Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

Gade, Dinakar; Hackebeil, Gabriel; Ryan, Sarah M.; Watson, Jean -Paul; Wets, Roger J.-B.; Woodruff, David L.

doi:10.1007/s10107-016-1000-z

Title: Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

Abstract

We present a method for computing lower bounds in the progressive hedging algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. In conclusion, we report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds.

Authors:

Gade, Dinakar ^[1]; Hackebeil, Gabriel ^[2]; Ryan, Sarah M. ^[3]; Watson, Jean -Paul ^[4]; Wets, Roger J.-B. ^[5]; Woodruff, David L. ^[5]

Sabre Holdings, Southlake, TX (United States)
Texas A & M Univ., College Station, TX (United States)
Iowa State Univ., Ames, IA (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Univ. of California, Davis, CA (United States)

Publication Date:: Sat Apr 02 00:00:00 EDT 2016

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1310314

Report Number(s):: SAND-2013-9195J
Journal ID: ISSN 0025-5610; PII: 1000

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Accepted Manuscript

Journal Name:: Mathematical Programming

Additional Journal Information:: Journal Volume: 157; Journal Issue: 1; Journal ID: ISSN 0025-5610

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; stochastic mixed-integer programming; decomposition algorithms; lower bounding

Citation Formats


                    Gade, Dinakar, Hackebeil, Gabriel, Ryan, Sarah M., Watson, Jean -Paul, Wets, Roger J.-B., and Woodruff, David L. Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs.  United States: N. p., 2016. 
Web.  doi:10.1007/s10107-016-1000-z.

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                    Gade, Dinakar, Hackebeil, Gabriel, Ryan, Sarah M., Watson, Jean -Paul, Wets, Roger J.-B., & Woodruff, David L. Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs.  United States.  https://doi.org/10.1007/s10107-016-1000-z

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                    Gade, Dinakar, Hackebeil, Gabriel, Ryan, Sarah M., Watson, Jean -Paul, Wets, Roger J.-B., and Woodruff, David L. Sat .  
"Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs".  United States.  https://doi.org/10.1007/s10107-016-1000-z.  https://www.osti.gov/servlets/purl/1310314.

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@article{osti_1310314,

  title        = {Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs},

  author       = {Gade, Dinakar and Hackebeil, Gabriel and Ryan, Sarah M. and Watson, Jean -Paul and Wets, Roger J.-B. and Woodruff, David L.},

  abstractNote = {We present a method for computing lower bounds in the progressive hedging algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. In conclusion, we report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds.},

  doi          = {10.1007/s10107-016-1000-z},

  journal      = {Mathematical Programming},

  number       = 1,

  volume       = 157,

  place        = {United States},

  year         = {Sat Apr 02 00:00:00 EDT 2016},

  month        = {Sat Apr 02 00:00:00 EDT 2016}

}

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Title: Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

Abstract

Citation Formats

A finite branch-and-bound algorithm for two-stage stochastic integer programs journal, June 2004

Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers book, January 2011

Dual decomposition in stochastic integer programming journal, February 1999

A Computationally Efficient Mixed-Integer Linear Formulation for the Thermal Unit Commitment Problem journal, August 2006

Toward scalable stochastic unit commitment: Part 2: solver configuration and performance assessment journal, April 2015

Stochastic Network Interdiction journal, April 1998

Scenario grouping in a progressive hedging-based meta-heuristic for stochastic network design journal, March 2014

Decomposition Principle for Linear Programs journal, February 1960

A flexible power system operations simulation model for assessing wind integration conference, July 2011

Scenario Cluster Decomposition of the Lagrangian dual in two-stage stochastic mixed 0–1 optimization journal, January 2013

Lagrangean decomposition: A model yielding stronger lagrangean bounds journal, June 1987

Integration of progressive hedging and dual decomposition in stochastic integer programs journal, May 2015

Pyomo: modeling and solving mathematical programs in Python journal, August 2011

A decomposition algorithm applied to planning the interdiction of stochastic networks journal, June 2005

Reformulation and sampling to solve a stochastic network interdiction problem journal, October 2008

Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming journal, January 1996

On parallelizing dual decomposition in stochastic integer programming journal, May 2013

A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems journal, June 2004

On the existence of optimal solutions to integer and mixed-integer programming problems journal, December 1974

Tight and Compact MILP Formulation for the Thermal Unit Commitment Problem journal, November 2013

Applying the progressive hedging algorithm to stochastic generalized networks journal, December 1991

Solving multistage stochastic networks: An application of scenario aggregation journal, October 1991

The Million-Variable “March” for Stochastic Combinatorial Optimization journal, July 2005

Multiarea Stochastic Unit Commitment for High Wind Penetration in a Transmission Constrained Network journal, June 2013

Scenarios and Policy Aggregation in Optimization Under Uncertainty journal, February 1991

