```Python
import numpy as np
from scipy.optimize import minimize

def complex_method(x):
var_a = sum(x**2 - 10 * np.cos(2 * np.pi * x))
var_b = sum(20 * np.log2(np.abs(x)) if x.all() != 0 else 0 for x in x)
var_c = sum(x//2 for x in x if x % 2 == 0)
return var_a + var_b - var_c

guess = np.array([100, 400, 300, -200, 500])

solution = minimize(complex_method, guess, method='BFGS',options={'gtol': 1e-6, 'disp': True})

print("Solution vector : ", solution.x)
print("Value of complex_method at solution vector : ", complex_method(solution.x))
```

Question

```Python
import numpy as np
from scipy.optimize import minimize

def complex_method(x):
    var_a = sum(x**2 - 10 * np.cos(2 * np.pi * x))
    var_b = sum(20 * np.log2(np.abs(x)) if x.all() != 0 else 0 for x in x)
    var_c = sum(x//2 for x in x if x % 2 == 0)
    return var_a + var_b - var_c

guess = np.array([100, 400, 300, -200, 500])

solution = minimize(complex_method, guess, method='BFGS',options={'gtol': 1e-6, 'disp': True})

print("Solution vector : ", solution.x)
print("Value of complex_method at solution vector : ", complex_method(solution.x))
```

Accepted Answer

## Complexity Estimation

### Time Complexity
1. The `sum(x**2 - 10 * np.cos(2 * np.pi * x))` computation is performed for each element in `x`. This operation has a time complexity of O(n), where n is the length of the input `x`.
2. The `sum(20 * np.log2(np.abs(x)) if x.all() != 0 else 0 for x in x)` computation is also performed for each element in `x`. This operation has a time complexity of O(n), where n is the length of the input `x`. The `np.log2` operation has a constant time complexity.
3. The `sum(x//2 for x in x if x % 2 == 0)` computation filters even elements in `x` and performs integer division by 2 for each filtered element. This operation has a time complexity of O(k), where k is the number of even elements in `x`.
4. The `minimize` function from `scipy.optimize` is called. The time complexity of this function depends on the optimization method used, but typically has a complexity of at least O(n^2), where n is the length of the input `x`.

Therefore, the overall time complexity of the `complex_method` function is O(n) + O(n) + O(k) + O(n^2), which can be simplified to O(n^2) since n >= k.

### Space Complexity
1. The `sum(x**2 - 10 * np.cos(2 * np.pi * x))` computation uses a temporary variable `var_a`. This variable has a space complexity of O(1), as it only holds a single scalar value.
2. The `sum(20 * np.log2(np.abs(x)) if x.all() != 0 else 0 for x in x)` computation does not require additional space beyond the input `x` and the temporary variables used within each generator expression.
3. The `sum(x//2 for x in x if x % 2 == 0)` computation uses a temporary variable `var_c`. This variable has a space complexity of O(1), as it only holds a single scalar value.
4. The `minimize` function may use additional space depending on the optimization method used, but in this case, it does not require any additional space beyond the input `x` and the temporary variables used during the optimization process.

Therefore, the overall space complexity of the `complex_method` function is O(1), since it does not require any additional space that grows with the size of the input.

### Potential Bottlenecks and Areas for Optimization
1. The `sum(x**2 - 10 * np.cos(2 * np.pi * x))` computation can be optimized by vectorizing the operations using NumPy functions. This can improve the performance by taking advantage of vectorized operations instead of looping over the elements of `x`.
2. The `sum(20 * np.log2(np.abs(x)) if x.all() != 0 else 0 for x in x)` computation can also be optimized by vectorizing the operations using NumPy functions. Additionally, it can be simplified by computing the logarithm only once for the entire array `x`.
3. The `sum(x//2 for x in x if x % 2 == 0)` computation can be optimized by using NumPy functions to perform the filtering and integer division operations. This can eliminate the need for a loop and improve the performance.
4. The choice of optimization method in the `minimize` function can have an impact on the performance. It may be worth exploring different methods to find the most efficient one for the specific problem at hand.

Overall, optimizing the vector computations and using NumPy functions instead of loops can significantly improve the performance of the `complex_method` function. Using a more efficient optimization method in the `minimize` function could also lead to faster convergence.

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