optimization - Find maximum of a function -

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i need find maximum of function:

a1^x1 * const1 + a2^x2 * const2 +....+ ak^xk * constk = qaulity

where xk>0 , xk integer. ak constant.

constraint: a1^x1 * const1*func(x1) + a2^x2 * const2*func(x2) +....+ ak^xk * constk*func(xk) < budget

where func discrete function:

func(x) {     switch(x)     {         case 1: return 423;         case 2: return 544;         ...         etc     } }

k may big(over 1000). x less 100. best method?

there techniques nelder-mead optimization (which believe gsl implements), techniques assume sort of special structure (i.e. convexity or continuity). depending on values of function, there may not exist unique optimum or optimum normal downhill method can find.


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