analytic_min-max_operators

Within the context of optimisation, analytic approximations of the min and max operators are very useful. I found a few proposed solutions on the MathOverflow and after analysing their numerical stability decided to formulate my own variant inspired by the infinity norm.

To improve numerical stability my method does the following:

1. Standardises random vectors: https://www.britannica.com/topic/standardized-random-variable
2. Maps vectors from R^n to R+^n, which is necessary for methods inspired by the infinity norm to work.

function analytic_min_max(X::Array{Float64, 1},N::Int64,case::Int64)

    """
        An analytic approximation to the min and max operators

        Inputs: 
            X: a vector from R^n where n is unknown

            N: an integer such that the approximation
               improves with increasing N.

            case: If case == 1 apply analytic_min(), otherwise 
                  apply analytic_max() if case == 2

        Output: 

            An approximation to min(X) if case == 1, and max(X) if 
            case == 2
    """

    if (case != 1)*(case != 2)

        return print("Error: case isn't well defined")

    else

        ## q is the degree of the approximation: 
        q = N*(-1)^case

        mu, sigma = mean(X), std(X)

        ## rescale vector so it has zero mean and unit variance:
        Z_score = (X.-mu)./sigma

        exp_sum = sum(exp.(Z_score*q))

        log_ = log(exp_sum)/q

        return (log_*sigma)+mu

    end

end

Discussion:

It took me about an hour to come up with my solution so I doubt this method is either original or state-of-the-art. If you know of a more robust method, feel free to join the discussion on the MathOverflow.

Relevant articles:

1. analytic approximations of the min and max operators
2. Differentiable approximations to the min and max operators