Set the AUGMECON multi-objective method

Configure a Problem object to be solved with the augmented epsilon-constraint method (AUGMECON).

AUGMECON is an exact multi-objective optimization method in which one objective is treated as the primary objective and the remaining objectives are converted into \(\varepsilon\)-constraints. In the augmented formulation, each secondary objective is associated with a non-negative slack variable, and the primary objective is augmented with a small reward term based on the normalized slacks. This augmentation is used to avoid weakly efficient solutions, following Mavrotas (2009).

This function does not solve the problem directly. It stores the AUGMECON configuration in x$data$method, to be used later by solve.

Usage

set_method_augmecon(
  x,
  primary,
  aliases = NULL,
  runs = NULL,
  grid = NULL,
  n_points = NULL,
  include_extremes = NULL,
  lexicographic = TRUE,
  lexicographic_tol = 1e-09,
  augmentation = 0.001,
  slack_upper_bound = 1e+06,
  control = NULL
)

Arguments

x

A Problem object.

primary

Character string giving the alias of the primary objective, that is, the objective optimized directly in the AUGMECON formulation.

aliases

Optional character vector of objective aliases to include in the method. If NULL, all registered objective aliases are used. The value of primary must be included in aliases.

runs

A run design created with set_runs_grid or set_runs_manual. For AUGMECON, set_runs_grid() requests automatic epsilon-level generation for secondary objectives, while set_runs_manual() requires columns named eps_<alias> for each secondary objective.

grid

Deprecated. Previous manual-grid argument. It must be a named list with one numeric vector per secondary objective. New code should use runs = set_runs_manual(...) instead.

n_points

Deprecated. Previous automatic-grid argument. New code should use runs = set_runs_grid(n = ...) instead.

include_extremes

Deprecated and ignored. Automatic run grids now always include boundary levels. New code should use runs = set_runs_grid(n = ...).

lexicographic

Logical. If TRUE, use lexicographic anchoring when computing extreme points for automatic grid construction.

lexicographic_tol

Non-negative numeric tolerance used in lexicographic anchoring.

augmentation

Positive numeric augmentation coefficient \(\rho\). The effective coefficient of each secondary slack is computed internally as \(\rho / R_k\), where \(R_k\) is the payoff-table range of the corresponding secondary objective.

slack_upper_bound

A single positive finite numeric value defining the upper bound of AUGMECON slack variables. Defaults to 1e6.

control

A control object created with set_runs_control. It controls how infeasible runs, runs without a solution, and unexpected errors are handled.

Value

The updated Problem object with the AUGMECON method configuration stored in x$data$method.

Details

Use this method when one objective should be optimized directly, the remaining objectives should be controlled through epsilon levels, and weakly efficient solutions should be reduced through the augmented formulation.

General idea

Suppose that \(m \ge 2\) objective functions have already been registered in the problem: $$ f_1(x), f_2(x), \dots, f_m(x). $$

AUGMECON selects one of them as the primary objective, say \(f_p(x)\), and treats the remaining \(m - 1\) objectives as secondary objectives.

For a fixed combination of epsilon levels, the method solves a single-objective subproblem of the form:

$$ \max \; f_p(x) + \rho \sum_{k \in \mathcal{S}} \frac{s_k}{R_k} $$

subject to

$$ f_k(x) - s_k = \varepsilon_k, \qquad k \in \mathcal{S}, $$

$$ s_k \ge 0, \qquad k \in \mathcal{S}, $$

together with all original feasibility constraints of the planning problem.

Here:

• \(f_p(x)\) is the primary objective,

• \(\mathcal{S}\) is the set of secondary objectives,

• \(\varepsilon_k\) is the imposed level for secondary objective \(k\),

• \(s_k\) is a non-negative slack variable,

• \(R_k\) is the payoff-table range used to normalize objective \(k\),

• \(\rho > 0\) is a small augmentation coefficient.

In the original AUGMECON formulation of Mavrotas (2009), the augmentation term ensures that, among solutions with the same primary objective value, the solver prefers those with larger normalized slack, thereby avoiding weakly efficient points and improving Pareto-front generation.

Secondary-objective equalities and slacks

The key difference between standard epsilon-constraint and AUGMECON is that the secondary objectives are written as equalities with slacks rather than as simple inequalities. For a maximization-type secondary objective, this takes the form:

$$ f_k(x) - s_k = \varepsilon_k, \qquad s_k \ge 0. $$

This implies: $$ f_k(x) \ge \varepsilon_k, $$

while explicitly measuring the excess above the imposed epsilon level through \(s_k\). The augmentation term then rewards such excess in normalized form.

