Multiple Objectives#

While typical optimization models have a single objective function, real-world optimization problems often have multiple, competing objectives. For example, in a production planning model, you may want to both maximize profits and minimize late orders, or in a workforce scheduling application, you may want to minimize the number of shifts that are short-staffed while also respecting worker’s shift preferences.

The main challenge you face when working with multiple, competing objectives is deciding how to manage the trade-offs between them. Gurobi provides tools that simplify the task: Gurobi allows you to blend multiple objectives, to treat them hierarchically, or to combine the two approaches. In a blended approach, you optimize a weighted combination of the individual objectives. In a hierarchical or lexicographic approach, you set a priority for each objective, and optimize in priority order. When optimizing for one objective, you only consider solutions that would not degrade the objective values of higher-priority objectives. Gurobi allows you to enter and manage your objectives, to provide weights for a blended approach, and to set priorities for a hierarchical approach.

Specifying Multiple Objectives#

Let us first discuss the interface for managing multiple objectives. An empty model starts with one objective function, which is initially just 0.0. We’ll refer to this as the primary objective. You can modify the primary objective in two ways: you can set the Obj attribute, or you can use the setObjective method from your language API (e.g., Model.setObjective in Python). In contrast to models with a single objective, where the primary objective can be linear, quadratic, or piecewise-linear, all objectives must be linear for multi-objective models. In general, attributes and methods that aren’t specific to multi-objective optimization will work with the primary objective function.

Every objective in a multi-objective model has the following settable attributes: ObjNCon with default value 0, ObjNPriority with default value 0, ObjNWeight with default value 1, ObjNRelTol with default value 0, ObjNAbsTol with default value 0, and ObjNName.

To provide additional objectives, use the setObjectiveN method from your language API (e.g. Model.setObjectiveN in Python). Objectives are numbered 0 through NumObj-1. The order of the objectives is arbitrary, but you must provide a unique index for each one (specified using the index argument to setObjectiveN). You can query the number of objectives in your model using the NumObj attribute. As noted above, all objectives, including the primary one, must be linear.

You can query and modify information about multiple objectives using the ObjNumber parameter, in conjunction with several model and variable attributes. For example, to retrieve the coefficient for variable x in objective \(2\), you’d set the ObjNumber parameter to \(2\), then query the ObjN attribute for x. Similarly, querying the ObjNName attribute after setting ObjNumber to \(3\) would give the name of objective \(3\).

We should note that there is one important exception to our statement above that the order of objectives is arbitrary: objective \(0\) is treated as the primary objective. One consequence is that the original objective automatically becomes objective \(0\) when you add a second objective. Another is that querying the ObjN attribute is equivalent to querying the Obj attribute when ObjNumber is \(0\).

Note that a model has a single objective sense (controlled by the ModelSense attribute). This means that you can’t maximize the first objective and minimize the second. However, you can achieve the same result with a simple trick. Each objective has a weight, and these weights are allowed to be negative. Minimizing an objective function is equivalent to maximizing the negation of that function.

You can change the number of objectives in your model as many times as you like (by modifying the NumObj attribute). When you increase the objective count, the newly added objectives and their associated attributes are set to 0. When you decrease the count, objectives beyond the new count are discarded. If you set the number of objectives to zero, the model becomes a pure feasibility problem.

We have extended the LP and MPS file formats, so writing a model with multiple objectives to a file will capture those objectives. Similarly, if you read a model file that contains multiple objectives, then NumObj and ObjN will capture the objectives stored in the file. See the file format section for details.

Working With Multiple Objectives#

Of course, specifying a set of objectives is only the first step in solving a multi-objective optimization problem. The next step is to indicate how the objectives should be combined. As noted earlier, we support two approaches: blended and hierarchical.

Blended Objectives#

A blending approach creates a single objective by taking a linear combination of your objectives. You provide a weight for each objective as an argument to setObjectiveN. Alternatively, you can use the ObjNWeight attribute, together with ObjNumber. The default weight for an objective is 1.0.

To give an example, if your model has two objectives, \(1 + x + 2y\) and \(y + 2z\), and if you give weights of \(-1\) and \(2\) to them, respectively, then Gurobi would solve your model with a blended objective of \(-1 \cdot (1 + x + 2y) + 2 \cdot (y + 2z) = -1 - x + 4z\).

You should avoid weights that are very large or very small. A very large weight (i.e., larger than \(10^6\)) may lead to very large objective coefficients, which can cause numerical difficulties. A very small weight (i.e., smaller than \(10^{-6}\)) may cause the contribution from that objective to the overall blended objective to be smaller than tolerances, which may lead to that objective being effectively ignored.

