I am concerned with the following optimization problem. I first state the problem and then briefly discus its background.
$$f(a_1,...,a_n)=\frac 1n \sum_1^n (a_i-b_i)^2$$
should be maximized subject to constraints: $$\sum a_i=1$$ $$\sum b_i=1$$ $$ 0\le a_i \le 1$$ $$ 0\le b_i \le 1$$
The background is a square contingency table from which only the marginal proportions on one side ($b_i$) are known. The function gives the variance across the margins, where $a_i$ denotes the corresponding marginal proportions that are unknown. I am interested in the maximum variance conditional on the marginal distribution of one side of the table.
Thank you for advice on how to approach this problem.
