**Task 4**

**Problem 1:** (Submit your response only as a maximum of a two-page PDF file with a font size of 11. Avoid copying from each other, this will be marked as negative)

The two-page document should contain results with an explanation. There is no need to provide the MATLAB code in this document.

Consider the following function:

𝑓(𝑥) = (𝑥1 − 4)2 + (𝑥2 − 4)2 + 10(5 − 𝑥1 − 𝑥2)2

**a.** Evaluate the gradient at each point in a two-dimensional grid in the space (−10 ≤ 𝑥1, 𝑥2 ≤ 10). Choose an appropriate grid spacing (at least 100 points). Plot each gradient vector as a small arrow at its 𝑥1, 𝑥2 location. Interpret your plot and comment on the likely location of a minimum point by observing the gradient vectors and the contours of 𝑓(𝑥). You can use the “quiver” command for plotting the vector field. [2 marks]

**b.** We wish to find the minimum of the above function:

**i.** Use the steepest decent method with Armijo condition.

**ii.** Use the Newton-Raphson Method.

**iii.** Use the quasi-Newton Method (BFGS).

In each case use the starting point as [10, 0]. Also, for each case, on the contours plot of the objective function, plot the sequence of points obtained during the optimization iterations. Use appropriate stopping criteria. Compare your results in the above three methods and also compare your optimum result with the results of fmincon.

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