Consider the more general version of theminimum$-$normproblem, also referred to as thelinearly constrainedleast squaresproblem $($or just constrained least squares problem$)$minimize$\backslash |$ Ax$$b$\backslash |2$subject toCx$=$dHerex, the variable to be found, is ann$-$vector. The problem data $($which are given$)$ are them$\backslash$times nmatrixA,them$-$vectorb, thep$\backslash$times nmatrixC, and thep$-$vectord, andCx$=$drepresentplinear equality constraints.$($a$)$ Derive the optimality conditions using Lagrange multipliers and write them in a matrix vector form.$($b$)$ Show that the optimal point is a unique minimum $($See section $16\mathrm{.}2$ in Boyd$$s book$)($c$)$ Generate a random $20\backslash$times $10$ matrixAand a random $5\backslash$times $10$ matrixC. Then generate random vectorsbanddof appropriate dimensions for the constrained least squares problem and compute the solution$$xby form and solvingKKTequations usingQRfactorization, using Matlab or Python.$($d$)$ Use Matlab$$sfmincon$()$or Python$$sscipy$.$optimize.minimize$()$to verify your solution. See thedocumentation on how to use these functions.$($e$)($Approximate Solutions using Penalty Method$)$ Form a new augmented objective function$\backslash |$ Ax$$b$\backslash |2+\backslash$rho $\backslash |$ Cx$$d$\backslash |2$and solve the problem as constrained optimization problem for$\backslash$rho $=1,10,100,1000,$ everytime using the solution of previous$\backslash$rho for the next run for the initial guessx$0$using $($i$)$ Gradient Descentalgorithm $($ii$)$ Newton$$s Method. You can use fixed step size or backtracking line search and comparewith the results obtained in the earlier parts. You will need to use some seeding method so the randomnumbers are used for matrices when comparing results, or just pick some fixed numbers for the matrices.The gradients and hessians have to be calculated for the augmented cost function.