$2.$ Global optimality of Particle Swarm Optimization Algorithm

$(50$ Marks$)$

The optimization algorithms tend to get trapped in local extrema. Hence, instead of providing globally optimal solutions, the optimization algorithms usually provide locally optimal solutions. One of the ways to test the ability of an optimization algorithm to provide the global solutions is by using multi$-$modal functions. In this homework assignment, you need to write a particle swarm optimization $($PSO$)$ code in Matlab $($or Python$).$ The code will be applied to three "two$-$dimensional" multi$-$modal functions $($these functions accept two scalar inputs$).$ The range of input variables within which the globally optimal solutions for respective functions exist is provided. The Matlab optimization code should output two values: i$)$ the point where the function achieves the minimum value; and ii$)$ the value of the function at that point. The functions used for testing the PSO algorithm are provided below:

I. Multi$-$modal Gaussian function as shown in the following figure with three minimums. A Matlab code for the function has been uploaded on the Canvas under the assignment section separately. This function takes a vector with two elements as inputs. Use a range of $0$ to $50$ for both inputs.