  We prove a general duality result for multi-stage portfolio optimization
problems in markets with proportional transaction costs. The financial market
is described by Kabanov's model of foreign exchange markets over a finite
probability space and finite-horizon discrete time steps. This framework allows
us to compare vector-valued portfolios under a partial ordering, so that our
model does not require liquidation into some numeraire at terminal time.
  We embed the vector-valued portfolio problem into the set-optimization
framework, and generate a problem dual to portfolio optimization. Using recent
results in the development of set optimization, we then show that a strong
duality relationship holds between the problems.
