MITESD_77S10_soln03 (1)

MITESD_77S10_soln03 (1) - 16.888/ESD 77 Multidisciplinary...

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16.888/ESD 77 Multidisciplinary System Design± Optimization:± Assignment 3 Part a) Solution ± Part A1 The absolute error between first and second derivative estimates vis-à-vis their analytical value at x =1 by suggested methods is shown in the figure below using log-log plots. For estimation of first derivatives, the forward difference is a first order method while central and complex step methods are second order. Therefore we expect reduced estimation error from the last two methods in general. Looking at the plots, the forward difference shows optimal stepsize for all three problems. Central difference method shows monotonic behavior for x 2 and the estimation error is essentially zero for complex step method for this function. This is the case because the function itself is of second order or quadratic. The error in central-difference method at very small stepsizes for this function is due to loss of precision in the subtraction step. Since there is no subtraction operation in complex step, the error remains essentially zero. For x 3 , both central difference and complex step methods shows optimal stepsizes, which one usually observes in more general cases. At very small stepsize, the error is primarily due to loss of precision and at much larger stepsize, the error is primarily due to approximation error.
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For the second derivative approximation, the absolute error for central-difference method grows at a much higher rate for very small stepsize while it levels off for complex step estimate. The optimal stepsize for second derivative approximation is much larger as compared to the same for first derivative approximation. For these three problems, a step size of 10 -6 seems adequate if for forward difference scheme is used for estimating first derivative. However a larger stepsize of 10 -4 or 10 -3 might be adequate for central difference or complex step schemes. For the second derivative, a step size of 10 -3 would be advisable for these problems. The choice of stepsize would also depend on the computational resources/time and the accuracy required. Part A2 The revenue from this flight is given by: J ( p , p , p ) = p D + p D + p D (1) 1 2 3 1 1 2 2 3 3 This function has to be maximized under the equality constraint (i.e., assuming all seats are occupied) or you can solve this as an inequality constraint as well, but one can observe that this would be active since this is the bounding constraint on the objective (e.g., to limit the revenue one needs to set a limit of the available capacity): 3 3 p i i a h = D i 150 = a e 150 = 0 (2) i i = 1 i = 1 Hence the formal problem statement can be written as: min J p . = 0 s t h p i 0 We can write the Lagrangian as follows by replacing the objective function by –J and converting into a minimization problem of (-J): L = − J + λ h (3) Applying KKT condition on the Lagrangian results in ( i , ) L p = 0 , which implies; p i = a i + , i (4) 3 p i i i a e a 150 = 0 (5) i = 1 2
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MITESD_77S10_soln03 (1) - 16.888/ESD 77 Multidisciplinary...

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