function $[$optimal$\_$x$,$ optimal$\_$y$]=$gradient$\_$opti$\_$line$\_$search$($h$,$ int, tol, w$)\%$ Function definitions f$1=$ @$($x$)$ x$^2+$ x $+1$; f$2=$ @$($x$)$ x $+1\mathrm{.}5$; $\%$ Cost function costFunc$=$@$($x$,$ w$)$ w$(1)*$f$1($x$)+$ w$(2)*$f$2($x$)$; $\%$ Gradient of the cost function w$.$r$.$t x gradient $=$ @$($x$,$ w$)2*$x$*$w$(1)+$ w$(1)+$ w$(2)$; $\%$ Initialize variables x $=$ int$(1)$; $\%$ Start at the beginning of interval alpha $=$ h; $\%$ Learning rate max$\_$iter $=1000$; $\%$ Maximum number of iterations iter $=0$; $\%$ Iteration counter $\%$ Gradient descent with line search while true $\%$ Calculate the gradient grad $=$ gradient$($x$,$ w$)$; $\%$ Update x using line search t $=$ alpha; $\%$ Start with the maximum step size while costFunc$($x$-$t$*$grad$,$w$)>$ costFunc$($x$,$ w$)-$ t$*$tol$*$grad$^2$ t $=$ t$/2$; $\%$ Reduce the step size

CW$1\_1$ Gradient Based Optimisation

Write a function that find the optimal value of the given cost function below. The cost function is a weight combination of the two functions and. Use a line search to improve the efficiency of the algorithm. Write the program so that the weightings and can be changed.

Use backward difference for the gradients. Take h as the maximum step size and use fractions of this for the line search. w is a vector of and $.$ and should sum to $1$ and both be positive.

$y_1=f_1(x)=x^2+x+1$

$y_2=f_2(x)=x+1\mathrm{.}5$

$cost=f_1{w}_1+f_2{w}_2$