Solving Infinite Horizon Stochastic Optimization Problems John R. Birg - PDF document

1Solving Infinite Horizon Stochastic Op
1Solving Infinite Horizon Stochastic Optimization Problems John R. BirgeNorthwestern University(joint work with Chris Donohue, Xiaodong Xu, and Gongyun Zhao)•(Very) long-term investor (example: university endowment

)•Payout from portfolio over time (want
)•Payout from portfolio over time (want to keep •Invest in various asset categories•Decisions:•How much to payout (consume)?•How to invest in asset categories?•Complication: restrictions on asset trades2•Notation:x

–current state (x u (oru) –current acti
–current state (x u (oru) –current action given x (u (oru–single period discount factorx,u–probability measure on next period state y c(x,u) –objective value for V(x) –value function of optimal expected future rewar

ds •Problem: Find V such that V(x) = max
ds •Problem: Find V such that V(x) = maxU(x)[V(y)] }for all x •Define an upper bound on the value function•Iteration k: upper bound VSolve for some x) = max,u) + Pxk,uxk,uk(y)]•Update uses an outer linear approximat

ion on U4•Value Iteration•Distributed
ion on U4•Value Iteration•Distributed Value IterationX infinitely often, (Here, random choice of x, use concavity.)•Deepest Cut to maximize Vof domain of V*)•Consider any x•Choose i and x–x|| •Suppose || (x) –V*(x

)|| ) –V*(x7•Constrain feasible region
)|| ) –V*(x7•Constrain feasible region to obtain convergence results•Accelerate the DC search problem to find •Accelerate overall algorithm using:•multiple simultaneous cuts?•nonlinear cuts?•bundles approach?•Can

formulate infinite-horizon investment pr
formulate infinite-horizon investment problem in stochastic programming •Solution with cutting plane method•Convergence with some conditions•Results for trade-restricted assets significantly different from market ass