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Numerical Optimization: Do we need to increase the finite difference step length for gradient and hessian evaluation as the dimension of the problem increases?

• I am trying to optimize a 6-D objective function using Trust region method. I am using Matlab's inbuilt function fmincon which requires me to calculate finite difference gradient and Hessian. The issue is that when I fixed three of the six parameters and optimized the 3-D function, a finite difference step length of deltax=0.01 gave marvelous results. However when I am trying to optimize in 6-D (i.e. keeping all the six parameters as variable), any finite difference step length below 1 does not solve the problem well, deltax=1 however works nicely in 6-D. Anyone who could help me figure out a possible explanation for this observation? Details of my code:- 1) I use numerical simulation to evaluate objective function (Quasi Monte Carlo Method) 2) Objective function, f = func(V), where V= (mu1+sigma1*s1)*a + (mu2+sigma2*s2)*b + (mu3+sigma3*s3)*c, ----->{mu1,mu2,mu3,sigma1,sigma2,sigma3} are the six parameters; ----->{s1,s2,s3} are quasi-random numbers in (0,1); ----->a,b,c are observed values 3) In 6-D case, V= (mu1+sigma1*s1)*a + (mu2+sigma2*s2)*b + (mu3+sigma3*s3)*c => inaccuracy in objective function value due to random numbers In 3-D case, V = mu1*a + mu2*b + mu3*c => no random numbers involved

• Answer:

  I don't think that the problem is dimensionality.  Rather your 6-d problem has a lot of noise and when you add noise optimization routines will break down, because you could accidentally get stuck in a local minimum.  Increasing the size of the finite difference step adds in a smoothing function.