A Decomposition Framework for Nonlinear Nonconvex Two-Stage Optimization

TL;DR

This paper introduces a novel decomposition framework for nonlinear nonconvex two-stage optimization problems, enabling the use of standard optimization tools through a smoothing technique that ensures differentiability of the second-stage solutions.

Contribution

It presents a new smoothing-based decomposition method for nonconvex two-stage problems, with convergence guarantees and demonstrated computational efficiency.

Findings

Framework effectively handles large-scale problems.

Ensures differentiability of second-stage solutions.

Proves convergence and local convergence speed.

Abstract

We propose a new decomposition framework for continuous nonlinear constrained two-stage optimization, where both first- and second-stage problems can be nonconvex. A smoothing technique based on an interior-point formulation renders the optimal solution of the second-stage problem differentiable with respect to the first-stage parameters. As a consequence, efficient off-the-shelf optimization packages can be utilized. We show that the solution of the nonconvex second-stage problem behaves locally like a differentiable function so that existing proofs can be applied to prove the convergence of the iterates to first-order optimal points for the first-stage. We also prove fast local convergence of the algorithm as the barrier parameter is driven to zero. Numerical experiments for large-scale instances demonstrate the computational advantages of the decomposition framework.