In implementation terms, the exact sign convention for each objective depends on whether it is internally treated as a minimization or maximization objective, but the method always preserves the same AUGMECON principle:

• one objective is optimized directly,

• all others are turned into constrained objectives,

• non-negative slacks measure controlled deviation from the imposed epsilon levels,

• the primary objective is augmented with a small slack-based reward.

Run designs

AUGMECON runs are specified through the runs argument. This argument must be created with either set_runs_grid or set_runs_manual.

set_runs_grid(n = ...) requests automatic generation of epsilon levels for the secondary objectives during solve. In that case, the method first computes extreme points and payoff-table ranges for the secondary objectives, and then generates n levels for each one.

Boundary epsilon levels are always included in automatic grids. Therefore, the lower and upper bounds of the automatically derived epsilon ranges are part of the generated run design.

set_runs_manual() allows users to provide explicit epsilon combinations. In manual AUGMECON runs, each row is one optimization run and columns must be named eps_<alias>, where <alias> is the alias of a secondary objective. For example, if primary = "benefit" and aliases = c("benefit", "cost", "loss"), the manual run table must contain columns eps_cost and eps_loss.

In set_runs_manual(), each row is used exactly as supplied. The function does not automatically create a Cartesian product of epsilon values. If a Cartesian product is desired, it should be created explicitly by the user, for example with expand.grid, and then passed to set_runs_manual().

The older arguments grid, n_points, and include_extremes are deprecated. They are still accepted for backwards compatibility and are internally converted to set_runs_grid() or set_runs_manual() designs. The deprecated include_extremes argument is ignored because automatic run grids now always include boundary levels.

Automatic epsilon grids

When runs = set_runs_grid(n = ...) is used, the epsilon design is not built immediately. Instead, it is constructed later during solve using extreme-point or payoff-table information.

For each secondary objective, set_runs_grid() generates a sequence of epsilon levels. With multiple secondary objectives, the final AUGMECON design is the Cartesian product of these sequences. Therefore, the number of runs can grow quickly as the number of secondary objectives increases.

If lexicographic = TRUE, extreme points are computed using lexicographic anchoring, which can improve payoff-table quality when objectives are tightly competing. The tolerance used for lexicographic anchoring is controlled by lexicographic_tol.

Manual epsilon runs

Manual run designs are the most explicit way to use AUGMECON, especially when more than two objectives are involved or when only selected epsilon combinations should be explored.

For example, with one primary objective and two secondary objectives, a manual run design may contain:

data.frame(
  eps_cost = c(4, 6, 8),
  eps_loss = c(0, 1, 1)
)

This creates three runs, not a full Cartesian grid. To create all combinations, use expand.grid() before calling set_runs_manual().

Normalization and augmentation

The augmentation term is scaled using the payoff-table ranges of the secondary objectives. If \(R_k\) denotes the range of secondary objective \(k\), then the effective coefficient applied to the slack is:

$$ \frac{\rho}{R_k}, $$

where \(\rho = \code{augmentation}\).

This normalization is important because different objectives may be measured on very different numerical scales. Without normalization, a slack belonging to a large-scale objective could dominate the augmentation term simply due to units.

In this implementation, the user supplies augmentation as the base coefficient \(\rho\), while the normalized slack coefficients are computed internally at solve time using the corresponding payoff-table ranges.

Failure handling

The control argument controls how failed runs are handled. It must be created with set_runs_control.

Some epsilon combinations may define infeasible subproblems. By default, failed runs can be retained in the returned SolutionSet with missing objective values, while feasible runs are preserved. Alternatively, users can request that the solve stops when an infeasible run, a run without a solution, or an unexpected error is encountered.

AUGMECON slack upper bound

slack_upper_bound defines an explicit upper bound for slack variables introduced by the AUGMECON formulation. The value should be sufficiently large to avoid excluding valid solutions, but unnecessarily large bounds can weaken the mixed-integer formulation and reduce numerical performance.

When possible, a problem-specific bound based on the ranges of the constrained objectives should be used.

Stored configuration

This function stores the method definition in x$data$method with:

• name = "augmecon",

• type = "augmecon",

• the primary objective alias,

• the full set of participating aliases,

• the set of secondary aliases,

• runs,

• lexicographic configuration,

• augmentation,

• slack_upper_bound,

• control.

The actual payoff table, grid construction, and subproblem solution loop are performed later by solve.

References

Mavrotas, G. (2009). Effective implementation of the \(\varepsilon\)-constraint method in multi-objective mathematical programming problems. Applied Mathematics and Computation, 213(2), 455–465.

