Last Friday was a great seminar of the Combinatorial Optimization group in Paris, celebrating the 85th birthday of Jack Edmonds, one of the founding researchers of combinatorial optimization, with the notable Blossom matching algorithm.

.@SoniaVanier opened the workshop and organized a great party at Sorbonne for Jack Edmonds. I had a great honor to be one of the speakers at this event #orms pic.twitter.com/oHwKvg43Zm — Ivana Ljubic (@ILjubic) May 3, 2019

Laurence Wolsey and Ivana Ljubic were both giving talks on applications and developments in Benders decompositions. It also made me want to refresh my knowledge of the subject and play a bit with a simple implementation.

Laurence Wolsey talks about Benders decomposition at the Jack Edmonds birthday workshop at Sorbonne #orms pic.twitter.com/K8hjdqKmwQ — Ivana Ljubic (@ILjubic) May 3, 2019

High-level idea

Problem decompositions are used on large-scale optimization problems with a particular structure. The decomposition turns a compact, hard-to-solve formulation into an easier one but of great size. In the case of Benders, great size means a number of constraints growing exponentially with the size of the input problem. Adding all constraints upfront would be too costly. Furthermore, in general, only a small fraction of these constraints will be active in a final solution, the associated algorithm is to generate them incrementally, re-solve the problem with the new constraint until no relevant constraint can be found anymore.

We can establish a more general pattern of on-the-fly addition of information to an optimization problem, which entails two components:

An incrementally-built problem, called Restricted Master Problem (RMP) in decomposition. An oracle or sub-problem, taking the problem state and building the new required structure (here a new constraint).

Sounds familiar? Benders can be seen as the “dual twin” of the Dantzig-Wolfe decomposition I had played with in a previous post.

Digging into the structure

Now that we have a general idea of the problem at hand, let’s see the specifics. Consider a problem such as: $$ \min_{x,y} f(y) + c^T x $$ s.t. $$ G(y) \in \mathcal{S}$$ $$ A x + D y \geq b $$ $$ x \in \mathbb{R}^{n_1}_{+}, y \in \mathcal{Y} $$

We will not consider the constraints specific to $y$ (the first row) nor the $y$-component of the objective. The key assumption of Benders is that if the $y$ are fixed, the problem on the $x$ variables is fast to solve. Lots of heuristics use this idea of “fix-and-optimize” to avoid incorporating the “hard” variables in the problem, Benders leverages several properties to bring the idea to exact methods (exact in the sense of proven optimality).

Taking the problem above, we can simplify the structure by abstracting away (i.e. projecting out) the $x$ part: $$ \min_{y} f(y) + \phi(y) $$ s.t. $$ G(y) \in \mathcal{S}$$ $$ y \in \mathcal{Y} $$

Where: $$ \phi(y) = \min_{x} \{c^T x, Ax \geq b - Dy, x \geq 0 \} $$

$\phi(y)$ is a non-smooth function, with $, dom\ \phi ,$ the feasible domain of the problem. If you are familiar with bilevel optimization, this could remind you of the optimal value function used to describe lower-level problems. We will call $SP$ the sub-problem defined in the function $\phi$.

The essence of Benders is to start from an outer-approximation (overly optimistic) by replacing $\phi$ with a variable $\eta$ which might be higher than the min value, and then add cuts which progressively constrain the problem. The initial outer-approximation is:

$$ \min_{y,\eta} f(y) + \eta $$ s.t. $$ G(y) \in \mathcal{S}$$ $$ y \in \mathcal{Y} $$

Of course since $\eta$ is unconstrained, the problem will start unbounded. What are valid cuts for this? Let us define the dual of the sub-problem $SP$, which we will name $DSP$: $$ \max_{\alpha} (b - Dy)^T \alpha $$ s.t. $$ A^T \alpha \leq c $$ $$ \alpha \geq 0 $$

Given that $\eta \geq min SP$, by duality, $\eta \geq max DSP$. Furthermore, by strong duality of linear problems, if $\eta = \min \max_{y} DSP$, it is exactly equal to the minimum of $\phi(y)$ and yields the optimal solution.

One thing to note about the feasible domain of $DSP$, it does not depend on the value of $y$. This means $z$ feasible for all values of the dual is equivalent to being feasible for all extreme points and rays of the dual polyhedron. Each of these can yield a new cut to add to the relaxed problem. For the sake of conciseness, I will not go into details on the case when the sub-problem is not feasible for a $y$ solution. Briefly, this is equivalent to the dual being unbounded, it thus defines an extreme ray which must be cut out. For more details, you can check these lecture notes.

A JuMP implementation

We will define a simple implementation using JuMP, a generic optimization modeling library on top of Julia, usable with various solvers. Since the master and sub-problem resolutions are completely independent, they can be solved in separated software components, even with different solvers. To highlight this, we will use SCIP to solve the master problem and COIN-OR’s Clp to solve the sub-problem.

