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Pant , Sridhar Krishnan. Efficient channel estimation algorithms for cooperative multiple-input multiple-output MIMO wireless communication networks Choo W. References Publications referenced by this paper. Practical Optimization. However, most practical optimization problems involve complex constraints.
A simple example of that is bound on the independent variable x. As we can see that this function is characterized by two minima, the result would be different if we only considered the positive values of x.
The code to search with bound is only slightly different from above. We could have had other complicated constraints in the problem. Suppose, we want the following conditions to be met along with the goal of finding the global minimum. Note, one of them is inequality and another is equality constraint.
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SciPy allows handling arbitrary constraints through the more generalized method optimize. The constraints have to be written in a Python dictionary following a particular syntax. The following code demonstrates the idea. After that, we can run the optimization by choosing a suitable method which supports constraints not all methods in the minimize function support constraint and bounds.
Furthermore, to use minimize we need to pass on an initial guess in the form of x0 argument. If we print the result, we see something different from the simple unconstrained optimization result.
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The optimization parameter success: False indicates it did not succeed in reaching the global minimum. The answer lies in the deep theory of the mathematical optimization and associated algorithm but it suffices to say that the initial guess played a big role. In general, a non-convex optimization problem has no mathematical guarantee to be solved successfully and the nature of our problem here is non-convex.
To know more about convexity of an optimization problem, see this video,. In general cases, we cannot do much. However, in this toy example, we already have the plot of the function and can eyeball the optimum solution. Therefore, we can just give a better initial guess to the algorithm.
What if we restrict the number of iterations performed by the algorithm? For demonstration purpose only, we severely limit the number of iteration to 3. The result is, as expected, not favorable. Note that the optimization came close to the global minimum, but did not quite reach it — of course, due to the fact that it was not allowed to iterate a sufficient number of times.
That is because of the fact that each iteration equates to computational and sometimes not computational but actual physical cost. This is a business aspect of the optimization process. In real life, we may not be able to run the optimization for a long period of time if the individual function evaluation costs significant resources. This kind of scenario arises when the optimization is done not involving simple mathematical evaluation but complex, time-consuming simulation or cost and labor intensive experimentation.
When each evaluation costs money or resources, then not only the choice of the algorithm but also the finer details become important to consider.
Although we considered all essential aspects of solving a standard optimization problem in the preceding sections, the example consisted of a simple single-variable, analytical function. But it does not have to be the case! SciPy methods work with any Python function — not necessarily a closed-form, single-dimensional mathematical function. Let us show an example with a multi-valued function. Often in a chemical or manufacturing process, multiple stochastic sub-processes are combined to give rise to a Gaussian mixture. It may be desirable to maximize the final resultant process output by choosing the optimum operating points in the individual sub-processes within certain process limits.
The trick is to use a vector as the input to the objective function and to make sure the objective function still returns a single scalar value. Also, because the optimization problem here is about maximization of the objective function, we need to change the sign and return the negative of the sum of the Gaussian functions as the result of the objective function. The same result['x'] stores the optimum setting of the individual processes as a vector. That is the only difference between optimizing a single-valued and a multivariate function is that we get back a vector instead of a scalar.
Needless to say that we can change the bounds here to reflect practical constraints. For example, if the sub-process settings can occupy only a certain range of values some must be positive, some must be negative, etc. Here, the solution is as follows. This is dictating to push the 3rd sub-process setting to the maximum possible value zero while adjusting the other two suitably. The constraints for multi-variate optimization are handled in a similar way as shown for the single-variable case.
For more detailed documentation and their usage, see the following links,. To be honest, there is no limit to the level of complexity you can push this approach as long as you can define a proper objective function which generates a scalar value and suitable bounds and constraints matching the actual problem scenario. The crux of almost all machine learning ML algorithms is to define a suitable error function or loss metric , iterate over the data, and find the optimum settings of the parameter of the ML model which minimizes the total error.
Often, the error is a measure of some kind of distance between the model prediction and the ground truth given data. Therefore, it is perfectly possible to use SciPy optimization routines to solve an ML problem.
This gives you a deep insight into the actual working of the algorithm as you have to construct the loss metric yourself and not depend on some ready-made, out-of-the-box function. Tuning parameters and hyperparameters of ML models is often a cumbersome and error-prone task.
Although there are grid-search methods available for searching the best parametric combination, some degree of automation can be easily introduced by running an optimization loop over the parameter space. The objective function, in this case, has to be some metric of the quality of the ML model prediction mean-square error, complexity measure, or F1 score for example. In many situations, you cannot have a nice, closed-form analytical function to use as the objective of an optimization problem.
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