[Paper Review] Robust optimization with incremental recourse

Ebrahim Nasrabadi, James B. Orlin|arXiv (Cornell University)|Dec 14, 2013

Risk and Portfolio Optimization26 references19 citations

TL;DR

This paper introduces a robust incremental optimization framework for uncertain linear programs, where decisions are made in two stages: an initial solution is chosen, then adjusted within a bounded distance after uncertain costs are revealed. The key contribution is proving that the robust incremental counterpart of a linear program is itself a linear program when the uncertainty set is polyhedral, solvable in polynomial time, while also establishing NP-hardness for discrete uncertainty sets and NP-completeness for robust incremental shortest path and minimum spanning tree problems under polyhedral uncertainty.

ABSTRACT

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem.

Motivation & Objective

• To develop a two-stage robust optimization model where decisions can be incrementally adjusted after uncertain cost parameters are revealed.
• To analyze the computational complexity of robust incremental variants of classical optimization problems such as shortest path and minimum spanning tree.
• To determine whether the robust incremental counterpart of a linear program remains tractable under different uncertainty set structures.
• To establish complexity bounds, including NP-hardness and NP-completeness, for specific robust incremental problems.

Proposed method

• Models the robust incremental problem as a bilevel optimization: minimize the worst-case cost of an initial solution plus its incremental adjustment, subject to a bounded change in solution space.
• Defines the incremental set 𝒮ₓ as all feasible solutions within a fixed distance K from the initial solution x, using a distance function F(x,y).
• For polyhedral uncertainty sets, shows that the robust incremental linear program can be reformulated as a single linear program, enabling polynomial-time solution.
• Applies Lagrangian relaxation and duality techniques to solve the incremental minimum spanning tree problem under cardinality-constrained uncertainty, leveraging matroid structure and strong duality.
• Uses the equivalence of optimization and separation to solve the adversarial problem in polynomial time when the uncertainty set is defined by a cardinality constraint on cost deviations.
• Employs a separation oracle for subtour elimination constraints in the minimum spanning tree formulation, enabling efficient solution via interior-point or cutting-plane methods.

Experimental results

Research questions

• RQ1 Is the robust incremental counterpart of a linear program solvable in polynomial time when the uncertainty set is polyhedral?
• RQ2 What is the computational complexity of the robust incremental shortest path problem under polyhedral uncertainty?
• RQ3 Is the robust incremental minimum spanning tree problem NP-hard or polynomial-time solvable under cardinality-constrained uncertainty?
• RQ4 Can the adversarial problem (worst-case cost selection) be solved efficiently in the robust incremental framework?
• RQ5 Are the decision versions of robust incremental problems Σ₂^p-complete, given that both the incremental and adversarial problems are in NP?

Key findings

• The robust incremental counterpart of a linear program is itself a linear program when the uncertainty set is polyhedral, and thus solvable in polynomial time.
• The robust incremental linear programming problem is NP-hard when the uncertainty set is discrete, establishing a lower bound on its complexity.
• The robust incremental shortest path problem is NP-complete even when only one new arc may be added to the initial path, under a polyhedral uncertainty set.
• The adversarial minimum spanning tree problem under a cardinality-constrained uncertainty set can be solved in polynomial time using duality and a separation oracle for subtour constraints.
• The incremental minimum spanning tree problem has the integrality property due to its representation as the intersection of a uniform matroid and a forest matroid, enabling efficient Lagrangian relaxation.
• The robust incremental minimum spanning tree problem remains open in terms of its exact complexity class, though its adversarial variant is polynomial-time solvable under the given uncertainty model.