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PlasmoAlgorithms is a collection of decomposition algorithms to solve mathematical programming models taking Plasmo graphs as input. Plasmo graphs allow to create graphs of models, in which each node represents a model, and the arcs represent linking constraints. Several hierarchical problems can be expressed in this way, for example: supply chain planning and scheduling, optimization of power systems, and stochastic programming problems.

The algorithms implemented are:

• Lagrange Decomposition
• Multilevel Benders Decomposition

Pkg.clone("https://github.com/jalving/Plasmo.jl.git")
Pkg.clone("https://github.com/bbrunaud/PlasmoAlgorithms.jl.git")

PlasmoAlgorithms is currently working with Plasmo version v0.0.1. In order to use the algorithms, after installation Plasmo must be downgraded using the following command in the Plasmo tree (/home/user/.julia/v0.6/Plasmo.jl/). PlasmoAlgorithms will be updated to use the latest version of Plasmo in August 2018.

$ git checkout v0.0.1

using JuMP
using Gurobi
using Plasmo
using PlasmoAlgorithms

## Model on x
# Min 16x[1] + 10x[2]
# s.t. x[1] + x[2] <= 1OutputFlag=0
#      x ∈ {0,1}
m1 = Model(solver=GurobiSolver(OutputFlag=0))
@variable(m1, xm[i in 1:2],Bin)
@constraint(m1, xm[1] + xm[2] <= 1)
@objective(m1, Max, 16xm[1] + 10xm[2])

## Model on y`
# Max  4y[2]
# s.t. y[1] + y[2] <= 1
#      8x[1] + 2x[2] + y[2] + 4y[2] <= 10
#      x, y ∈ {0,1}
m2 = Model(solver=GurobiSolver(OutputFlag=0))
@variable(m2, xs[i in 1:2],Bin)
@variable(m2, y[i in 1:2], Bin)
@constraint(m2, y[1] + y[2] <= 1)
@constraint(m2, 8xs[1] + 2xs[2] + y[2] + 4y[2] <= 10)
@objective(m2, Max, 4y[2])

## Plasmo Graph
g = PlasmoGraph()
g.solver = GurobiSolver(OutputFlag=0)
n1 = add_node(g)
setmodel(n1,m1)
n2 = add_node(g)
setmodel(n2,m2)

## Linking
@linkconstraint(g, [i in 1:2], n1[:xm][i] == n2[:xs][i])

solution = lagrangesolve(g,options...)

solution = bendersolve(g,options...)

The algorithms will return a solution object with all relevant information about the solution process

The Lagrangean decomposition algorithm will dualize all linking constraints for any arbitrary graph. It could be a tree, it could be a sequence of nodes connected (e.g. temporal decomposition), or it may even contain cycles.

lagrangesolve(g::PlasmoGraph;update_method,ϵ,timelimit,lagrangeheuristic,initialmultipliers,,α,δ,maxnoimprove,cpbound), solves the input graph using the lagrangean decomposition algorithm

• update_method Multiplier update method
  • allowed values: :subgradient, :probingsubgradient, :marchingstep, :intersectionstep, :cuttingplanes
  • default: :subgradient
• ϵ Convergence tolerance
  • default: 0.001
• timelimit Algorithm time limit in seconds
  • default: 3600 (1 hour)
• lagrangeheuristic Function to solve the lagrangean heuristic. PlasmoAlgorithms provides 2 heuristic functions: fixbinaries, fixintegers
  • default: fixbinaries
• initialmultipliers initialization method for lagrangean multipliers. When :relaxation is selected the algorithm will use the multipliers from the LP relaxation
  • allowed values: :zero,:relaxation
  • default: zero
• α Initial value for the step parameter in subgradient methods
  • default: 2
• δ Shrinking factor for α
  • default: 0.5
• maxnoimprove Number of iterations without improvement before shrinking α
  • default: 3

It supports the following methods for updating the lagrangean multipliers:

• Subgradient
• Probing Subgradient
• Marching Step
• Intersection Step (experimental)
• Interactive
• Cutting Planes
• Cutting planes with trust region
• Levels