# Solutions

This section of the manual describes how to access a solved solution to a problem. It uses the following model as an example:

model = Model(GLPK.Optimizer)
@variable(model, x >= 0)
@variable(model, y[[:a, :b]] <= 1)
@objective(model, Max, -12x - 20y[:a])
@expression(model, my_expr, 6x + 8y[:a])
@constraint(model, my_expr >= 100)
@constraint(model, c1, 7x + 12y[:a] >= 120)
optimize!(model)
print(model)

# output

Max -12 x - 20 y[a]
Subject to
6 x + 8 y[a] ≥ 100.0
c1 : 7 x + 12 y[a] ≥ 120.0
x ≥ 0.0
y[a] ≤ 1.0
y[b] ≤ 1.0

## Solutions summary

solution_summary can be used for checking the summary of the optimization solutions.

julia> solution_summary(model)
* Solver : GLPK

* Status
Termination status : OPTIMAL
Primal status      : FEASIBLE_POINT
Dual status        : FEASIBLE_POINT
Message from the solver:
"Solution is optimal"

* Candidate solution
Objective value      : -205.14285714285714
Objective bound      : Inf
Dual objective value : -205.1428571428571

* Work counters
Solve time (sec)   : 0.00008

julia> solution_summary(model, verbose=true)
* Solver : GLPK

* Status
Termination status : OPTIMAL
Primal status      : FEASIBLE_POINT
Dual status        : FEASIBLE_POINT
Result count       : 1
Has duals          : true
Message from the solver:
"Solution is optimal"

* Candidate solution
Objective value      : -205.14285714285714
Objective bound      : Inf
Dual objective value : -205.1428571428571
Primal solution :
x : 15.428571428571429
y[a] : 1.0
y[b] : 1.0
Dual solution :
c1 : 1.7142857142857142

* Work counters
Solve time (sec)   : 0.00008

## Why did the solver stop?

Usetermination_status to understand why the solver stopped.

julia> termination_status(model)
OPTIMAL::TerminationStatusCode = 1

The MOI.TerminationStatusCode enum describes the full list of statuses that could be returned.

Common return values include MOI.OPTIMAL, MOI.LOCALLY_SOLVED, MOI.INFEASIBLE, MOI.DUAL_INFEASIBLE, and MOI.TIME_LIMIT.

Info

A return status of MOI.OPTIMAL means the solver found (and proved) a globally optimal solution. A return status of MOI.LOCALLY_SOLVED means the solver found a locally optimal solution (which may also be globally optimal, but it could not prove so).

Warning

A return status of MOI.DUAL_INFEASIBLE does not guarantee that the primal is unbounded. When the dual is infeasible, the primal is unbounded if there exists a feasible primal solution.

Use raw_status to get a solver-specific string explaining why the optimization stopped:

julia> raw_status(model)
"Solution is optimal"

## Primal solutions

### Primal solution status

Use primal_status to return an MOI.ResultStatusCode enum describing the status of the primal solution.

julia> primal_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

Other common returns are MOI.NO_SOLUTION, and MOI.INFEASIBILITY_CERTIFICATE. The first means that the solver doesn't have a solution to return, and the second means that the primal solution is a certificate of dual infeasbility (a primal unbounded ray).

You can also use has_values, which returns true if there is a solution that can be queried, and false otherwise.

julia> has_values(model)
true

### Objective values

The objective value of a solved problem can be obtained via objective_value. The best known bound on the optimal objective value can be obtained via objective_bound. If the solver supports it, the value of the dual objective can be obtained via dual_objective_value.

julia> objective_value(model)
-205.14285714285714

julia> objective_bound(model)  # GLPK only implements objective bound for MIPs
Inf

julia> dual_objective_value(model)
-205.1428571428571

### Primal solution values

If the solver has a primal solution to return, use value to access it:

julia> value(x)
15.428571428571429

Broadcast [value](@ref) over containers:

julia> value.(y)
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, Symbol[:a, :b]
And data, a 2-element Array{Float64,1}:
1.0
1.0

value also works on expressions:

julia> value(my_expr)
100.57142857142857

and constraints:

julia> value(c1)
120.0
Info

Calling value on a constraint returns the constraint function evaluated at the solution.

## Dual solutions

### Dual solution status

Use dual_status to return an MOI.ResultStatusCode enum describing the status of the dual solution.

julia> dual_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

Other common returns are MOI.NO_SOLUTION, and MOI.INFEASIBILITY_CERTIFICATE. The first means that the solver doesn't have a solution to return, and the second means that the dual solution is a certificate of primal infeasbility (a dual unbounded ray).

