LP-Based Approximations for Disjoint Bilinear and Two-Stage Adjustable Robust Optimization

We consider the class of disjoint bilinear programs $$ \max \; \{ \mathbf {x}^T\mathbf {y} \mid \mathbf {x} \in \mathcal {X}, \;\mathbf {y} \in \mathcal {Y}\}$$ where $$\mathcal {X}$$ and $$\mathcal {Y}$$ are packing polytopes. We present an $$O(\frac{\log \log m_1}{\log m_1} \frac{\log \log m_2}{\log m_2})$$ -approximation algorithm for this problem where $$m_1$$ and $$m_2$$ are the number of packing constraints in $$\mathcal {X}$$ and $$\mathcal {Y}$$ respectively. In particular, we show that there exists a near-optimal solution $$(\mathbf {x}, \mathbf {y})$$ such that $$\mathbf {x}$$ and $$\mathbf {y}$$ are “near-integral”. We give an LP relaxation of this problem from which we obtain the near-optimal near-integral solution via randomized rounding. As an application of our techniques, we present a tight approximation for the two-stage adjustable robust optimization problem with covering constraints and right-hand side uncertainty where the separation problem is a bilinear optimization problem. In particular, based on the ideas above, we give an LP restriction of the two-stage problem that is an $$O(\frac{\log n}{\log \log n} \frac{\log L}{\log \log L})$$ -approximation where L is the number of constraints in the uncertainty set. This significantly improves over state-of-the-art approximation bounds known for this problem.