# Maximizing strictly increasing convex function

Let the objective be to maximize the sum of $f_i(x_i)$ where all $f_i$ are strictly increasing convex functions. Maximizing a convex function is hard as a local maximum might not be a global one. However for the special case described above, since $f_i$ are strictly increasing, from what I understand, the local maximum should be the global one.

If so, are there any special techniques that can optimize the objective (or maybe a piecewise approximation of it) subject to linear inequality constraints quickly?

Thanks

• If all $f_i$ are strictly convex then a local maximum must be the global maximum. If all $f_i$ are strictly increasing then the maximum is unbounded. To maximize you would have to set $x_i=\infty$ for all $x_i$. – James King Sep 19 '11 at 10:00
• @James - The OP asked for maximizing w/ respect to linear inequality constraints. – Opt Sep 19 '11 at 21:07
• Note: for a CONVEX function f_i, the local minimum would be the same as the global minimum. This is not true because you want to MAXIMIZE the sum of f_i(x_i). BTW, exactly the reverse would be true if f_i were concave, i.e., maximization would be easy, but minimization would be hard. – user8040 Jan 20 '12 at 17:17
• Use the Lagrange multiplier, you'll solve a similar problem with the same results. – luke14free Jan 23 '12 at 8:18
• Local maxima are not necessarily the global maximum in your situation. For example, consider maximizing 2x^2+y^2 subject to x+y≤1, x≥0, y≥0. Point (1,0) is a local maximum but not the global maximum. – Tsuyoshi Ito Jan 25 '12 at 1:25