Multi-Stage Adjustable Robust Mixed-Integer Optimization via Iterative Splitting of the Uncertainty Set

Abstract

In this paper we propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problems finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem's computational complexity stays at the same level as for the static robust problem. This holds also in the non-fixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results how to split the uncertainty set by identifying sets of uncertain parameter scenarios to be divided for an improvement in the worst-case objective value. Based on this theory, we propose several splitting heuristics. Numerical examples entailing a capital budgeting and a lot sizing problem illustrate the advantages of the proposed approach.

Multistage Robust Mixed-Integer Optimization with Adaptive Partitions

“…The method presented in Postek and Den Hertog (2014) shares some similarities with the algorithm we propose. Both describe an iterative partitioning method for AMIO that uses active uncertain parameters to inform which partitions to use.…”

K-Adaptability in Two-Stage Robust Binary Programming

“…It follows from [14,Lemma 1] that the first of the two factors in the last expression can be made arbitrarily small by choosing > 0 appropriately. The second product term, on the other hand, is bounded over (x, {y k } k ) ∈ dom (DP K ) since X and Y are bounded, while the set of optimal solutions (β , θ ) to problem (18 ) can without loss of generality be bounded uniformly over (x, {y k } k ) ∈ dom (DP K ). For sufficiently small > 0, we can thus upper bound the difference ϕ − ϕ uniformly over (x, {y k } k ) ∈ dom (DP K ) by an arbitrarily small constant, which concludes the proof.…”

“…Another early approach for solving dynamic robust integer programs uses a fixed tessellation of the support set Ξ into subcells and restricts the continuous and binary recourse decisions to affine and constant functions of ξ over each cell, respectively [12,21]. This approach has recently been refined by allowing for an adaptive tessellation of the support set [8,18]. All solution schemes discussed so far result in conservative (upper bound) approximations for dynamic robust MILPs.…”

Adjustable Robust Strategies for Flood Protection

“…Postek and den Hertog (2016) provide theoretical results on how to do it. This approach relies on the observation that for different constraints in the problem, different scenarios r j are the worst-case scenarios, i.e., maximizing the left hand side.…”

“…This is the approach taken in Postek and den Hertog (2016) and Bertsimas and Dunning (2014) where, having determined the splits of the uncertainty set, a different constant decision is applied for each part of the uncertainty set. The essence of this approach lies in determining the conditions that a splitting needs to satisfy in order to improve on the decisions' adaptivity.…”