Uncertainty Management in the Unit Commitment Problem journal, May 2009

Toward scalable, parallel progressive hedging for stochastic unit commitment conference, January 2013

A stochastic programming approach for supply chain network design under uncertainty journal, November 2005

Dynamic Testing of Wholesale Power Market Designs: An Open-Source Agent-Based Framework journal, August 2007

A stochastic model for the unit commitment problem journal, January 1996

PySP: modeling and solving stochastic programs in Python journal, March 2012

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems journal, July 2010

Tight and compact MILP formulation for the thermal unit commitment problem conference, July 2015

Integer and Combinatorial Optimization journal, September 1989

The million-variable "march" for stochastic combinatorial optimization text, January 2004

A branch-and-price algorithm for multi-stage stochastic integer programming with application to stochastic batch-sizing problems text, January 2002

Integer and Combinatorial Optimization journal, February 1990

Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers January 2010

Stochastic dual dynamic integer programming journal, March 2018

A Progressive Hedging based branch-and-bound algorithm for mixed-integer stochastic programs journal, June 2018

Progressive hedging for stochastic programs with cross-scenario inequality constraints journal, November 2019

A finite $$\epsilon $$-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables journal, August 2019

Optimising data-driven network under limited resource: a partial diversification approach journal, August 2018

Combining penalty-based and Gauss-Seidel methods for solving stochastic mixed-integer problems journal, March 2018

Modeling and Solution Techniques Used for Hydro Generation Scheduling journal, July 2019

A finite branch-and-bound algorithm for two-stage stochastic integer programs
journal, June 2004

Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers
book, January 2011

Dual decomposition in stochastic integer programming
journal, February 1999

A Computationally Efficient Mixed-Integer Linear Formulation for the Thermal Unit Commitment Problem
journal, August 2006

Toward scalable stochastic unit commitment: Part 2: solver configuration and performance assessment
journal, April 2015

Stochastic Network Interdiction
journal, April 1998

Scenario grouping in a progressive hedging-based meta-heuristic for stochastic network design
journal, March 2014

Decomposition Principle for Linear Programs
journal, February 1960

A flexible power system operations simulation model for assessing wind integration
conference, July 2011

Scenario Cluster Decomposition of the Lagrangian dual in two-stage stochastic mixed 0–1 optimization
journal, January 2013

Lagrangean decomposition: A model yielding stronger lagrangean bounds
journal, June 1987

Integration of progressive hedging and dual decomposition in stochastic integer programs
journal, May 2015

Pyomo: modeling and solving mathematical programs in Python
journal, August 2011

A decomposition algorithm applied to planning the interdiction of stochastic networks
journal, June 2005

Reformulation and sampling to solve a stochastic network interdiction problem
journal, October 2008

Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming
journal, January 1996

On parallelizing dual decomposition in stochastic integer programming
journal, May 2013

A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems
journal, June 2004

On the existence of optimal solutions to integer and mixed-integer programming problems
journal, December 1974

Tight and Compact MILP Formulation for the Thermal Unit Commitment Problem
journal, November 2013

Applying the progressive hedging algorithm to stochastic generalized networks
journal, December 1991

Solving multistage stochastic networks: An application of scenario aggregation
journal, October 1991

The Million-Variable “March” for Stochastic Combinatorial Optimization
journal, July 2005

Multiarea Stochastic Unit Commitment for High Wind Penetration in a Transmission Constrained Network
journal, June 2013

Scenarios and Policy Aggregation in Optimization Under Uncertainty
journal, February 1991

Uncertainty Management in the Unit Commitment Problem
journal, May 2009

Toward scalable, parallel progressive hedging for stochastic unit commitment
conference, January 2013

A stochastic programming approach for supply chain network design under uncertainty
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Dynamic Testing of Wholesale Power Market Designs: An Open-Source Agent-Based Framework
journal, August 2007

A stochastic model for the unit commitment problem
journal, January 1996

PySP: modeling and solving stochastic programs in Python
journal, March 2012

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems
journal, July 2010

Tight and compact MILP formulation for the thermal unit commitment problem
conference, July 2015

Integer and Combinatorial Optimization
journal, September 1989

The million-variable "march" for stochastic combinatorial optimization
text, January 2004

A branch-and-price algorithm for multi-stage stochastic integer programming with application to stochastic batch-sizing problems
text, January 2002

Integer and Combinatorial Optimization
journal, February 1990

Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers
January 2010

Stochastic dual dynamic integer programming
journal, March 2018

A Progressive Hedging based branch-and-bound algorithm for mixed-integer stochastic programs
journal, June 2018

Progressive hedging for stochastic programs with cross-scenario inequality constraints
journal, November 2019

A finite $$\epsilon $$-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables
journal, August 2019

Optimising data-driven network under limited resource: a partial diversification approach
journal, August 2018

Combining penalty-based and Gauss-Seidel methods for solving stochastic mixed-integer problems
journal, March 2018

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