1) 6.19. a) An amphibian vehicle needs to travel from point A(on land) to point B (in water), as shown in the figure below. The speed at which the vehicle travels on land and in water are v1 and v2respectively. Suppose the vehicle traverses a path that minimizes the total time taken to travel from A to B. Using classical optimization technique ,show that for the optimal path above ,tha angles 1 and 2 in the below satisfy Snells Law: sin 1/sin 2 = v1/v2. b) solve the following using two phase (phase I and phase II) Simplex Algorithm maximize f(x)=4X1 + 3X2 , subject to 3X1 + 4X2 12, 3X1 + 3X2 10, 4X1 + 2X2 8, X1 + X2 1, X1, X2 0. 2) Consider the following NLP problem Maximize Z = a1 X1 + X2, subject to G1(X): X1^2 + X2^2 25 G2(X): X1 -X2 1 G3(X): X10 G4(X): X20 a) Write down the KKT conditions b) Assuming G1(X) and G2(X) are active at the optimal point, find a candidate optimal solution. c) Determine the range of values of a1 for which the above solution become the optimal solution. 3) Consider the following problem problem: Minimize f(X1,X2) = X1^2 + X2^2, subject to H(X) : X1^2 = X2, G(X) : X1 + 2(X2) 2 a) First solve it using graphical method b) Suppose we wish to solve it using a penalty function approach. First define an augmented cost function, P(X,R), by introducing suitable penalty function for violation of constraints. Then minimize P(X,R) as an unconstrained optimization problem. Show how the solution varies as a function of R. Please explain how the solution depends on R and how R should be chosen to reach the true constrained minimum