# Solutions

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

julia> begin
model = Model(HiGHS.Optimizer)
set_silent(model)
@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)
end
Max -12 x - 20 y[a]
Subject to
6 x + 8 y[a] ≥ 100
c1 : 7 x + 12 y[a] ≥ 120
x ≥ 0
y[a] ≤ 1
y[b] ≤ 1

## Solutions summary

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

julia> solution_summary(model)
* Solver : HiGHS

* Status
Result count       : 1
Termination status : OPTIMAL
Message from the solver:
"kHighsModelStatusOptimal"

* Candidate solution (result #1)
Primal status      : FEASIBLE_POINT
Dual status        : FEASIBLE_POINT
Objective value    : -2.05143e+02
Objective bound    : -0.00000e+00
Relative gap       : Inf
Dual objective value : -2.05143e+02

* Work counters
Solve time (sec)   : 6.70987e-04
Simplex iterations : 2
Barrier iterations : 0
Node count         : -1

julia> solution_summary(model; verbose = true)
* Solver : HiGHS

* Status
Result count       : 1
Termination status : OPTIMAL
Message from the solver:
"kHighsModelStatusOptimal"

* Candidate solution (result #1)
Primal status      : FEASIBLE_POINT
Dual status        : FEASIBLE_POINT
Objective value    : -2.05143e+02
Objective bound    : -0.00000e+00
Relative gap       : Inf
Dual objective value : -2.05143e+02
Primal solution :
x : 1.54286e+01
y[a] : 1.00000e+00
y[b] : 1.00000e+00
Dual solution :
c1 : 1.71429e+00

* Work counters
Solve time (sec)   : 6.70987e-04
Simplex iterations : 2
Barrier iterations : 0
Node count         : -1

## 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 OPTIMAL, LOCALLY_SOLVED, INFEASIBLE, DUAL_INFEASIBLE, and TIME_LIMIT.

Info

A return status of OPTIMAL means the solver found (and proved) a globally optimal solution. A return status of 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 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)
"kHighsModelStatusOptimal"

## 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 NO_SOLUTION, and 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 infeasibility (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)  # HiGHS only implements objective bound for MIPs
-0.0

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 over containers:

julia> value.(y)
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, [:a, :b]
And data, a 2-element Vector{Float64}:
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 NO_SOLUTION, and 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 infeasibility (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, [:a, :b]
And data, a 2-element Vector{Float64}:
-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, [:a, :b]
And data, a 2-element Vector{Float64}:
0.5714285714285694
-0.0

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

julia> begin
if termination_status(model) == OPTIMAL
println("Solution is optimal")
elseif termination_status(model) == 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) == FEASIBLE_POINT
println("  primal solution: x = ", value(x))
end
if dual_status(model) == FEASIBLE_POINT
println("  dual solution: c1 = ", dual(c1))
end
end
Solution is optimal
objective value = -205.14285714285714
primal solution: x = 15.428571428571429
dual solution: c1 = 1.7142857142857142

## OptimizeNotCalled errors

Due to differences in how solvers cache solutions internally, modifying a model after calling optimize! will reset the model into the MOI.OPTIMIZE_NOT_CALLED state. If you then attempt to query solution information, an OptimizeNotCalled error will be thrown.

If you are iteratively querying solution information and modifying a model, query all the results first, then modify the problem.

julia> model = Model(HiGHS.Optimizer);

julia> set_silent(model)

julia> @variable(model, x >= 0);

julia> optimize!(model)

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

julia> set_upper_bound(x, 1)

julia> x_val = value(x)
┌ Warning: The model has been modified since the last call to optimize! (or optimize! has not been called yet). If you are iteratively querying solution information and modifying a model, query all the results first, then modify the model.
└ @ JuMP ~/work/JuMP.jl/JuMP.jl/src/optimizer_interface.jl:712
ERROR: OptimizeNotCalled()
Stacktrace:
[...]

julia> termination_status(model)
OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0

do

julia> model = Model(HiGHS.Optimizer);

julia> set_silent(model)

julia> @variable(model, x >= 0);

julia> optimize!(model);

julia> x_val = value(x)
0.0

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

julia> set_upper_bound(x, 1)

julia> set_lower_bound(x, x_val)

julia> termination_status(model)
OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0

If you know that your particular solver supports querying solution information after modifications, you can use direct_model to bypass the MOI.OPTIMIZE_NOT_CALLED state:

julia> model = direct_model(HiGHS.Optimizer());

julia> set_silent(model)

julia> @variable(model, x >= 0);

julia> optimize!(model)

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

julia> set_upper_bound(x, 1)

julia> x_val = value(x)
0.0

julia> set_lower_bound(x, x_val)

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

Be careful doing this. If your particular solver does not support querying solution information after modification, it may silently return incorrect solutions or throw an error.

## Accessing attributes

MathOptInterface defines many 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.

