  This paper studies some unconventional utility maximization problems when the
ratio type relative portfolio performance is periodically evaluated over an
infinite horizon. Meanwhile, the agent is prohibited from short-selling stocks.
Our goal is to understand the impact of the periodic reward structure on the
long-run constrained portfolio strategy. For power and logarithmic utilities,
we can reformulate the original problem into an auxiliary one-period
optimization problem. To cope with the auxiliary problem with no short-selling,
the dual control problem is introduced and studied, which gives the
characterization of the candidate optimal portfolio within one period. With the
help of the results from the auxiliary problem, the value function and the
optimal constrained portfolio for the original problem with periodic evaluation
can be derived and verified, allowing us to discuss some financial implications
under the new performance paradigm.
