Combining Progressive Hedging with a Frank-Wolfe Method to Compute Lagrangian Dual Bounds in Stochastic Mixed-Integer Programming

@article{Boland2018CombiningPH,
  title={Combining Progressive Hedging with a Frank-Wolfe Method to Compute Lagrangian Dual Bounds in Stochastic Mixed-Integer Programming},
  author={Natashia Boland and Jeffrey Christiansen and Brian C. Dandurand and Andrew Craig Eberhard and Jeff T. Linderoth and James R. Luedtke and Fabricio Oliveira},
  journal={SIAM J. Optim.},
  year={2018},
  volume={28},
  pages={1312-1336}
}
We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new… 

Figures and Tables from this paper

A Progressive Hedging Based Branch-and-Bound Algorithm for Stochastic Mixed-Integer Programs
TLDR
This paper presents a new framework that shows how PH can be utilized while guaranteeing convergence to globally optimal solutions of stochastic mixed-integer convex programs.
A Progressive Hedging based branch-and-bound algorithm for mixed-integer stochastic programs
TLDR
This paper presents a new framework that shows how PH can be utilized while guaranteeing convergence to globally optimal solutions of mixed-integer stochastic convex programs.
Asynchronous Projective Hedging for Stochastic Programming ∗
This paper proposes a decomposition algorithm for multistage stochastic programming that resembles the progressive hedging method of Rockafellar and Wets, but is capable of asynchronous parallel
A converging Benders’ decomposition algorithm for two-stage mixed-integer recourse models
TLDR
The solution method is a Benders’ decomposition, in which it iteratively construct tighter approximations of the expected second-stage cost function using a new family of optimality cuts, derived by parametrically solving extended formulations of the second- stage problems using deterministic mixed-integer programming techniques.
A Frank–Wolfe Progressive Hedging Algorithm for Improved Lower Bounds in Stochastic SCUC
TLDR
A novel Frank–Wolfe-based simplicial decomposition method is applied in conjunction with the PHA to improve the quality of dual bounds and the convergence characteristics in solving the S-SCUC.
The p-Lagrangian relaxation for nonconvex MIQCQP problems with complicating constraints
TLDR
This paper presents a novel technique to solve nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) with separable structures, such as those arising in deterministic equivalent representations of two-stage stochastic programming problems, and presents an appealing alternative algorithm that allows for overcoming computational performance issues.
Projective Hedging Algorithms for Distributed Optimization under Uncertainty *
We propose a decomposition algorithm for multistage stochastic programming that resembles the progressive hedging method of Rockafellar and Wets, but is provably ca-pable of several forms of
On Generating Lagrangian Cuts for Two-Stage Stochastic Integer Programs
TLDR
Computer results demonstrate that the proposed method improves the Benders relaxation significantly faster than previous methods for generating Lagrangian cuts and, when used within a branch-and-cut algorithm, significantly reduces the size of the search tree for three classes of test problems.
Approximating the Lagrangian Dual of a Stochastic Integer Program via Fenchel Cuts
TLDR
Computational results demonstrate that the proposed decomposition method based on generation of Fenchel cuts gives strong lower bounds at the root node and significantly reduces the size of the branch-and-cut tree for two classes of test problems.
Computational Optimization and Applications A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Algorithms and Software
This paper presents a tutorial on the state-of-the-art methodologies for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this
...
...

References

SHOWING 1-10 OF 71 REFERENCES
Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs
TLDR
This work presents a method for computing lower bounds in the progressive hedging algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs, and explores the relationship between key PHA parameters and the quality of the resulting lower bounds.
Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming
TLDR
This article introduces the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems and introduces the notion of integer convergence for progressive hedging.
Scalable Heuristics for a Class of Chance-Constrained Stochastic Programs
We describe computational procedures for solving a wide-ranging class of stochastic programs with chance constraints where the random components of the problem are discretely distributed. Our
On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
TLDR
This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm, which allows the unification and generalization of a variety of convex programming algorithms.
Progressive Hedging Innovations for a Class of Stochastic Resource Allocation Problems
Progressive hedging (PH) is a scenario-based decomposition technique for solving stochastic programs. While PH has been successfully applied to a number of problems, a variety of issues arise when
Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse
TLDR
It is demonstrated that using split cuts within the cut-and-project framework can significantly improve the performance of Benders decomposition and yield stronger relaxations in general when using multiple split disjunctions.
Simplicial decomposition in nonlinear programming algorithms
TLDR
Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Carathéodory's theorem, which allows the direct application of any unrestricted optimization method in the master program to find constrained maximizers for it.
...
...