Numerically find the minimum optimal feasible design (vector x) for each of the following problems using any gradient based search technique. for each run record the starting point and iteration history (plot a graph with objective function on y axis and iteration number on x axis), record also the final point at which the iteration terminated and whether or not the final solution is feasible. Do ten runs for each problem. Discuss your result.

1)The banana function: In this problem there are two design variables with lower and upper limits of [-5,5] the function has a global minimum at [1,1] with an optimal function value of zero.

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

2)Eggrate function: In this problem there are two design variables with lower and upper limits of [-2pi,2pi] the function has a global minimum at [0,0] with an optimal function value of zero.

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

3)Golinski speed reducer: This represent the design of a simple gearbox such as might be used in light airplanes. The objective is to minimze the speed reducer weight. A known feasible solution is a 2994.34 gearbox with the following seven design variables (3.5000, 0.7000, 17, 7.3000, 7.7153, 3.3502 and 5.2867). This is a feasible solution with four active constraint but is it an optimal solution?