AAE550_hw3 - NELDERMEADSIMPLEX

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NELDER-MEAD SIMPLEX Use the Nelder-Mead Simplex to find the minimum of the 5-D version ( = 5) of the Ackley  Function: 1. Formulate an appropriate objective function for the Nelder-Mead Simplex to account  for the bounds on the design variables. Recall that function and / or slope continuity is  not a requirement when developing your penalty function. Pseudo objective function:      In our case we will use the exterior penalty function with  Results: Pseudo objective function: And  10                     2. Solve the problem starting from x0 = [5 5 5 5 5] T . Be sure to use the first x* as x0 for a  re-start of the algorithm. Summarize these runs in a table like the one shown below. X0 Functions  evaluation x* f(x*) iteration exitflag Run1 5 222 4.9862144 8 12.632269 3 131 1 5 4.9861381 3 5 4.9862555 6 5 4.9862329 5 5 4.9863011
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AAE 550 HW#3 2 Run2 4.9862144 8 167 4.9861589 4 12.632269 104 1 4.9861381 3 4.9862161 5 4.9862555 6 4.9861597 8 4.9862329 5 4.9861585 1 4.9863011 2 4.9861674 1 3. Solve the problem again starting from x0 = [0.5 0.5 0.5 0.5 0.5]T. As before, use the first  x* as x0 for a re-start of the algorithm. Summarize these runs in a table like the one  shown below. X0 Functions  evaluation x* f(x*) iteration exitflag Run1 0.5 301 0.91748864 1.64622371 183 1 0.5 5.8437E-05 0.5 -1.8011E- 05 0.5 3.6801E-05 0.5 -8.6266E- 06 Run2 0.91748864 24 0.91757823 1.6462237    10 1 5.8437E-05 5.902E-05 -1.8011E- 05 -1.8191E- 05 3.6801E-05 3.7168E-05 -8.6266E- 06 -8.7127E- 06
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AAE 550 HW#3
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4. Solve the problem again starting from x0 = [-0.1 -0.1 -0.1 -0.1 -0.1]T. Be sure to use the  first x* as x0 for a restart of the algorithm. Summarize these runs in a table like the one  shown below. X0 Functions  evaluation x*e-4 f(x*) iteratio n exitfla g Run 1 -0.1 250 - 0.12752068 9.66E- 05 158 1 -0.1 0.01181752 -0.1 - 0.44888641 -0.1 0.2530493 -0.1 - 0.09697103 Run 2 - 0.12752068 6 - 0.12752068 9.66E- 05 1 1 0.01181752 0.01181752 - 0.44888641 - 0.44888641 0.2530493 0.2530493 - 0.09697103 - 0.09697103 5. Recall that Nelder-Mead Simplex is generally better at non-smooth optimization than it  is at global optimization. Do your results indicate this? The Nelder-Mead Simplex will be ineffective when the objective function has many local  minimum. In our results, when we chose [-0.1 -0.1 -0.1 -0.1 -0.1] T  as an initial value, we got a  value very close to the global minimum. However, for  ? 0 =[5 5 5 5 5] T  and  ? 0=[0.5 0.5 0.5 0.5  0.5] T , we find local minima. Therefore our results indicate that Nelder-Mead Simplex is generally  better at non-smooth optimization than it is at global optimization because we were only able to  obtain local minimum. Matlab Code:
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This note was uploaded on 02/14/2012 for the course IE 530 taught by Professor Ravindran during the Spring '97 term at Purdue University-West Lafayette.

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AAE550_hw3 - NELDERMEADSIMPLEX

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