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 $(\tilde{\mathbf{x}}, \tilde{\mathbf{y}})$ such that $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$ are ``near-integral". We give an LP relaxation of the problem from which we obtain the near-optimal near-integral solution via randomized rounding. We show that our relaxation is tightly related to the widely used reformulation linearization technique (RLT). 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. Furthermore, we show that our LP restriction gives a feasible affine policy for the two-stage robust problem with the same (or better) objective value. As a consequence, affine policies give an $O(\frac{\log n}{\log \log n} \frac{\log L}{\log \log L})$-approximation of the two-stage problem, significantly generalizing the previously known bounds on their performance.