Comparison of Optimization Algorithms Problem A: Consider the following three optimization problems: Numerically find the minimum (=optimal) feasible design vector x for each of the three problems usi

Comparison of Optimization Algorithms 

Problem A: Consider the following three optimization problems: Numerically find the minimum (=optimal) feasible design vector x for each of the three problems using a gradient-based search technique of your choice 

 For each run (2runs) record the starting point you used, the iteration history (objective value on y-axis and iteration number on x-axis), the final point at which the algorithm terminated and whether or not the final solution is feasible

Problem B: Repeat problem a but this time using a heuristic technique of your choice Explain how you “tuned” the heuristic algorithm. compare convergence history of both method and frequency at which the technique gets trapped in a local maximum

1. The Rosenbrock Function 

This function is known as the “banana function” because of its shape; it is described mathematically in Equation 1. In this problem, there are two design variables with lower and upper limits of [−5,5]. The Rosenbrock function has a known global minimum at [1,1] with an optimal function value of zero. 

Minimize f(x) = 100 (x2 −(x1)^2)^2  + (1−x1) ^2 

2. The Eggcrate Function:

 This function is described mathematically in Equation 2. In this problem, there are two design variables with lower and upper bounds of [−2π,2π]. The Eggcrate function has a known global minimum at [0,0] with an optimal function value of zero. 

Minimize f(x) = (x 1 )^2 + (x2)^2 +25 ((sin^2) x1 +(sin^2) x2)

3.Golinski’s Speed Reducer :

This hypothetical problem represents the design of a simple gearbox such as might be used  in a light airplane between the engine and propeller to allow each to rotate at its most efficient speed. 

The objective is to minimize the speed reducer’s weight while satisfying the 11 constraints imposed by gear and shaft design practices.

 A known feasible solution obtained by a sequential quadratic programming (SQP) approach (Matlab’s fmincon) is a 2994.34 kg gearbox with the following values for the seven design variables: [3.5000,0.7000,17,7.3000,7.7153,3.3502,5.2867]. This is a feasible solution with four active constraints, but is it an optimal solution?