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I am interested in computing the maximum of the following function

$E[ ((XB) - E(XB))^TR ] $ subject to the constraint $E(XB)=1$ where $X$ is a $n\times m$ random matrix, $B$ is a $m \times 1$ variable vector, and $R$ is a $n\times 1$ random vector. $T$ is the transpose operator.

I would like to find the value of $B$ that maximizes this function.What are my options in terms of solutions to this problem?


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Expected value is linear, so if $X$ is the only random quantity in the expression $E[((XB) - E(XB))^T R] = 0$.

EDIT: with the question changed to make $R$ random as well, the objective becomes $E(B^T X^T R) - E(XB)^T E(R) = B^T C = C^T B$ where $C = E(X^T R) - E(X^T) E(R)$, while the constraint is $E(X) B = 1$ (I assume that's a vector of all $1$'s?).

Case 1: If $1$ is not in the column space of the matrix $E(X)$, the problem is infeasible: there are no solutions to the constraint.

Case 2: If $1$ is in the column space of $E(X)$ and $C^T$ is in the row space of $E(X)$, the objective is constant on the solution space.
Any solution of the constraint maximizes it.

Case 3: If $1$ is in the column space of $E(X)$ and $C^T$ is not in the row space of $E(X)$, the objective is unbounded. You can take a vector $V$ such that $E(X) V = 0$ and $C^T V > 0$, and by adding a suitable multiple of $V$ to any solution of $E(X) B = 1$ you get a solution with $C^T X$ arbitrarily large.

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Oops I apologize R is a $n \times 1$ random vector, so the function is $E(B^TX^TR) - E(XB)^TE(R)$ subject to $E(XB)=1$ – peterdevi Aug 30 '12 at 20:02

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