direction of descent of f(x1,x2)=2x1x2+1+x2{"version":"1.1","math":"f(x_1, x_2)=frac{2x_1}{x_2+1} + x_2 "} at the point x(0)=[01]⊤{"version":"1.1","math":"( x^{(0)}=left[begin{array}{cc} 0 & 1 end{array}right]^{top})"} or not? Justify your answer. If yes, then why? If not, then why not? Problem 3. (10 pts) Let A = [ 4 − 2 0 2 1 − 1 ] . {"version":"1.1","math":"[A = begin{bmatrix}4 & -2 & 0\2 & 1 & -1end{bmatrix}. ]"} (5 pts) Find the nullspace of A {"version":"1.1","math":"( A)"}. (5 pts) Find the nullspace of A ⊤ {"version":"1.1","math":"( A^top )"}. Problem 4. (20 pts) Given the following function, f = f ( x 1 , x 2 ) = x 1 2 x 2 + x 1 x 2 3 . {"version":"1.1","math":"f=f(x_1,x_2)=x_1^2x_2+x_1x_2^3. "} (5 pts) In what direction does the function f{"version":"1.1","math":"( f)"} increase most rapidly at the point x ( 0 ) = [ 2 1 ] ⊤ {"version":"1.1","math":"( x^{(0)}=left[begin{array}{cc} 2 & 1 end{array}right]^{top})"}? (5 pts) What is the rate of increase of f{"version":"1.1","math":"(f )"} at the point x(0){"version":"1.1","math":"( x^{(0)})"} in the direction of maximum increase of f{"version":"1.1","math":"(f)"}? (10 pts) Find the rate of increase of f {"version":"1.1","math":"(f)"} at the point x ( 0 ) {"version":"1.1","math":"( x^{(0)} )"} in the direction d = [ 3 4 ] ⊤ {"version":"1.1","math":"( d=begin{bmatrix} 3 & 4 end{bmatrix}^top )"}. Problem 5. (15 pts) (5 pts) Find the factor by which the uncertainty range is reduced when using the Fibonacci method. Assume that the last step has the form 1−ρN−1=F2F3=23,{"version":"1.1","math":"1-rho_{N-1}=frac{F_2}{F_3}=frac{2}{3}, "} where N−1{"version":"1.1","math":"( N-1)"} is the number of steps performed in the uncertainty range reduction process. (10 pts) It is known that the minimizer of a certain unimodal function f(x){"version":"1.1","math":"( f(x))"} is located in the interval [ − 5 , 10 ] {"version":"1.1","math":"(left[begin{array}{cc} -5,& 10 end{array}right])"}. What is the minimal number of iterations of the Fibonacci method required to box in the minimizer within the range 1.0 {"version":"1.1","math":"(1.0)"}? Assume that the last useful value of the factor reducing the uncertainty range is 2/3{"version":"1.1","math":"( 2/3)"}, that is, 1−ρN−1=F2F3=23.{"version":"1.1","math":"1-rho_{N-1}=frac{F_2}{F_3}=frac{2}{3}. "} Problem 6. (15 pts) Find strict local minimizers and maximizers of the function f ( x 1 , x 2 ) = 1 2 x 1 2 − x 1 + 1 3 x 2 3 − 4 x 2 . {"version":"1.1","math":"f(x_1,x_2)=frac{1}{2}x_1^2-x_1+frac{1}{3}x_2^3-4x_2."} Problem 7. (20 pts) Consider the following optimization problem: maximize − | x 1 | − | x 2 | − | x 3 | subject to [ 1 1 − 1 0 − 1 0 ] [ x 1 x 2 x 3 ] = [ 2 1 ] . {"version":"1.1","math":"begin{eqnarray} mbox{maximize}&{}& -|x_1|-|x_2|-|x_3|\ mbox{subject to}&{}& {}\ &{}& left[begin{array}{ccc} 1 & 1 & -1\ 0 & -1 & 0 end{array}right]left[begin{array}{c} x_1\ x_2\ x_3 end{array}right]=left[begin{array}{c} 2\ 1 end{array}right]. end{eqnarray}"} (10 pts) Convert this problem into a linear programming problem and solve it; (10 pts) Construct its dual program and solve it. Problem 8. (20 pts) Use a simplex method to