Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Decomposition Framework for Nonlinear Nonconvex Two-Stage Optimization

Yuchen Lou, Xinyi Luo, Andreas Wächter, Ermin Wei

TL;DR

The paper develops a log-barrier smoothing-based decomposition framework for nonlinear nonconvex two-stage optimization, enabling differentiable treatment of the second-stage and the use of off-the-shelf second-order solvers in a distributed setting. It introduces two smoothing variants (objective and solution smoothing), proves global convergence for fixed and vanishing smoothing parameters, and demonstrates fast local convergence via an extrapolation step without requiring strong LICQ. The framework is validated on large-scale nonconvex QCQP instances, showing substantial scalability and parallel efficiency while achieving results consistent with monolithic solvers. This work advances practical solving of complex two-stage problems by uniting smoothness-based theory with robust, scalable optimization infrastructure.

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.

A Decomposition Framework for Nonlinear Nonconvex Two-Stage Optimization

TL;DR

The paper develops a log-barrier smoothing-based decomposition framework for nonlinear nonconvex two-stage optimization, enabling differentiable treatment of the second-stage and the use of off-the-shelf second-order solvers in a distributed setting. It introduces two smoothing variants (objective and solution smoothing), proves global convergence for fixed and vanishing smoothing parameters, and demonstrates fast local convergence via an extrapolation step without requiring strong LICQ. The framework is validated on large-scale nonconvex QCQP instances, showing substantial scalability and parallel efficiency while achieving results consistent with monolithic solvers. This work advances practical solving of complex two-stage problems by uniting smoothness-based theory with robust, scalable optimization infrastructure.

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.
Paper Structure (35 sections, 8 theorems, 63 equations, 6 figures, 4 algorithms)

This paper contains 35 sections, 8 theorems, 63 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related research
  3. Contributions and outline
  4. Notation
  5. Smoothing the second-stage problem
  6. Objective smoothing
  7. Solution smoothing
  8. Challenges of nonconvex subproblems
  9. Original formulation without smoothing
  10. Nonconvex subproblem with smoothing
  11. Example \ref{['ex:bilinear']}
  12. Example \ref{['ex:curvature']}
  13. Decomposition algorithm
  14. Algorithm for the master problem
  15. Interior-point method for the subproblems
  16. ...and 20 more sections

Key Result

Lemma 5.4

\newlabellem:unif_bound_IFT0 Suppose Assumption assump:full_recourse and assump:lipschitz hold. Let $\bar{x}\in\mathbb{R}^{n_0}$ and $\bar{v}_i\in\mathbb{R}^{n_{v_i}}$ such that $(\bar{x},\bar{v}_i)\in \mathcal{S}_i$ is nondegenerate. Then there exists $r_1>0$ such that there is a unique $C^1$ sol

Figures (6)

  • Figure 1: Plot of $\hat{f}_1^{{\text{sol}}}(x;\mu)$ (left) and plot of $\hat{f}_1^{{\text{obj}}}(x;\mu)$ (right), $x\in[0.1,2]$, $\mu=1$.
  • Figure 1: Left: Feasible region of the subproblem in Example \ref{['ex:bilinear']} for different values of $x$. Right: Solution maps and $\hat{f}_1(x)$ Example \ref{['ex:bilinear']}.
  • Figure 1: Wallclock computation times for QCQP instances.
  • Figure 2: Plot of solution maps (left) and $\hat{f}_1(x)$ (right) for Example \ref{['ex:curvature']}.
  • Figure 3: Solution maps (left) and smoothed master problem objective (left) for Example \ref{['ex:bilinear']}.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Example 2.1
  • Example 3.1
  • Example 3.2
  • Definition 5.3
  • Lemma 5.4
  • Proof 1
  • Proposition 5.6
  • Proof 2
  • Remark 5.7
  • Lemma 5.8: Newton's region of attraction with warm start
  • ...and 13 more