Consider the following optimization problem minxsigmani1xi

Consider the following optimization problem min/x{sigma^n_i=1|x_i| : x Element R^n As written, this problem appears nonsmoothed, as the objective function is not, differentiable at 0. Let f(x) = sigma^n_i=1 |x_i|l. Rewrite the problem as the minimization of the scalar alpha Element R over the variables (x, alpha) Element R^n x R subject to the new constraint alpha greaterthanorequalto f(x). (Why is this the same problem?) Let beta Element R^n be a vector of variables. Prove the single (nonsmoothed) constraint beta_i greaterthanorequalto |x_i| can be written as two (smooth) constraints beta_i greaterthanorequalto x_i and beta_i greaterthanorequalto -x_i. Prove that the problem can be reformulated as a smooth problem with 2n variables.

Solution

Ans:(Part1)

The given optimization problem is nonsmooth as the objective function is notdifferentiable at 0.This is the most difficult type of optimization problem to solve.This type of problem normally is, or must be assumed to be non-convex.

NSO can be solved by using smooth gradient based method ,Powells method,Bundles method.

Thus we reformulate the problem as shown above.