solve the problem, maximize x 1 + 2 x 2 subject to x 1 − x 2 + x 3 = − 2 x 1 + x 2 + x 4 = 6 x 1 , x 2 , x 3 , x 4 ≥ 0. {"version":"1.1","math":"[ begin{eqnarray} mbox{maximize}&{}& quad x_1 + 2 x_2\ mbox{subject to}&{}& quad x_1 - x_2 +x_3= -2\ &{}& quad x_1 + x_2 +x_4 = 6\ &{}& quad x_1, x_2,x_3,x_4ge 0. end{eqnarray} ]"} Problem 9. (20 pts) Consider the following model of a discrete-time system, x [ k + 1 ] = 2 x [ k ] + u [ k ] , x [ 0 ] = 10.5 , 0 ≤ k ≤ 2 . {"version":"1.1","math":"x[k+1]=2x[k]+u[k],quad x[0]=10.5,quad 0le kle 2 . "} Use the Lagrange multiplier approach to calculate the optimal control sequence {u[0],u[1],u[2]}{"version":"1.1","math":"{u[0], u[1], u[2] } "} that transfers the initial state x[0]{"version":"1.1","math":"( x[0])"} to x [ 3 ] = 0 {"version":"1.1","math":"(x[3]=0)"} while minimizing the performance index J=12∑k=02u[k]2.{"version":"1.1","math":"J=frac{1}{2}sum_{k=0}^2 u[k]^2 . "} Problem 10. (20 pts) The point x ∗ = [ − 5 74 − 5 37 155 74 30 37 ] ⊤ {"version":"1.1","math":"[ x^=left[begin{array}{cccc} -frac{5}{74} & -frac{5}{37} & frac{155}{74} & frac{30}{37} end{array}right]^{top} ]"} satisfies the first-order necessary condition to be an extremizer, that is, either a minimizer or a maximizer, of the following optimization problem: extremize 1 2 ( x 1 2 + x 2 2 + x 3 2 + x 4 2 ) subject to x 1 + 2 x 2 + 3 x 3 + 5 x 4 = 10 x 1 + 2 x 2 + 5 x 3 + 6 x 4 = 15. {"version":"1.1","math":"begin{eqnarray} mbox{extremize}&{}& frac{1}{2}left(x_1^2 + x_2^2 + x_3^2 + x_4^2 right)\ mbox{subject to}&{}& x_1+2x_2+3x_3+5x_4=10\ &{}& x_1+2x_2+ 5x_3 + 6x_4=15. end{eqnarray}"} Is the point x ∗ {"version":"1.1","math":"(x^)"} a relative minimizer, strict relative minimizer, relative maximizer, strict relative maximizer, or neither? Justify your answer. Problem 11. (20 pts) Consider the optimization problem: extremize ( x 1 − 2 ) 2 + ( x 2 − 1 ) 2 subject to x 2 − x 1 2 ≥ 0 2 − x 1 − x 2 ≥ 0 x 1 ≥ 0. {"version":"1.1","math":"begin{eqnarray} mbox{extremize}&{}&quad (x_1-2)^2 + (x_2-1)^2\ mbox{subject to}&{}&quad x_2 - x_1^2 ge 0\ &{}& quad 2-x_1-x_2 ge 0\&{}&quad x_1 ge 0. end{eqnarray}"} The point x ∗ = [ 0 0 ] ⊤ {"version":"1.1","math":"( x^=begin{bmatrix} 0 & 0 end{bmatrix}^top )"}satisfies the KKT conditions. (10 pts) Does x ∗ {"version":"1.1","math":"( x^ )"} satisfy the FONC for minimum or maximum? What are the KKT multipliers? (10 pts) Does x ∗ {"version":"1.1","math":"( x^*)"} satisfy the SOSC? Carefully justify your answer. Problem 12. (20 pts) (10 pts) For what values of the parameter α∈R{"version":"1.1","math":"( alpha in mathbb{R})"} the function, f ( x 1 , x 2 , x 3 ) = x 1 2 + x 2 2 + 2 x 3 2 − 2 x 1 x 3 + 2 α x 2 x 3 , {"version":"1.1","math":"[ f(x_1,x_2,x_3)=x_1^2+x_2^2 + 2x_3^2-2x_1x_3+ 2alpha x_2x_3 , ]"} is convex? (10 pts) Determine the range of x∈R{"version":"1.1","math":"( x in mathbb{R} )"} for which the function f ( x ) = 2 x e − 3 x {"version":"1.1","math":"f(x)= 2xe^{-3x} "} is concave. *** Congratulations, you are almost done with Final Exam. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope: Final Exam Click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.