Hierarchical Objectives#

A hierarchical or lexicographic approach assigns a priority to each objective, and optimizes for the objectives in decreasing priority order. During each of these optimization passes, it finds the best solution for the current objective, but only from among those that would not degrade the solution quality for higher-priority objectives. You provide the priority for each objective as an argument to setObjectiveN. Alternatively, you can use the ObjNPriority attribute. Priorities are integral, not continuous. Larger values indicate higher priorities. The default priority for an objective is 0.

To give an example, if your model has two objectives, with priorities \(10\) and \(5\), and objective weights 1.0 and -1.0. Assuming the optimal solution for the first objective has value \(100\), then the solver will find the solution that optimizes \(-1\) times the second objective from among all solutions with objective \(100\) for the first objective.

Allowing Multiple-Objective Degradation#

By default, our hierarchical approach won’t allow later objectives to degrade earlier objectives, subject to the user-given ending gap conditions for the optimization problem. More precisely, the base value used to define what solutions are acceptable for lower priorities objectives – for a minimization problem – is computed as:

\[\mathrm{base\_value} = \max\{bestsol, bestbound + |bestsol|*rgap, bestbound + agap\},\]

where bestsol is the value of the best incumbent solution, bestbound is the value of the best proven lower bound for the problem, rgap is the relative MIP gap, and agap is the absolute MIP gap, and the set of feasible solutions for the next objective will consider solutions whose objective value is at most that value.

This behavior can be relaxed for MIPs through a pair of tolerances: a relative and an absolute tolerance. These are provided as arguments to setObjectiveN, or they can be set using attributes ObjNRelTol and ObjNAbsTol. By setting one of these for a particular objective, you can indicate that later objectives are allowed to degrade this objective by the specified relative or absolute amount, respectively. In our earlier example, if the optimal value for the first objective is \(100\), and if we set ObjNAbsTol for this objective to \(20\), then the second optimization pass would find the best solution for the second objective from among all solutions with objective \(120\) or better for the first objective. Note that if you modify both tolerances, later optimizations would use the looser of the two values (i.e., the one that allows the larger degradation).

Objective degradations are handled differently for multi-objective LP models. For LP models, solution quality for higher-priority objectives is maintained by fixing some variables to their values in previous optimal solutions. These fixings are decided using variable reduced costs. The value of the ObjNAbsTol parameter indicates the amount by which a fixed variable’s reduced cost is allowed to violate dual feasibility, whereas the ObjNRelTol parameter is simply ignored. If you want the MIP behavior, where the degradation is controlled more directly, you can add a dummy binary variable to the model, thus transforming it into a MIP. Solving the resulting multi-objective MIP will be much more time consuming than solving the original multi-objective LP.

Combining Blended and Hierarchical Objectives#

Every objective in a multi-objective model has both a weight and a priority, which allows you to seamlessly combine blended and hierarchical approaches. To understand how this works, we should first provide more detail on how hierarchical objectives are handled.

When you specify a different priority for each of \(n\) objectives, the solver performs \(n\) separate optimization passes. In each pass, in decreasing priority order, it optimizes for the current objective multiplied by its ObjNWeight attribute, while imposing constraints that ensure that the quality of higher-priority objectives isn’t degraded by more than the specified tolerances.

If you give the same priority to multiple objectives, then they will be handled in the same optimization pass, resulting in fewer than \(n\) total passes for \(n\) objectives. More precisely, one optimization pass is performed per distinct priority value, in order of decreasing priority, and all objectives with the same priority are blended together, using the weights for those objectives. This gives you quite a bit of flexibility when combining the blended and hierarchical approaches.

One subtle point when blending multiple objectives within a single level in a hierarchical approach relates to the handling of degradations from lower-priority levels. The objective degradation allowed after a blended optimization pass is the maximum absolute and relative degradations allowed by each of the participating objectives. For example, if we have three objectives with ObjNPriority equal to \(\{2, 2, 1\}\), and ObjNRelTol equal to \(\{0.10, 0.05, 0.00\}\) and ObjNAbsTol equal to \(\{0, 1, 2\}\), and if the best solution for the first priority objective is \(10\), then the allowed degradation for the first priority objective is \(\max\{10 \cdot 0.10, 10 \cdot 0.05, 0, 1\}~=~1\).

