  In this paper we consider the problem of minimising drawdown in a portfolio
of financial assets. Here drawdown represents the relative opportunity cost of
the single best missed trading opportunity over a specified time period. We
formulate the problem (minimising average drawdown, maximum drawdown, or a
weighted combination of the two) as a nonlinear program and show how it can be
partially linearised by replacing one of the nonlinear constraints by
equivalent linear constraints.
  Computational results are presented (generated using the nonlinear solver
SCIP) for three test instances drawn from the EURO STOXX 50, the FTSE 100 and
the S&P 500 with daily price data over the period 2010-2016. We present results
for long-only drawdown portfolios as well as results for portfolios with both
long and short positions. These indicate that (on average) our minimal drawdown
portfolios dominate the market indices in terms of return, Sharpe ratio,
maximum drawdown and average drawdown over the (approximately 1800 trading day)
out-of-sample period.