You can also use has_duals, which returns true if there is a solution that can be queried, and false otherwise.

julia> has_duals(model)
true

### Dual solution values

If the solver has a dual solution to return, use dual to access it:

julia> dual(c1)
1.7142857142857142

Query the duals of variable bounds using LowerBoundRef, UpperBoundRef, and FixRef:

julia> dual(LowerBoundRef(x))
0.0

julia> dual.(UpperBoundRef.(y))
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, Symbol[:a, :b]
And data, a 2-element Array{Float64,1}:
-0.5714285714285694
0.0
Warning

JuMP's definition of duality is independent of the objective sense. That is, the sign of feasible duals associated with a constraint depends on the direction of the constraint and not whether the problem is maximization or minimization. This is a different convention from linear programming duality in some common textbooks. If you have a linear program, and you want the textbook definition, you probably want to use shadow_price and reduced_cost instead.

julia> shadow_price(c1)
1.7142857142857142

julia> reduced_cost(x)
0.0

julia> reduced_cost.(y)
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, Symbol[:a, :b]
And data, a 2-element Array{Float64,1}:
0.5714285714285694
-0.0

The recommended workflow for solving a model and querying the solution is something like the following:

if termination_status(model) == MOI.OPTIMAL
println("Solution is optimal")
elseif termination_status(model) == MOI.TIME_LIMIT && has_values(model)
println("Solution is suboptimal due to a time limit, but a primal solution is available")
else
error("The model was not solved correctly.")
end
println("  objective value = ", objective_value(model))
if primal_status(model) == MOI.FEASIBLE_POINT
println("  primal solution: x = ", value(x))
end
if dual_status(model) == MOI.FEASIBLE_POINT
println("  dual solution: c1 = ", dual(c1))
end

# output

Solution is optimal
objective value = -205.14285714285714
primal solution: x = 15.428571428571429
dual solution: c1 = 1.7142857142857142
Warning

Querying solution information after modifying a solved model is undefined behavior, and solvers may throw an error or return incorrect results. Modifications include adding, deleting, or modifying any variable, objective, or constraint. Instead of modify-then-query, query the results first, then modify the problem. For example:

model = Model(GLPK.Optimizer)
@variable(model, x >= 0)
optimize!(model)
set_lower_bound(x, 1)
@show value(x)
# Good:
x_val = value(x)
set_lower_bound(x, 1)
@show x_val

## Accessing attributes

MathOptInterface defines a large number of model attributes that can be queried. Some attributes can be directly accessed by getter functions. These include:

## Sensitivity analysis for LP

Given an LP problem and an optimal solution corresponding to a basis, we can question how much an objective coefficient or standard form right-hand side coefficient (c.f., normalized_rhs) can change without violating primal or dual feasibility of the basic solution.

Note that not all solvers compute the basis, and for sensitivity analysis, the solver interface must implement MOI.ConstraintBasisStatus.

To give a simple example, we could analyze the sensitivity of the optimal solution to the following (non-degenerate) LP problem:

model = Model(GLPK.Optimizer)
@variable(model, x[1:2])
set_lower_bound(x, -0.5)
set_upper_bound(x, 0.5)
@constraint(model, c1, x + x <= 1)
@constraint(model, c2, x - x <= 1)
@objective(model, Max, x)
print(model)

# output

Max x
Subject to
c1 : x + x ≤ 1.0
c2 : x - x ≤ 1.0
x ≥ -0.5
x ≤ 0.5

To analyze the sensitivity of the problem we could check the allowed perturbation ranges of, e.g., the cost coefficients and the right-hand side coefficient of the constraint c1 as follows:

julia> optimize!(model)

julia> value.(x)
2-element Array{Float64,1}:
1.0
0.0

julia> report = lp_sensitivity_report(model);

julia> x1_lo, x1_hi = report[x]
(-1.0, Inf)

julia> println("The objective coefficient of x could decrease by $(x1_lo) or increase by$(x1_hi).")
The objective coefficient of x could decrease by -1.0 or increase by Inf.

julia> x2_lo, x2_hi = report[x]
(-1.0, 1.0)

julia> println("The objective coefficient of x could decrease by $(x2_lo) or increase by$(x2_hi).")
The objective coefficient of x could decrease by -1.0 or increase by 1.0.

julia> c_lo, c_hi = report[c1]
(-1.0, 1.0)

julia> println("The RHS of c1 could decrease by $(c_lo) or increase by$(c_hi).")
The RHS of c1 could decrease by -1.0 or increase by 1.0.

The range associated with a variable is the range of the allowed perturbation of the corresponding objective coefficient. Note that the current primal solution remains optimal within this range; however the corresponding dual solution might change since a cost coefficient is perturbed. Similarly, the range associated with a constraint is the range of the allowed perturbation of the corresponding right-hand side coefficient. In this range the current dual solution remains optimal, but the optimal primal solution might change.

