Scalarization-based methods¶

In DESDEO, scalarization-based methods are implemented in a modular fashion. This means that no method is provided per se, but instead the different components required to build the method are provided. This idea is best illustrated by examples.

Example: The achievement scalarizing function method¶

In the achievement scalarizing function (ASF) method, a scalarization function is solved iteratively with new preference information provided by a decision maker. The preference information consists in this case of a reference point containing aspiration levels.

Suppose we have an instance of Problem defined as problem:

In [1]:

from desdeo.api.db_models import Problem

problem: Problem = Problem(...)

from desdeo.api.db_models import Problem problem: Problem = Problem(...)

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[1], line 3
      1 from desdeo.api.db_models import Problem
----> 3 problem: Problem = Problem(...)

TypeError: __init__() takes 1 positional argument but 2 were given

Now, to "implement" the ASF method, we first need preference information for the decision maker. Suppose z_pref is a list of aspiration levels.

In [2]:

z_pref = [...]

z_pref = [...]

The ASF is defined as follows:

$$ S_\text{AS}(F(\mathbf{x}); \mathbf{\bar{z}}, \mathbf{z}^\star, \mathbf{z}^\text{nad}) = \underset{i=1,\ldots,k}{\text{max}} \left[ \frac{f_i(\mathbf{x}) - \bar{z}_i}{z^\text{nad}_i - (z_i^\star - \delta)} \right] + \rho\sum_{i=1}^{k} \frac{f_i(\mathbf{x})}{z_i^\text{nad} - (z_i^\star - \delta)}, $$

where the variables (F(\mathbf{x}) = [f_1(\mathbf{x}),\dots,f_k(\mathbf{x})], \mathbf{z}^\star, \mathbf{z}^\text{nad}) are assumed to be available through the instance of problem. The variable (\mathbf{\bar{z}}) is our reference point, which is provided by a decision maker.

Now, to create the ASF method, we first create the ASF and then add it to the instance of problem:

!!! TODO fix this

In [3]:

problem = create_asf(problem=problem, symbol="s_asf", ref_point=z, delta=1e-6, rho=1e-6)

problem = create_asf(problem=problem, symbol="s_asf", ref_point=z, delta=1e-6, rho=1e-6)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[3], line 1
----> 1 problem = create_asf(problem=problem, symbol="s_asf", ref_point=z, delta=1e-6, rho=1e-6)

NameError: name 'create_asf' is not defined

In the above example, we create an instance of the ASF where the variables of the function have been replaced by the corresponding variables found in the instance of problem. When creating scalarization functions of any type, at least the problem and a symbol must be supplied. In this example, the argument z (reference point) is also needed. The symbol is important because it is used later in solvers to specify which scalarization functions should be optimized. In the ASF example, delta and rho are optional parameters of the scalarization function. The number of parameters found in scalarization functions varies.

Note

When creating scalarization functions of any kind, the `tag` argument is very important because it is used to specify to solvers which scalarization functions should be optimized. The `tag` must be unique inside the instance of problem, i.e., no other field of a `problem` dataclass should share the same symbol.

To solve the scalarization function, we can use the solver. For instance, the scipy_solver can utilize the minimizer routine defined in scipy.optimize.minimize. To solve the problem utilizing scipy_solver, we do:

In [4]:

scipy_solver(problem, objective_tag="s_asf", kwargs={...})

scipy_solver(problem, objective_tag="s_asf", kwargs={...})

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[4], line 1
----> 1 scipy_solver(problem, objective_tag="s_asf", kwargs={...})

NameError: name 'scipy_solver' is not defined

As mentioned, the tag defined previously is now important so that we can tell the solver what to use as an objective when solving scalarization functions. The kwargs are optional arguments to control the internal solver, in this case, the routine scipy.optimize.minimize...