See also

set_runs_grid, set_runs_manual, set_runs_control, set_method_epsilon_constraint, set_method_weighted_sum, solve

Examples

# Small toy problem
pu_tbl <- data.frame(
  id = 1:4,
  cost = c(1, 2, 3, 4)
)

feat_tbl <- data.frame(
  id = 1:2,
  name = c("feature_1", "feature_2")
)

dist_feat_tbl <- data.frame(
  pu = c(1, 1, 2, 3, 4),
  feature = c(1, 2, 2, 1, 2),
  amount = c(5, 2, 3, 4, 1)
)

actions_df <- data.frame(
  id = c("conservation", "restoration"),
  name = c("conservation", "restoration")
)

effects_df <- data.frame(
  pu = c(1, 2, 3, 4, 1, 2, 3, 4),
  action = c("conservation", "conservation", "conservation", "conservation",
             "restoration", "restoration", "restoration", "restoration"),
  feature = c(1, 1, 1, 1, 2, 2, 2, 2),
  benefit = c(2, 1, 0, 1, 3, 0, 1, 2),
  loss = c(0, 0, 1, 0, 0, 1, 0, 0)
)

x <- create_problem(
  pu = pu_tbl,
  features = feat_tbl,
  dist_features = dist_feat_tbl,
  cost = "cost"
) |>
  add_actions(actions_df, cost = c(conservation = 1, restoration = 2)) |>
  add_effects(effects_df) |>
  add_objective_max_benefit(alias = "benefit") |>
  add_objective_min_cost(alias = "cost") |>
  add_objective_min_loss(alias = "loss")

# Automatic epsilon grids generated later during solve()
x1 <- set_method_augmecon(
  x,
  primary = "benefit",
  aliases = c("benefit", "cost"),
  runs = set_runs_grid(n = 5),
  lexicographic = TRUE,
  augmentation = 1e-3
)

x1$data$method
#> $name
#> [1] "augmecon"
#> 
#> $type
#> [1] "augmecon"
#> 
#> $primary
#> [1] "benefit"
#> 
#> $aliases
#> [1] "benefit" "cost"   
#> 
#> $secondary
#> [1] "cost"
#> 
#> $runs
#> $type
#> [1] "grid"
#> 
#> $n
#> [1] 5
#> 
#> $include_extremes
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunGrid"   "RunDesign"
#> 
#> $lexicographic
#> [1] TRUE
#> 
#> $lexicographic_tol
#> [1] 1e-09
#> 
#> $augmentation
#> [1] 0.001
#> 
#> $slack_upper_bound
#> [1] 1e+06
#> 
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> $control
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunsControl" "MOControl"   "list"       
#> 

# Manual runs for one secondary objective
aug_runs <- data.frame(
  eps_cost = c(4, 6, 8)
)

x2 <- set_method_augmecon(
  x,
  primary = "benefit",
  aliases = c("benefit", "cost"),
  runs = set_runs_manual(aug_runs),
  augmentation = 1e-3
)

x2$data$method
#> $name
#> [1] "augmecon"
#> 
#> $type
#> [1] "augmecon"
#> 
#> $primary
#> [1] "benefit"
#> 
#> $aliases
#> [1] "benefit" "cost"   
#> 
#> $secondary
#> [1] "cost"
#> 
#> $runs
#> $type
#> [1] "manual"
#> 
#> $values
#>   eps_cost
#> 1        4
#> 2        6
#> 3        8
#> 
#> attr(,"class")
#> [1] "RunManual" "RunDesign"
#> 
#> $lexicographic
#> [1] TRUE
#> 
#> $lexicographic_tol
#> [1] 1e-09
#> 
#> $augmentation
#> [1] 0.001
#> 
#> $slack_upper_bound
#> [1] 1e+06
#> 
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> $control
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunsControl" "MOControl"   "list"       
#> 

# Manual runs for two secondary objectives
aug_runs_3obj <- data.frame(
  eps_cost = c(4, 6, 8),
  eps_loss = c(0, 1, 1)
)

x3 <- set_method_augmecon(
  x,
  primary = "benefit",
  aliases = c("benefit", "cost", "loss"),
  runs = set_runs_manual(aug_runs_3obj),
  augmentation = 1e-3
)

x3$data$method
#> $name
#> [1] "augmecon"
#> 
#> $type
#> [1] "augmecon"
#> 
#> $primary
#> [1] "benefit"
#> 
#> $aliases
#> [1] "benefit" "cost"    "loss"   
#> 
#> $secondary
#> [1] "cost" "loss"
#> 
#> $runs
#> $type
#> [1] "manual"
#> 
#> $values
#>   eps_cost eps_loss
#> 1        4        0
#> 2        6        1
#> 3        8        1
#> 
#> attr(,"class")
#> [1] "RunManual" "RunDesign"
#> 
#> $lexicographic
#> [1] TRUE
#> 
#> $lexicographic_tol
#> [1] 1e-09
#> 
#> $augmentation
#> [1] 0.001
#> 
#> $slack_upper_bound
#> [1] 1e+06
#> 
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> $control
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunsControl" "MOControl"   "list"       
#> 

