42,005 research outputs found
Multi-stage Adjustable Robust Mixed-Integer Optimization via Iterative Splitting of the Uncertainty set
In this paper we propose a methodology for constructing decision rules for in- teger 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
Multi-stage Adjustable Robust Mixed-Integer Optimization via Iterative Splitting of the Uncertainty set (Revision of CentER Discussion Paper 2014-056)
In this paper we propose a methodology for constructing decision rules for integer and continuous decision variables in multi period 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
On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)
The "scenario approach" provides an intuitive method to address chance
constrained problems arising in control design for uncertain systems. It
addresses these problems by replacing the chance constraint with a finite
number of sampled constraints (scenarios). The sample size critically depends
on Helly's dimension, a quantity always upper bounded by the number of decision
variables. However, this standard bound can lead to computationally expensive
programs whose solutions are conservative in terms of cost and violation
probability. We derive improved bounds of Helly's dimension for problems where
the chance constraint has certain structural properties. The improved bounds
lower the number of scenarios required for these problems, leading both to
improved objective value and reduced computational complexity. Our results are
generally applicable to Randomized Model Predictive Control of chance
constrained linear systems with additive uncertainty and affine disturbance
feedback. The efficacy of the proposed bound is demonstrated on an inventory
management example.Comment: Accepted for publication at Automatic
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