univariate nonlinear optimization with quadratic constraint in R

Question

I have a quadratic function f where, f = function (x) {2+.1*x+.23*(x*x)} . Let's say I have another quadratic fn g where g = function (x) {3+.4*x-.60*(x*x)}

Now, I want to maximize f given the constraints 1. g>0 and 2. 600<x<650

I have tried the packages optim , constrOptim and optimize . optimize does one dim. optimization, but without constraints and constrOptim I couldn't understand. I need to this using R. Please help.

PS In this example, the values may be erratic as I have given two random quadratic functions, but basically I want maximization of a quadratic fn given a quadratic constraint.

Answer 1

If you solve g(x)=0 for x by the usual quadratic formula then that just gives you another set of bounds on x . If your x^2 coefficent is negative then g(x) > 0 between the solutions, otherwise g(x)>0 outside the solutions, so within (-Inf, x1) and (x2, Inf).

In this case, g(x)>0 for -1.927 < x < 2.59 . So in this case both your constraints cannot be simultaneously achieved ( g(x) is LESS THAN 0 for 600<x<650 ).

But supposing your second condition was 1 < x < 5 , then you'd just combine the solution from g(x)>0 with that interval to get 1 < x < 2.59 , and then maximise f in that interval using standard univariate optimisation.

And you don't even need to run an optimisation algorithm. Your target f is quadratic. If the coefficient of x^2 is positive the maximum is going to be at one of your limits of x, so you only have a small number of values to try. If the coefficient of x^2 is -ve then the maximum is either at a limit or at the point where f(x) peaks (solve f'(x)=0 ) if that is within your limits.

So you can do this precisely, there's just a few conditions to test and then some intervals to compute and then some values of f at those interval limits to calculate.