A solver for Mixed-Integer Convex Optimization that uses Frank-Wolfe methods for convex relaxations and a branch-and-bound algorithm.
The Boscia.jl solver combines (a variant of) the Frank-Wolfe algorithm with a branch-and-bound like algorithm to solve mixed-integer convex optimization problems of the form
min_{x ∈ C, x_I ∈ Z^n} f(x),
where f is a differentiable convex function, C is a convex and compact set, and I is a set of indices of integral variables.
They are especially useful when we have a method to optimize a linear function over C and the integrality constraints in a compuationally efficient way.
C is specified using the MathOptInterface API or any DSL like JuMP implementing it.
A paper presenting the package with mathematical explanations and numerous examples can be found here:
Convex integer optimization with Frank-Wolfe methods: 2208.11010
Boscia.jl uses FrankWolfe.jl for solving the convex subproblems, Bonobo.jl for managing the search tree, and oracles optimizing linear functions over the feasible set, for instance calling SCIP or any MOI-compatible solver to solve MIP subproblems.
Add the Boscia stable release with:
import Pkg
Pkg.add("Boscia")Or get the latest master branch with:
import Pkg
Pkg.add(url="https://github.com/ZIB-IOL/Boscia.jl", rev="main")Here is a simple example to get started. For more examples, see the examples folder in the package.
using Boscia
using FrankWolfe
using Random
using SCIP
using LinearAlgebra
import MathOptInterface
const MOI = MathOptInterface
n = 6
const diffw = 0.5 * ones(n)
o = SCIP.Optimizer()
MOI.set(o, MOI.Silent(), true)
x = MOI.add_variables(o, n)
for xi in x
MOI.add_constraint(o, xi, MOI.GreaterThan(0.0))
MOI.add_constraint(o, xi, MOI.LessThan(1.0))
MOI.add_constraint(o, xi, MOI.ZeroOne())
end
lmo = FrankWolfe.MathOptLMO(o)
function f(x)
return sum(0.5*(x.-diffw).^2)
end
function grad!(storage, x)
@. storage = x-diffw
end
x, _, result = Boscia.solve(f, grad!, lmo, verbose = true)
Boscia Algorithm.
Parameter settings.
Tree traversal strategy: Move best bound
Branching strategy: Most infeasible
Absolute dual gap tolerance: 1.000000e-06
Relative dual gap tolerance: 1.000000e-02
Frank-Wolfe subproblem tolerance: 1.000000e-05
Total number of varibales: 6
Number of integer variables: 0
Number of binary variables: 6
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Iteration Open Bound Incumbent Gap (abs) Gap (rel) Time (s) Nodes/sec FW (ms) LMO (ms) LMO (calls c) FW (Its) #ActiveSet Discarded
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
* 1 2 -1.202020e-06 7.500000e-01 7.500012e-01 Inf 3.870000e-01 7.751938e+00 237 2 9 13 1 0
100 27 6.249998e-01 7.500000e-01 1.250002e-01 2.000004e-01 5.590000e-01 2.271914e+02 0 0 641 0 1 0
127 0 7.500000e-01 7.500000e-01 0.000000e+00 0.000000e+00 5.770000e-01 2.201040e+02 0 0 695 0 1 0
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Postprocessing
Blended Pairwise Conditional Gradient Algorithm.
MEMORY_MODE: FrankWolfe.InplaceEmphasis() STEPSIZE: Adaptive EPSILON: 1.0e-7 MAXITERATION: 10000 TYPE: Float64
GRADIENTTYPE: Nothing LAZY: true lazy_tolerance: 2.0
[ Info: In memory_mode memory iterates are written back into x0!
----------------------------------------------------------------------------------------------------------------
Type Iteration Primal Dual Dual Gap Time It/sec #ActiveSet
----------------------------------------------------------------------------------------------------------------
Last 0 7.500000e-01 7.500000e-01 0.000000e+00 1.086583e-03 0.000000e+00 1
----------------------------------------------------------------------------------------------------------------
PP 0 7.500000e-01 7.500000e-01 0.000000e+00 1.927792e-03 0.000000e+00 1
----------------------------------------------------------------------------------------------------------------
Solution Statistics.
Solution Status: Optimal (tree empty)
Primal Objective: 0.75
Dual Bound: 0.75
Dual Gap (relative): 0.0
Search Statistics.
Total number of nodes processed: 127
Total number of lmo calls: 699
Total time (s): 0.58
LMO calls / sec: 1205.1724137931035
Nodes / sec: 218.96551724137933
LMO calls / node: 5.503937007874016