(Disclaimer: I also posted to mathworks in the hope of a speedy answer. I hope this is not frowned upon.)
I want to minimize a very well-behaved real-valued objective function $f$ from R^I. Feasible points are inside the convex hull of a finite set of points x^1 through x^n. I am only interested in finding the point y where f(y) is maximized, and do not care about the N weights (in the N-dimensional simplex) that allow me to write y as a convex combination of the x^i.
Of course, I could simply write the optimization as:
popt = fmincon(@(p) f(x'*p),[],[],ones(1,N),1,zeros(N,1),[]);
y = x'*popt;
However, if the points that span the convex hull are nearly multicollinear, then this compound function (as a function of p) is very flat around the optimum. I suspect that this is slowing down fmincon a lot, and would help it "solve the problem" in the R^I space rather than over the N dimensional simplex. What steps could I take to achieve this?
