Publications

Second-Order Multiplier Updates to Accelerate Admm Methods in Optimization Under Uncertainty

Rodriguez, Jose S.; Hackebeil, Gabriel; Siirola, John D.; Zavala, Victor M.; Laird, Carl D.

There is a need for efficient optimization strategies to efficiently solve large-scale, nonlinear optimization problems. Many problem classes, including design under uncertainty are inherently structured and can be accelerated with decomposition approaches. This paper describes a second-order multiplier update for the alternating direction method of multipliers (ADMM) to solve nonlinear stochastic programming problems. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for the method of multipliers (MM). We interpret ADMM as a decomposable version of MM and propose modifications to the multiplier update of the standard ADMM scheme based on improvements observed in MM. The modifications to the ADMM algorithm seek to accelerate solutions of optimization problems for design under uncertainty and the numerical effectiveness of the approaches is demonstrated on a set of ten stochastic programming problems. Practical strategies for improving computational performance are discussed along with comparisons between the algorithms. We observe that the second-order update achieves convergence in fewer unconstrained minimizations for MM on general nonlinear problems. In the case of ADMM, the second-order update reduces significantly the number of subproblem solves for convex quadratic programs (QPs).