Higher derivatives beyond second order do not provide meaningful information as concerning the optimality of a solution. Nothing tells you whether a solution is global or local unfortunately.

Temitayo Bankole , help my thinking here,

a) it should be possible to still move/ walk from local solutions to other, better local solutions, or the global solution.

perhaps this should be restated as a question: when is it possible to still move/ walk from a local solution, to another better local solution, or the global solution.

b) changing the right number of coefficients, in the right way, would permit this.

c) derivatives higher than order 2, can help with b).

(2nd order derivatives would move the gradient to 0; higher order derivatives can help to shift the gradient altogether)

Derivatives at the local optimum (or 'at a point' in general) is a very local property. It is hard to see how it can help moving 'away' from your current optimum. There may be some merit in thinking whether using higher derivatives at multiple sampled points in the neighbourhood would help you in escaping the local optimum - I have not seen any published heuristic work in that direction. I am not an expert in global optimization though.

i currently have an optimization problem, that solver can only solve up to a point - a local solution. and i know the global solution, so i know that solver only finds a local solution.

i am referring to excel's solver. the NLP algorithm does ok, but it is a local solution. the evolutionary algorithm struggles.

the problem is not 100% convex non-linear.

it was of the form: min P = sum((P1-X1)^2)

but i managed to turn it into min P = sum(P1-X1), by working with absolutes

the equation of P1 renders it more non-linear than linear. X1 is a reference or constant.

if i calculate higher order PDs, i find additional possible steps, and i can move on - the algorithm can continue. at this stage, i calculate all PDs of a particular order, and use the first PD that has a positive step. i don't search around.

any idea on what would be the best - most efficient - way to utilize such higher order PDs for the next step. i am not sure whether simply taking the next possible step is the most efficient.

I would suggest to make your initial point (initial solution) better. Probably your solver from current initial point only finds local optima.

yes, the NLP algorithm is sensitive to the initial solution.

yet, i see this, or would take this, as a consequence and result of the nature of the optimization problem, and the (ability of the) algorithm.

the ideal case would be an algorithm that is not, or less, sensitive to the initial solution.