Instructions: On a separate sheet of paper, answer each of…
Instructions: On a separate sheet of paper, answer each of 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) Consider the function f(x1,x2)=2x1x2+1+x2{“version”:”1.1″,”math”:”f(x_1, x_2) = \frac{2x_1}{x_2+1} + x_2 “} and the point x(0)=⊤{“version”:”1.1″,”math”:”\( x^{(0)}=\left^{\top}\)”}. 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=⊤{“version”:”1.1″,”math”:”\( d=\left^{\top}\)”} a direction of ascent 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)=⊤{“version”:”1.1″,”math”:”\( x^{(0)}=\left^{\top}\)”} or not? Justify your answer. If yes, then why? If not, then why not? Problem 3. (10 pts) Consider the function f(x)=(Ax)⊤(Bx),{“version”:”1.1″,”math”:” f(x)=(A x)^{\top}(B x), “} where A={“version”:”1.1″,”math”:”\( A=\left\)”}, B={“version”:”1.1″,”math”:”\( B=\left\)”} , and x=⊤{“version”:”1.1″,”math”:”\( x=\left^\top\)”}. (5 pts) Find ∇f(x){“version”:”1.1″,”math”:”\( \nabla f(x)\)”}. (5 pts) Find the Hessian F(x){“version”:”1.1″,”math”:”\( F( x)\)”} of f(x){“version”:”1.1″,”math”:”\(f(x)\)”}. Problem 4. (20 pts) Given the following function, f=f(x1,x2)=ex2cosx1−ex1cosx2.{“version”:”1.1″,”math”:”f=f(x_1,x_2)=e^{x_2}\cos x_1-e^{x_1}\cos x_2. “} (10 pts) In what direction does the function f{“version”:”1.1″,”math”:”\( f\)”} increase most rapidly at the point x(0)=⊤{“version”:”1.1″,”math”:”\( x^{(0)}=\left^{\top}\)”}? (10 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”:”\(f\)”}? 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 {“version”:”1.1″,”math”:”\(\left\)”}. 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) minimize‖x+x0‖2subject tox=1,{“version”:”1.1″,”math”:”\begin{eqnarray*} \mbox{minimize}&{}& \|x+x_0\|_2\\ \mbox{subject to}&{}&{}\\ {}&{}& \leftx=1, \end{eqnarray*}”} where x0=⊤.{“version”:”1.1″,”math”:”x_0=\left^{\top}. “} Problem 7. (20 pts) (10 pts) Convert the optimization problem, minimize|x1|+|x2|+|x3|subject to=,{“version”:”1.1″,”math”:”\begin{eqnarray*} \mbox{minimize}&{}& |x_1|+|x_2|+|x_3|\\ \mbox{subject to}&{}& {}\\ &{}& \left\left=\left, \end{eqnarray*}”} into a linear programming problem and solve it. Hint: Introduce two sets of non-negative variables: xi+≥0{“version”:”1.1″,”math”:”\(x_i^+\ge 0\)”} and xi−≥0,{“version”:”1.1″,”math”:”\( x_i^- \ge 0, \) “} i=1,2,3.{“version”:”1.1″,”math”:”\( i=1,2,3. \)”} Then represent the optimization problem using the above variables. Only one xi+{“version”:”1.1″,”math”:”\( x_i^+ \)”} and xi−{“version”:”1.1″,”math”:”\( x_i^- \)”} can be non-zero at a time. If xi≥0{“version”:”1.1″,”math”:”\( x_i \ge 0 \)”} then xi=xi+{“version”:”1.1″,”math”:”\( x_i=x_i^+ \)”} and xi−=0.{“version”:”1.1″,”math”:”\( x_i^- =0. \)”} On the other hand, if xi