Tip

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

julia> begin
model = Model(HiGHS.Optimizer)
set_silent(model)
@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)
end
Max x
Subject to
c1 : x + x ≤ 1
c2 : x - x ≤ 1
x ≥ -0.5
x ≤ 0.5

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

julia> optimize!(model)

julia> value.(x)
2-element Vector{Float64}:
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, for example, 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, for example, 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, that is, a subset of the constraints that create this infeasibility. Depending on the solver, this can also be called an IIS (irreducible inconsistent subsystem).

If supported by the solver, use compute_conflict! to trigger the computation of a conflict. Once this process is finished, query the MOI.ConflictStatus attribute to check if a conflict was found.

If found, copy the IIS to a new model using copy_conflict, which you can then print or write to a file for easier debugging:

julia> using JuMP

julia> import Gurobi

julia> model = Model(Gurobi.Optimizer)
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: EMPTY_OPTIMIZER
Solver name: Gurobi

julia> set_silent(model)

julia> @variable(model, x >= 0)
x

julia> @constraint(model, c1, x >= 2)
c1 : x ≥ 2.0

julia> @constraint(model, c2, x <= 1)
c2 : x ≤ 1.0

julia> optimize!(model)

julia> compute_conflict!(model)

julia> if get_attribute(model, MOI.ConflictStatus()) == MOI.CONFLICT_FOUND
iis_model, _ = copy_conflict(model)
print(iis_model)
end
Feasibility
Subject to
c1 : x ≥ 2.0
c2 : x ≤ 1.0

If you need more control over the list of constraints that appear in the conflict, iterate over the list of constraints and query the MOI.ConstraintConflictStatus attribute:

julia> list_of_conflicting_constraints = ConstraintRef[]
ConstraintRef[]

julia> for (F, S) in list_of_constraint_types(model)
for con in all_constraints(model, F, S)
if get_attribute(con, MOI.ConstraintConflictStatus()) == MOI.IN_CONFLICT
push!(list_of_conflicting_constraints, con)
end
end
end

julia> list_of_conflicting_constraints
2-element Vector{ConstraintRef}:
c1 : x ≥ 2.0
c2 : x ≤ 1.0

## 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, for example, primal_status, dual_status, value, dual, and solution_summary all take an additional keyword argument result which can be used to specify which result to return.

Warning

Even if termination_status is 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.

julia> using JuMP

julia> import MultiObjectiveAlgorithms as MOA

julia> import HiGHS

julia> model = Model(() -> MOA.Optimizer(HiGHS.Optimizer));

julia> set_attribute(model, MOA.Algorithm(), MOA.Dichotomy())

julia> set_silent(model)

julia> @variable(model, x1 >= 0)
x1

julia> @variable(model, 0 <= x2 <= 3)
x2

julia> @objective(model, Min, [3x1 + x2, -x1 - 2x2])
2-element Vector{AffExpr}:
3 x1 + x2
-x1 - 2 x2

julia> @constraint(model, 3x1 - x2 <= 6)
3 x1 - x2 ≤ 6

julia> optimize!(model)

julia> solution_summary(model; result = 1)
* Solver : MOA[algorithm=MultiObjectiveAlgorithms.Dichotomy, optimizer=HiGHS]

* Status
Result count       : 3
Termination status : OPTIMAL
Message from the solver:
"Solve complete. Found 3 solution(s)"

* Candidate solution (result #1)
Primal status      : FEASIBLE_POINT
Dual status        : NO_SOLUTION
Objective value    : [0.00000e+00,0.00000e+00]
Objective bound    : [0.00000e+00,-9.00000e+00]
Relative gap       : Inf
Dual objective value : -9.00000e+00

* Work counters
Solve time (sec)   : 1.82720e-03
Simplex iterations : 1
Barrier iterations : 0
Node count         : -1

julia> for i in 1:result_count(model)
println("Solution \$i")
println("   x = ", value.([x1, x2]; result = i))
println(" obj = ", objective_value(model; result = i))
end
Solution 1
x = [0.0, 0.0]
obj = [0.0, 0.0]
Solution 2
x = [0.0, 3.0]
obj = [3.0, -6.0]
Solution 3
x = [3.0, 3.0]
obj = [12.0, -9.0]
Tip

The Multi-objective knapsack tutorial provides more examples of querying multiple solutions.

## 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(HiGHS.Optimizer);

julia> set_silent(model)

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:
x integer         => 0.1
c1 : x + y ≤ 1.95 => 0.01

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}()

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

julia> optimize!(model)

julia> primal_feasibility_report(model; atol = 0.0)
Dict{Any, Float64}()

Calling primal_feasibility_report without the point argument is useful when primal_status is FEASIBLE_POINT or NEARLY_FEASIBLE_POINT, and you want to assess the solution quality.

Warning

To apply primal_feasibility_report to infeasible models, you must also provide a candidate point (solvers generally do not provide one). To diagnose the source of infeasibility, see Conflicts.

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

You can also use the functional form, where the first argument is a function that maps variables to their primal values:

julia> optimize!(model)

julia> primal_feasibility_report(v -> value(v), model)
Dict{Any, Float64}()