30 research outputs found
Generalized Decision Rule Approximations for Stochastic Programming via Liftings
Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem.
Generalized decision rule approximations for stochastic programming via liftings
Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higher-dimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem
Constant Depth Decision Rules for multistage optimization under uncertainty
In this paper, we introduce a new class of decision rules, referred to as
Constant Depth Decision Rules (CDDRs), for multistage optimization under linear
constraints with uncertainty-affected right-hand sides. We consider two
uncertainty classes: discrete uncertainties which can take at each stage at
most a fixed number d of different values, and polytopic uncertainties which,
at each stage, are elements of a convex hull of at most d points. Given the
depth mu of the decision rule, the decision at stage t is expressed as the sum
of t functions of mu consecutive values of the underlying uncertain parameters.
These functions are arbitrary in the case of discrete uncertainties and are
poly-affine in the case of polytopic uncertainties. For these uncertainty
classes, we show that when the uncertain right-hand sides of the constraints of
the multistage problem are of the same additive structure as the decision
rules, these constraints can be reformulated as a system of linear inequality
constraints where the numbers of variables and constraints is O(1)(n+m)d^mu N^2
with n the maximal dimension of control variables, m the maximal number of
inequality constraints at each stage, and N the number of stages.
As an illustration, we discuss an application of the proposed approach to a
Multistage Stochastic Program arising in the problem of hydro-thermal
production planning with interstage dependent inflows. For problems with a
small number of stages, we present the results of a numerical study in which
optimal CDDRs show similar performance, in terms of optimization objective, to
that of Stochastic Dual Dynamic Programming (SDDP) policies, often at much
smaller computational cost