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Data-Driven Distributionally Robust Mixed-Integer Control through Lifted Control Policy

Xutao Ma, Chao Ning, Wenli Du, Yang Shi

TL;DR

This work tackles finite-horizon distributionally robust mixed-integer control for uncertain linear systems by introducing a novel DR-LCP framework that leverages a lifted disturbance representation. By restricting policies to lifted control policies and deriving equivalent reformulations under Wasserstein-type ambiguity sets, the method provides tractable, high-quality approximate solutions for problems with both continuous and integer controls. The paper delivers an asymptotic performance alignment with additive policy spaces and a non-asymptotic bound that scales with the breakpoint density, and it validates the approach on an inventory control case where DR-LCP outperforms existing methods. Overall, the DR-LCP approach combines rigorous DRO theory with practical lifted-policy design to enable scalable, robust mixed-integer control under distributional uncertainty, with potential extensions to distributed computation for long horizons.

Abstract

This paper investigates the finite-horizon distributionally robust mixed-integer control (DRMIC) of uncertain linear systems. However, deriving an optimal causal feedback control policy to this DRMIC problem is computationally formidable for most ambiguity sets. To address the computational challenge, we propose a novel distributionally robust lifted control policy (DR-LCP) method to derive a high-quality approximate solution to this DRMIC problem for a rich class of Wasserstein metric-based ambiguity sets, including the Wasserstein ambiguity set and its variants. In theory, we analyze the asymptotic performance and establish a tight non-asymptotic bound of the proposed method. In numerical experiments, the proposed DR-LCP method empirically demonstrates superior performance compared with existing methods in the literature.

Data-Driven Distributionally Robust Mixed-Integer Control through Lifted Control Policy

TL;DR

This work tackles finite-horizon distributionally robust mixed-integer control for uncertain linear systems by introducing a novel DR-LCP framework that leverages a lifted disturbance representation. By restricting policies to lifted control policies and deriving equivalent reformulations under Wasserstein-type ambiguity sets, the method provides tractable, high-quality approximate solutions for problems with both continuous and integer controls. The paper delivers an asymptotic performance alignment with additive policy spaces and a non-asymptotic bound that scales with the breakpoint density, and it validates the approach on an inventory control case where DR-LCP outperforms existing methods. Overall, the DR-LCP approach combines rigorous DRO theory with practical lifted-policy design to enable scalable, robust mixed-integer control under distributional uncertainty, with potential extensions to distributed computation for long horizons.

Abstract

This paper investigates the finite-horizon distributionally robust mixed-integer control (DRMIC) of uncertain linear systems. However, deriving an optimal causal feedback control policy to this DRMIC problem is computationally formidable for most ambiguity sets. To address the computational challenge, we propose a novel distributionally robust lifted control policy (DR-LCP) method to derive a high-quality approximate solution to this DRMIC problem for a rich class of Wasserstein metric-based ambiguity sets, including the Wasserstein ambiguity set and its variants. In theory, we analyze the asymptotic performance and establish a tight non-asymptotic bound of the proposed method. In numerical experiments, the proposed DR-LCP method empirically demonstrates superior performance compared with existing methods in the literature.

Paper Structure

This paper contains 17 sections, 10 theorems, 77 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Lifted Control Policy
  4. Lifting of Disturbance
  5. Continous and Integer Lifted Control Policy
  6. Performance Analysis
  7. Asymptotic Performance Analysis
  8. Non-asymptotic Performance Analysis
  9. Solution Methodology
  10. Equivalent Reformulation of Objective Function
  11. Wasserstein Ambiguity Set
  12. Mixed Wasserstein-Moment Ambiguity Set
  13. Event-Wise Wasserstein Ambiguity Set
  14. Equivalent Reformulation of Constraint
  15. Numerical Experiment
  16. ...and 2 more sections

Key Result

Proposition 1

Let $G_{t,i}(\cdot)$ and $\varXi_{t,i}^{\ast}$ be given in Definition def1. Then, the lifted support $\varXi_{t,i}^{\ast}$ is the union of $p_{t,i}$ line segments $K_{t,i,j}^{\ast},j=1,\cdots,p_{t,i}$ with the following analytical formulation: where $G_{t,i}^{-}( w_{t,i,j}):=\lim_{\xi\nearrow w_{t,i,j}}G_{t,i}(\xi)$ is the left limit of lifting function $G_{t,i}$ at point $w_{t,i,j}$.

Figures (2)

  • Figure 1: Illustration of the LCP. (In this example, the support $\varXi_{t',i}=[0,3]$ is divided by two breakpoints $w_{t',i,1}=1$ and $w_{t',i,2}=2$. As for the control policy, we take $\boldsymbol{y}_{t,j,t',i}=[2.5,-0.5,-2.5]^T,{y}_{t,j,t',i}^0=1.0$ for continuous control and $\boldsymbol{z}_{t,j,t',i}=[-2,1]^T,{z}_{t,j,t',i}^0=3$ for integer control.)
  • Figure 2: Closed-loop performance.

Theorems & Definitions (29)

  • Definition 1
  • Remark 1
  • Proposition 1
  • proof
  • Definition 2
  • Remark 2
  • Definition 3: Lifted Control Policy Space
  • Definition 4
  • Definition 5: Distance between continuous control policies
  • Definition 6: Additive Control Policy Space
  • ...and 19 more