If the problem is degenerate, there are multiple optimal bases and hence these ranges might not be as intuitive and seem too narrow. E.g., a larger cost coefficient perturbation might not invalidate the optimality of the current primal solution. Moreover, if a problem is degenerate, due to finite precision, it can happen that, e.g., a perturbation seems to invalidate a basis even though it doesn't (again providing too narrow ranges). To prevent this, increase the atol keyword argument to lp_sensitivity_report. Note that this might make the ranges too wide for numerically challenging instances. Thus, do not blindly trust these ranges, especially not for highly degenerate or numerically unstable instances.

## Conflicts

When the model you input is infeasible, some solvers can help you find the cause of this infeasibility by offering a conflict, i.e., a subset of the constraints that create this infeasibility. Depending on the solver, this can also be called an IIS (irreducible inconsistent subsystem).

The function compute_conflict! is used to trigger the computation of a conflict. Once this process is finished, the attribute MOI.ConflictStatus returns a MOI.ConflictStatusCode.

If there is a conflict, you can query from each constraint whether it participates in the conflict or not using the attribute MOI.ConstraintConflictStatus, which returns a MOI.ConflictParticipationStatusCode.

To create a new model containing only the constraints that participate in the conflict, use copy_conflict. It may be helpful to write this model to a file for easier debugging using write_to_file.

For instance, this is how you can use this functionality:

using JuMP
model = Model() # You must use a solver that supports conflict refining/IIS
# computation, like CPLEX or Gurobi
@variable(model, x >= 0)
@constraint(model, c1, x >= 2)
@constraint(model, c2, x <= 1)
optimize!(model)

# termination_status(model) will likely be MOI.INFEASIBLE,
# depending on the solver

compute_conflict!(model)
if MOI.get(model, MOI.ConflictStatus()) != MOI.CONFLICT_FOUND
error("No conflict could be found for an infeasible model.")
end

# Both constraints should participate in the conflict.
MOI.get(model, MOI.ConstraintConflictStatus(), c1)
MOI.get(model, MOI.ConstraintConflictStatus(), c2)

# Get a copy of the model with only the constraints in the conflict.
new_model, reference_map = copy_conflict(model)

## Multiple solutions

Some solvers support returning multiple solutions. You can check how many solutions are available to query using result_count.

Functions for querying the solutions, e.g., primal_status and value, all take an additional keyword argument result which can be used to specify which result to return.

Warning

Even if termination_status is MOI.OPTIMAL, some of the returned solutions may be suboptimal! However, if the solver found at least one optimal solution, then result = 1 will always return an optimal solution. Use objective_value to assess the quality of the remaining solutions.

using JuMP
model = Model()
@variable(model, x[1:10] >= 0)
# ... other constraints ...
optimize!(model)

if termination_status(model) != MOI.OPTIMAL
error("The model was not solved correctly.")
end

an_optimal_solution = value.(x; result = 1)
optimal_objective = objective_value(model; result = 1)
for i in 2:result_count(model)
@assert has_values(model; result = i)
println("Solution $(i) = ", value.(x; result = i)) obj = objective_value(model; result = i) println("Objective$(i) = ", obj)
if isapprox(obj, optimal_objective; atol = 1e-8)
print("Solution \$(i) is also optimal!")
end
end

## Checking feasibility of solutions

To check the feasibility of a primal solution, use primal_feasibility_report, which takes a model, a dictionary mapping each variable to a primal solution value (defaults to the last solved solution), and a tolerance atol (defaults to 0.0).

The function returns a dictionary which maps the infeasible constraint references to the distance between the primal value of the constraint and the nearest point in the corresponding set. A point is classed as infeasible if the distance is greater than the supplied tolerance atol.

julia> model = Model(GLPK.Optimizer);

julia> @variable(model, x >= 1, Int);

julia> @variable(model, y);

julia> @constraint(model, c1, x + y <= 1.95);

julia> point = Dict(x => 1.9, y => 0.06);

julia> primal_feasibility_report(model, point)
Dict{Any,Float64} with 2 entries:
c1 : x + y ≤ 1.95 => 0.01
x integer         => 0.1

julia> primal_feasibility_report(model, point; atol = 0.02)
Dict{Any,Float64} with 1 entry:
x integer => 0.1

If the point is feasible, an empty dictionary is returned:

julia> primal_feasibility_report(model, Dict(x => 1.0, y => 0.0))
Dict{Any,Float64} with 0 entries

To use the primal solution from a solve, omit the point argument:

julia> optimize!(model)

julia> primal_feasibility_report(model)
Dict{Any,Float64} with 0 entries

Pass skip_mising = true to skip constraints which contain variables that are not in point:

julia> primal_feasibility_report(model, Dict(x => 2.1); skip_missing = true)
Dict{Any,Float64} with 1 entry:
x integer => 0.1