[Paper Review] Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making

Weijun Xie, Jie Zhang+1 more|arXiv (Cornell University)|Mar 2, 2020

Risk and Portfolio Optimization47 references4 citations

TL;DR

This paper proposes distributionally robust bottleneck combinatorial problems (DRBCP) under Wasserstein ambiguity sets to address uncertainty in stochastic bottleneck optimization. It develops equivalent mixed-integer program reformulations for both uncertainty quantification (DRBCP-U) and decision-making (DRBCP-D), showing that the Wasserstein radius can be set in the order of $ O(N^{-1/2}) $, improving robustness with minimal conservatism. The approach is extended to $ ext{DR} ext{-} ext{BCP-} ext{D} $ and $ ext{DR} ext{-} ext{BCP-} ext{D} $, with a decision-robust model formulated as a mixed-integer second-order cone program (MISOCP).

ABSTRACT

This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max-min-max form. Fortunately, the alternative forms of the bottleneck combinatorial problems from their blockers allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-and-passenger matching that minimizes the unfairness. This gives rise to the decision-making DRBCP (denoted by DRBCP-D). For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixed-integer program.

Motivation & Objective

• To develop a data-driven robust optimization framework for bottleneck combinatorial problems under distributional uncertainty.
• To provide accurate worst-case value estimation (DRBCP-U) for applications like multi-hop wireless networks with stochastic link costs.
• To support fair decision-making (DRBCP-D) in ride-sharing and organ allocation by minimizing the worst-case bottleneck cost.
• To establish theoretical guarantees on the Wasserstein radius that balance robustness and conservatism.
• To extend the framework to $ ext{DR} ext{-} ext{BCP-} ext{D} $, where the objective is to minimize the sum of the $ heta $ largest costs.

Proposed method

• Uses a Wasserstein ambiguity set centered at the empirical distribution to model distributional uncertainty in stochastic bottleneck problems.
• Derives equivalent deterministic reformulations for DRBCP-U and DRBCP-D using clutter and blocking system duality, enabling solution via mixed-integer programs.
• Establishes a connection between DRBCP and its sampling average approximation (SAA) counterpart to justify a $ O(N^{-1/2}) $ Wasserstein radius.
• Proposes a decision-robust model for small samples that minimizes empirical variance over an indifferent solution space defined by the SAA solution.
• Reformulates the decision-robust model as a mixed-integer second-order cone program (MISOCP) using epigraphic and big-M reformulations.
• Applies the framework to $ ext{DR} ext{-} ext{BCP-} ext{D} $, where the objective is to minimize the worst-case sum of the $ heta $ largest costs, with a similar reformulation approach.

Experimental results

Research questions

• RQ1 How can the worst-case value of a stochastic bottleneck combinatorial problem be reliably quantified under distributional ambiguity?
• RQ2 What is the optimal decision for minimizing the worst-case bottleneck cost when the true cost distribution is ambiguous?
• RQ3 Can the Wasserstein radius be chosen conservatively while still ensuring statistical confidence in the robust solution?
• RQ4 How can robustness be maintained in small-sample regimes where the SAA solution may be unreliable?
• RQ5 Can the framework be extended to problems minimizing the sum of the $ heta $ largest costs rather than just the maximum?

Key findings

• DRBCP-U and DRBCP-D admit equivalent mixed-integer program (MIP) reformulations using duality in clutter and blocking systems.
• The Wasserstein radius can be set in the order of $ O(N^{-1/2}) $, which is less conservative than prior results and ensures statistical confidence.
• The optimal solution to DRBCP-D is also optimal for its sampling average approximation (SAA) counterpart under the same radius.
• For small samples, the proposed decision-robust model minimizes empirical variance over an indifferent solution space and is reformulated as a mixed-integer second-order cone program (MISOCP).
• The decision-robust model for $ ext{DR} ext{-} ext{BCP-} ext{D} $ ensures that the solution is robust to distributional shifts with high probability, under sub-Gaussian cost assumptions.
• Theoretical confidence bounds show that the true worst-case value lies within $ 2 heta $ of the DRBCP-D solution with high probability, given the chosen radius.