# 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
```

## Check if an optimal solution exists

Use `is_solved_and_feasible`

to check if the solver found an optimal solution:

```
julia> is_solved_and_feasible(model)
true
```

By default, `is_solved_and_feasible`

returns `true`

for both global and local optima. Pass `allow_local = false`

to check if the solver found a globally optimal solution:

```
julia> is_solved_and_feasible(model; allow_local = false)
true
```

Pass `dual = true`

to check if the solver found an optimal dual solution in addition to an optimal primal solution:

```
julia> is_solved_and_feasible(model; dual = true)
true
```

If this function returns `false`

, use the functions mentioned below like `solution_summary`

, `termination_status`

, `primal_status`

, and `dual_status`

to understand what solution (if any) the solver found.

## 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?

Use`termination_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`

.

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).

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
```

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
```

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
```

## Recommended workflow

You should always check whether the solver found a solution before calling solution functions like `value`

or `objective_value`

.

A simple approach for small scripts and notebooks is to use `is_solved_and_feasible`

:

```
julia> function solve_and_print_solution(model)
optimize!(model)
if !is_solved_and_feasible(model; dual = true)
error(
"""
The model was not solved correctly:
termination_status : $(termination_status(model))
primal_status : $(primal_status(model))
dual_status : $(dual_status(model))
raw_status : $(raw_status(model))
""",
)
end
println("Solution is optimal")
println(" objective value = ", objective_value(model))
println(" primal solution: x = ", value(x))
println(" dual solution: c1 = ", dual(c1))
return
end
solve_and_print_solution (generic function with 1 method)
julia> solve_and_print_solution(model)
Solution is optimal
objective value = -205.14285714285714
primal solution: x = 15.428571428571429
dual solution: c1 = 1.7142857142857142
```

For code like libraries that should be more robust to the range of possible termination and result statuses, do some variation of the following:

```
julia> function solve_and_print_solution(model)
status = termination_status(model)
if status in (OPTIMAL, LOCALLY_SOLVED)
println("Solution is optimal")
elseif status in (ALMOST_OPTIMAL, ALMOST_LOCALLY_SOLVED)
println("Solution is optimal to a relaxed tolerance")
elseif status == TIME_LIMIT
println(
"Solver stopped due to a time limit. If a solution is available, " *
"it may be suboptimal."
)
elseif status in (
ITERATION_LIMIT, NODE_LIMIT, SOLUTION_LIMIT, MEMORY_LIMIT,
OBJECTIVE_LIMIT, NORM_LIMIT, OTHER_LIMIT,
)
println(
"Solver stopped due to a limit. If a solution is available, it " *
"may be suboptimal."
)
elseif status in (INFEASIBLE, LOCALLY_INFEASIBLE)
println("The problem is primal infeasible")
elseif status == DUAL_INFEASIBLE
println(
"The problem is dual infeasible. If a primal feasible solution " *
"exists, the problem is unbounded. To check, set the objective " *
"to `@objective(model, Min, 0)` and re-solve. If the problem is " *
"feasible, the primal is unbounded. If the problem is " *
"infeasible, both the primal and dual are infeasible.",
)
elseif status == INFEASIBLE_OR_UNBOUNDED
println(
"The model is either infeasible or unbounded. Set the objective " *
"to `@objective(model, Min, 0)` and re-solve to disambiguate. If " *
"the problem was infeasible, it will still be infeasible. If the " *
"problem was unbounded, it will now have a finite optimal solution.",
)
else
println(
"The model was not solved correctly. The termination status is $status",
)
end
if primal_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT)
println(" objective value = ", objective_value(model))
println(" primal solution: x = ", value(x))
elseif primal_status(model) == INFEASIBILITY_CERTIFICATE
println(" primal certificate: x = ", value(x))
end
if dual_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT)
println(" dual solution: c1 = ", dual(c1))
elseif dual_status(model) == INFEASIBILITY_CERTIFICATE
println(" dual certificate: c1 = ", dual(c1))
end
return
end
solve_and_print_solution (generic function with 1 method)
julia> solve_and_print_solution(model)
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 `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.

For example, instead of:

```
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 `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
```

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`

.

Read the Sensitivity analysis of a linear program for more information on sensitivity analysis.

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[2], -0.5)
set_upper_bound(x[2], 0.5)
@constraint(model, c1, x[1] + x[2] <= 1)
@constraint(model, c2, x[1] - x[2] <= 1)
@objective(model, Max, x[1])
print(model)
end
Max x[1]
Subject to
c1 : x[1] + x[2] ≤ 1
c2 : x[1] - x[2] ≤ 1
x[2] ≥ -0.5
x[2] ≤ 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]]
(-1.0, Inf)
julia> println("The objective coefficient of x[1] could decrease by $(x1_lo) or increase by $(x1_hi).")
The objective coefficient of x[1] could decrease by -1.0 or increase by Inf.
julia> x2_lo, x2_hi = report[x[2]]
(-1.0, 1.0)
julia> println("The objective coefficient of x[2] could decrease by $(x2_lo) or increase by $(x2_hi).")
The objective coefficient of x[2] 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.

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 : 0
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]
```

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.

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}()
```