Neur2RO: Neural Two-Stage Robust Optimization

Robust optimization provides a mathematical framework for modeling and solving decision-making problems under worst-case uncertainty. This work addresses two-stage robust optimization (2RO) problems (also called adjustable robust optimization), wherein first-stage and second-stage decisions are made before and after uncertainty is realized, respectively. This results in a nested min-max-min optimization problem which is extremely challenging computationally, especially when the decisions are discrete. We propose Neur2RO, an efficient machine learning-driven instantiation of column-and-constraint generation (CCG), a classical iterative algorithm for 2RO. Specifically, we learn to estimate the value function of the second-stage problem via a novel neural network architecture that is easy to optimize over by design. Embedding our neural network into CCG yields high-quality solutions quickly as evidenced by experiments on two 2RO benchmarks, knapsack and capital budgeting. For knapsack, Neur2RO finds solutions that are within roughly $2\%$ of the best-known values in a few seconds compared to the three hours of the state-of-the-art exact branch-and-price algorithm; for larger and more complex instances, Neur2RO finds even better solutions. For capital budgeting, Neur2RO outperforms three variants of the $k$-adaptability algorithm, particularly on the largest instances, with a 5 to 10-fold reduction in solution time. Our code and data are available at https://github.com/khalil-research/Neur2RO

Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations

In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding whether the uncertainty set is contained within the projection of the feasible region onto the uncertainty-space. Moreover, the considered sets are assumed to be described by polynomials. For answering this question, two very general approaches using methods from polynomial optimization are presented - one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a nonconvex feasibility problem where the feasible set is described by a composition of polynomials with the absolute value function. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, the problem can be decided by eliminating the absolute value functions and solving the resulting linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. It turns out that for instances where robust feasibility or infeasibility can be decided successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically is sufficient

Robust global supply chain planning under uncertainty

The New World Economy presents business organizations with some special challenges that they have never met before, when they manage their activities in the global supply chain network. Business managers find that traditional managerial approaches, techniques and principles are no longer effective in dealing with these challenges. This dissertation is a study of how to solve new problems emerging in the global supply chain network. Three main issues identified in the global supply chain network are: production loading problems for global manufacturing, logistics problems for global road transport and container loading problems for global air transport. These problems involve a higher level of uncertainty and risk. Three types of dual-response strategies have been developed to hedge the uncertainty and short lead time in the above three problems. These strategies are: a dual-response production loading strategy for global manufacturing, a dual-response logistics strategy for global road transport and a dual-response container loading strategy for global air transport. In order to implement these strategies, the two-stage stochastic recourse programming models have been formulated. The computational results show that the two-stage stochastic recourse models have an advantage in comparison to the corresponding deterministic models for the three issues. However, the two-stage stochastic recourse models lack the ability of handling risk, which is particularly important in today's highly-competitive environment. We thus develop a robust optimization framework for dealing with uncertainty and risk. The robust optimization framework consists of a robust optimization model with solution robustness, a robust optimisation model with model robustness and a robust optimization model with trade-off between solution robustness and model robustness. Each type of the robust optimization models represents a different measure of performance in terms of risk and cost. A series of experiments demonstrate that the robust optimization models can create a global supply chain planning system with more flexibility, reliability, agility, responsiveness and lower risk