Interior point methods (IPM) have been recognized as an efficient approach for the solution of large scale stochastic programming problems by exploiting the block-angular structure of the augmented system resulting from this problem class. Stochastic programming problems, however, have exploitable structure beyond simple matrix shape: namely the scenarios are typically a discrete sampling of an underlying (continuous) probability distribution. An appealing way of exploiting this would be to initially use a coarser discretization, i.e. less scenarios to obtain an approximate solution, which could then be used to warmstart the solver on the full problem. Unfortunately IPMs are well known for their difficulties in exploiting warm-start information.
In this paper we present a multi-step warm-start scheme for stochastic programming problems, where a sequence of problems defined over scenario trees of differing sizes is given and an IPM warm-start point is constructed by successively finding approximations to the central path of the problems defined over the given trees. We analyse the resulting algorithm, argue that it yields improved complexity over either the coldstart or a naive two-step scheme and give numerical results.
Stochastic programming, Interior point methods, Warm-starting, Structure exploitation
Written: 30 June 2009
Revised: 3 December 2010
Submitted for publication