We can start by importing the required packages:

using JuMP import SCIP import Clp using LinearAlgebra : dot

Defining and solving dual sub-problems

Let us store static sub-problem data in a structure:

struct SubProblemData b :: Vector { Float64 } D :: Matrix { Float64 } A :: Matrix { Float64 } c :: Vector { Float64 } end

And the dual sub-problem is entirely contained in another structure:

struct DualSubProblem data :: SubProblemData α :: Vector {VariableRef} m :: Model end function DualSubProblem(d :: SubProblemData, m :: Model) α = @variable (m, α[i = 1 : size(d . A, 1 )] >= 0 ) @constraint (m, dot(d . A, α) .<= d . c) return DualSubProblem(d, α, m) end

The DualSubProblem is constructed from the static data and a JuMP model. We mentioned that the feasible space of the sub-problem is independent of the value of $y$, thus we can add the constraint right away. Only to optimize it do we require the $\hat{y}$ value, which is used to set the objective. We can then either return a feasibility cut or optimality cut depending on the solution status of the dual sub-problem:

function JuMP . optimize!(sp :: DualSubProblem, yh) obj = sp . data . b .- sp . data . D * yh @objective (sp . m, Max, dot(obj, sp . α)) optimize!(sp . m) st = termination_status(sp . m) if st == MOI . OPTIMAL α = JuMP . value . (sp . α) return ( : OptimalityCut, α) elseif st == MOI . DUAL_INFEASIBLE return ( : FeasibilityCut, α) else error( "DualSubProblem error: status $status " ) end end

Iterating on the master problem

The main part of the resolution holds here in three steps.

Initialize a master problem with variables $(y,\eta)$ Optimize and pass the $\hat{y}$ value to the sub-problem. Get back a dual value $\alpha$ from the dual sub-problem Is the constraint generated by the $\alpha$ value already respected?

If yes, the solution is optimal.

If no, add the corresponding cut to the master problem, return to 2.

function benders_optimize!(m :: Model, y :: Vector {VariableRef}, sd :: SubProblemData, sp_optimizer, f :: Union { Function , Type }; eta_bound :: Real = - 1000.0 ) subproblem = Model(with_optimizer(sp_optimizer)) dsp = DualSubProblem(sd, subproblem) @variable (m, η >= eta_bound) @objective (m, Min, f(y) + η) optimize!(m) st = MOI . get(m, MOI . TerminationStatus()) # restricted master has a solution or is unbounded nopt_cons, nfeas_cons = ( 0 , 0 ) @info "Initial status $st " cuts = Tuple { Symbol , Vector { Float64 }}[] while (st == MOI . DUAL_INFEASIBLE) || (st == MOI . OPTIMAL) optimize!(m) st = MOI . get(m, MOI . TerminationStatus()) ŷ = JuMP . value . (y) η0 = JuMP . value(η) (res, α) = optimize!(dsp, ŷ) if res == : OptimalityCut @info "Optimality cut found" if η0 ≥ dot(α, (dsp . data . b - dsp . data . D * ŷ)) break else nopt_cons += 1 @constraint (m, η ≥ dot(α, (dsp . data . b - dsp . data . D * y))) end else @info "Feasibility cut found" nfeas_cons += 1 @constraint (m, 0 ≥ dot(α, (dsp . data . b - dsp . data . D * y))) end push!(cuts, (res, α)) end return (m, y, cuts, nopt_cons, nfeas_cons) end

Note that we pass the function an already-built model with variable $y$ defined. This allows for a prior flexible definition of constraints of the type: $$y \in \mathcal{Y}$$ $$G(y) \in \mathcal{S}$$

Also, we return the $\alpha$ values found by the sub-problems and the number of cuts of each type. Finally, one “hack” I’m using is to give an arbitrary lower bound on the $\eta$ value, making it (almost) sure to have a bounded initial problem and thus a defined initial solution $y$.

We will re-use the small example from the lecture notes above:

function test_data() c = [ 2. , 3. ] A = [ 1 2 ; 2 - 1 ] D = zeros( 2 , 1 ) .+ [ 1 , 3 ] b = [ 3 , 4 ] return SimpleBenders . SubProblemData(b, D, A, c) end data = test_data() # objective function on y f(v) = 2 v[ 1 ] # initialize the problem m = Model(with_optimizer(SCIP . Optimizer)) @variable (m, y[j = 1 : 1 ] >= 0 ) # solve and voilà (m, y, cuts, nopt_cons, nfeas_cons) = SimpleBenders . benders_optimize!(m, y, data, () -> Clp . Optimizer(LogLevel = 0 ), f)

The full code is available on Github, run it, modify it and don’t hesitate to submit pull requests and issues, I’m sure there are :)

Benders is a central pillar for various problems in optimization, research is still very active to bring it to non-linear convex or non-convex sub-problems where duality cannot be used. If you liked this post or have questions, don’t hesitate to react or ping me on Twitter.