# Cartesian epsilon design created explicitly by the user
aug_cartesian <- expand.grid(
  eps_cost = c(4, 6, 8),
  eps_loss = c(0, 1),
  KEEP.OUT.ATTRS = FALSE
)

x4 <- set_method_augmecon(
  x,
  primary = "benefit",
  aliases = c("benefit", "cost", "loss"),
  runs = set_runs_manual(aug_cartesian),
  augmentation = 1e-3
)

x4$data$method
#> $name
#> [1] "augmecon"
#> 
#> $type
#> [1] "augmecon"
#> 
#> $primary
#> [1] "benefit"
#> 
#> $aliases
#> [1] "benefit" "cost"    "loss"   
#> 
#> $secondary
#> [1] "cost" "loss"
#> 
#> $runs
#> $type
#> [1] "manual"
#> 
#> $values
#>   eps_cost eps_loss
#> 1        4        0
#> 2        6        0
#> 3        8        0
#> 4        4        1
#> 5        6        1
#> 6        8        1
#> 
#> attr(,"class")
#> [1] "RunManual" "RunDesign"
#> 
#> $lexicographic
#> [1] TRUE
#> 
#> $lexicographic_tol
#> [1] 1e-09
#> 
#> $augmentation
#> [1] 0.001
#> 
#> $slack_upper_bound
#> [1] 1e+06
#> 
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> $control
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunsControl" "MOControl"   "list"       
#> 

# Backwards-compatible deprecated usage
x5 <- set_method_augmecon(
  x,
  primary = "benefit",
  aliases = c("benefit", "cost", "loss"),
  grid = list(
    cost = c(4, 6, 8),
    loss = c(0, 1)
  ),
  augmentation = 1e-3
)
#> Warning: `grid/n_points/include_extremes` is deprecated. Use `runs = set_runs_grid(...) or runs = set_runs_manual(...)` instead.

x5$data$method
#> $name
#> [1] "augmecon"
#> 
#> $type
#> [1] "augmecon"
#> 
#> $primary
#> [1] "benefit"
#> 
#> $aliases
#> [1] "benefit" "cost"    "loss"   
#> 
#> $secondary
#> [1] "cost" "loss"
#> 
#> $runs
#> $type
#> [1] "manual"
#> 
#> $values
#>   eps_cost eps_loss
#> 1        4        0
#> 2        6        0
#> 3        8        0
#> 4        4        1
#> 5        6        1
#> 6        8        1
#> 
#> attr(,"class")
#> [1] "RunManual" "RunDesign"
#> 
#> $lexicographic
#> [1] TRUE
#> 
#> $lexicographic_tol
#> [1] 1e-09
#> 
#> $augmentation
#> [1] 0.001
#> 
#> $slack_upper_bound
#> [1] 1e+06
#> 
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> $control
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunsControl" "MOControl"   "list"       
#> 

# Control failure handling and the AUGMECON slack upper bound
x6 <- set_method_augmecon(
  x,
  primary = "benefit",
  aliases = c("benefit", "cost"),
  runs = set_runs_manual(data.frame(eps_cost = c(4, 6, 8))),
  augmentation = 1e-3,
  slack_upper_bound = 1e6,
  control = set_runs_control(
    stop_on_infeasible = FALSE,
    stop_on_no_solution = FALSE,
    stop_on_error = TRUE
  )
)

x6$data$method
#> $name
#> [1] "augmecon"
#> 
#> $type
#> [1] "augmecon"
#> 
#> $primary
#> [1] "benefit"
#> 
#> $aliases
#> [1] "benefit" "cost"   
#> 
#> $secondary
#> [1] "cost"
#> 
#> $runs
#> $type
#> [1] "manual"
#> 
#> $values
#>   eps_cost
#> 1        4
#> 2        6
#> 3        8
#> 
#> attr(,"class")
#> [1] "RunManual" "RunDesign"
#> 
#> $lexicographic
#> [1] TRUE
#> 
#> $lexicographic_tol
#> [1] 1e-09
#> 
#> $augmentation
#> [1] 0.001
#> 
#> $slack_upper_bound
#> [1] 1e+06
#> 
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> $control
#> $stop_on_infeasible
#> [1] FALSE
#> 
#> $stop_on_no_solution
#> [1] FALSE
#> 
#> $stop_on_error
#> [1] TRUE
#> 
#> attr(,"class")
#> [1] "RunsControl" "MOControl"   "list"       
#>