Querying multi-objective results#

Once you have found one or more solutions to your multi-objective model, you can query the achieved objective value for each objective on each solution. Specifically, if you set the ObjNumber parameter to choose an objective, and the SolutionNumber parameter to choose a solution, then the ObjNVal attribute will give the value of the chosen objective on the chosen solution. This is illustrated in the following Python example:

import gurobipy as gp
from gurobipy import GRB

# Read and solve a model with multiple objectives
m = gp.read('input.mps')
m.optimize()

# Get the set of variables
x = m.getVars()

# Ensure status is optimal
assert m.Status == GRB.Status.OPTIMAL

# Query number of multiple objectives, and number of solutions
nSolutions  = m.SolCount
nObjectives = m.NumObj
print('Problem has', nObjectives, 'objectives')
print('Gurobi found', nSolutions, 'solutions')

# For each solution, print value of first three variables, and
# value for each objective function
solutions = []
for s in range(nSolutions):
  # Set which solution we will query from now on
  m.params.SolutionNumber = s

  # Print objective value of this solution in each objective
  print('Solution', s, ':', end='')
  for o in range(nObjectives):
    # Set which objective we will query
    m.params.ObjNumber = o
    # Query the o-th objective value
    print(' ', m.ObjNVal, end='')

  # Print first three variables in the solution
  n = min(len(x),3)
  for j in range(n):
    print(x[j].VarName, x[j].Xn, end='')
  print('')

  # Query the full vector of the s-th solution
  solutions.append(m.getAttr('Xn',x))

Additional Details#

Multi-Objective Environments#

As we progress from higher-priority objectives to lower-priority objectives in a hierarchical multi-objective model, we won’t necessarily solve each pass to exact optimality. By default, termination criteria (e.g. TimeLimit, SolutionLimit, etc.) are controlled by the parameters defined in the model environment. However, we provide a feature called multi-objective environments that allows you to create a Gurobi environment for each optimization pass and set parameters on those environments. Those settings will only affect the corresponding pass of the multi-objective optimization. Thus, for example, if the TimeLimit parameter for the model is 100, but you use a multi-objective environment to set the parameter to 10 for a particular optimization pass, then the multi-objective optimization will spend at most 10 seconds on that particular pass (and at most 100 seconds in total).

To create a multi-objective environment for a particular optimization pass, use the getMultiobjEnv method from your language API (e.g. Model.getMultiobjEnv in Python). The index argument gives the index of the optimization pass that you want to control.

Please note that optimization passes and objectives are not quite synonymous, due the possibility of blending objectives at the same priority level. Multi-objective environment 0 is always tied to the highest priority (possibly blended) objective, while multi-objective environment 1 is always tied to the second highest priority objective (if any). For details on how multiple objectives with the same priority are treated, please refer to the hierarchical objectives section.

Once you create multi-objective environments, they will be used for every subsequent multi-objective optimization on that model. Use the discardMultiobjEnvs method from your language API (e.g. Model.discardMultiobjEnvs in Python) to revert back to default multi-objective optimization behavior.

Please note that parameter values are copied from the model when the multi-objective environment is created. Therefore, changes to parameter values on the model have no effect on multi-objective environments that have already been created. This is a frequent source of confusion.

Other Details#

We haven’t attempted to generalize the notions of dual solutions or simplex bases for continuous multi-objective models, so you can’t query attributes such as Pi, RC, VBasis, or CBasis for multi-objective solutions. Because of this, we’ve concluded that the most consistent result to return for attribute IsMIP is 1. Note, however, that several MIP-specific attributes such as ObjBound, ObjBoundC and MIPGap don’t make sense for multi-objective models and are also not available.

Gurobi will only solve multi-objective models with strictly linear objectives. If the primary objective is quadratic or piecewise linear, the solve call will return an error.

When solving a continuous multi-objective model using a hierarchical approach, you have a choice of which optimization algorithm to use for the different passes (primal simplex, dual simplex, or barrier). The first pass will always use the algorithm specified in the Method parameter. The algorithm for subsequent passes is controlled by the MultiObjMethod parameter. This parameter has no effect for multi-objective MIP models. Note you can get finer-grained control over the algorithm choice using our multi-objective environment feature, by setting the Method parameter for individual objectives.

For the hierarchical approach, Gurobi will perform a conservative presolve step at the beginning of the multi-objective optimization, and a more aggressive presolve step at the beginning of each pass (assuming presolve hasn’t been turned off). You can optionally perform a more aggressive presolve step at the beginning of the multi-objective optimization by setting parameter MultiObjPre to value 2. This can help performance, but it makes a few simplifying assumptions that could lead to small degradations in the values achieved for lower-priority objectives.

The log file when using a hierarchical approach will show optimization progress for each pass in the process. You’ll see log lines that look like this:

Multi-objectives: optimize objective 1 (Obj1Name) ...
...
Multi-objectives: optimize objective 2 (weighted) ...
...

For further details, please see section Multi-Objective Logging.

Callbacks are available for multi-objective optimization, but they are a bit more involved than those for single-objective optimization. When you are solving for a specific objective (either in one phase of a hierarchical optimization or when solving a blended objective), you will receive callbacks from the algorithm that solves that model: MIP callbacks if the model is a MIP, and simplex or barrier callbacks if the model is continuous. For a hierarchical objective, you will also get a MULTIOBJ callback at the end of each phase that allows you to query the current solution, the number of solutions found, and the number of objectives that have been solved for at that point. Refer to the Callback discussion for further details.

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