This is a least-square fitting problem, for which there are a couple Julia packages: GitHub - JuliaNLSolvers/LsqFit.jl: Simple curve fitting in Julia, https://github.com/matthieugomez/LeastSquaresOptim.jl … e.g. try the Levenberg–Marquardt algorithm in LsqFit.jl.
fminsearch in Matlab or Octave is a generic derivative-free optimization function using the Nelder–Mead simplex algorithm. There are many, many general optimization packages in Julia: https://github.com/JuliaNLSolvers/Optim.jl, https://github.com/JuliaNonconvex/Nonconvex.jl, GitHub - JuliaOpt/NLopt.jl: Package to call the NLopt nonlinear-optimization library from the Julia language, https://github.com/robertfeldt/BlackBoxOptim.jl, https://github.com/emmt/OptimPack.jl, not to mention higher-level “modeling” packages like https://github.com/jump-dev/JuMP.jl. There are several implementations of Nelder–Mead among those packages, but in general I would tend not to use Nelder–Mead as my first (or second, or third) choice of algorithms these days.