An Optimization Problem with Applications to Optimal Design Theory

Abstract

The problem of minimizing $\sum^n_{i = 1}f(x_i)$ subject to the constraints $\sum^n_{i = 1}x_i = A, \sum^n_{i = 1}g(x_i) = B$ and $x_i \geq 0$ is solved. The solutions are different depending upon whether $(\operatorname{sgn} g")f"/g"$ is an increasing or decreasing function. The result is used to show that for certain designs, if they are optimal with respect to two criteria, then they are also optimal with respect to many other criteria.