Multi-objective transformation

The multi-objective transformation is a technique to solve constrained problem by transforming the initial single (or multi) objective constrained problem into a multi-objective unconstrained problem with the number of objectives equal to the sum of the number of objectives and the number of constraints. In this tutorial, we are going to solve a single objective constrained problem using this technique.


The multi-objective transformation consists in treating each constraints of a constrained optimization problem as objectives of a multi-objective problem, the first objective(s) being the cost function(s) itself. Three different implementation of this method are available in PaGMO/PyGMO.

The first one called OBJ_CSTRS splits all the constraints into different objectives, generating a problem with nobj+m objectives (where nobj is the number of objectives and m is the number of constraints). The transformation adopted is the one of Coello Coello in the CHVEGA algorithm where the nobj+j objective is the j-th constraints violation (if the j-th constraint is violated), otherwise it is the overall number of violated constraints, otherwise if the solution is feasible it is the aggregation of the objectives. This formulation makes sense if the multiobjective algorithm is based on a population decomposition according to the different objectives as VEGA or PADE. The second one called OBJ_CSTRSVIO has nobj+1 objectives, the first nobj are the objective functions to be optimized and the last objective is the sum of the constraints violations. The third methodology has nobj+2 objectives, the first nobj are the objective functions, the second is the sum of inequality constraints violation and the third one is the sum of equality constraints violation.

In all cases, when a constraint is satisfied, the associated objective is either a measure of the number of the violated constraints, if at least one constraint is violated, or the objective function by itself. That way, each objectives commonly helps to reduce the number of constraints violation to find feasible solutions.


The problem considered here is the problem g04 from the Congress on Evolutionary Computation 2006 (CEC2006). This problem has a quadratic objective function with six non linear inequality constraints. In the optimum two constraints are active.

To solve it, in PaGMO/PyGMO, the meta-problem that modifies the original constrained problem into a multi-objective one is used. Then a multi-objective optimizer called VEGA is used to solve it. This optimizer is a simple extention of simple genetic algorithm to solve multi-objective algorithms. The population considered is composed by 280 individulas (40 individuals per objectives: (6+1) * 40 = 280 since VEGA equally divides the original population into a number of subpopulations equal to the number of objectives).

Step by step, the PyGMO library is imported and the populations size and the number of generation for the algorithm are initialized.

In [1]: from PyGMO import *
In [2]: pop_size = 280
In [3]: n_gen = 500

Then the problem, the meta-problem (using the first methodology OBJ_CSTRS) and the algorithm are created.

In [4]: prob = problem.cec2006(4)
In [5]: prob_mo = problem.con2mo(prob,problem.con2mo.method.OBJ_CSTRS)
In [6]: algo_mo = algorithm.vega(gen = n_gen)

The next step consists in creating the population and evolving it.

In [7]: pop = population(prob_mo, pop_size)
In [8]: pop = algo_mo.evolve(pop)

From that point, the obtained solutions can be printed.

In [9]: print(pop.champion.x)
In [10]: print(prob.objfun(pop.champion.x))
In [11]: print(prob.compute_constraints(pop.champion.x))
Out [1]:
(78.09947909827132, 33.30065259242254, 30.894527032856043, 44.013876005779935, 35.07762608948753)
(-0.2613258099525382, -91.73867419004746, -11.284338550618045, -8.715661449381955, -4.980246388885227, -0.019753611114772696)

The solution found by this algorithm is close to the optimum value:

In [12: print(prob.best_x)
In [13]: print(prob.best_f)
In [14]: print(prob.best_c)
Out [2]:
((78.0, 33.0, 29.9952560256816, 45.0, 36.77581290578821),)
((0.0, -92.0, -11.159499691073137, -8.840500308926863, -4.9999999999999964, -3.552713678800501e-15),)

Due to the stochastic behavior of the algorithm in a single run it might not converge to the exact optimum. We invite you to play with the population size and the number of generations to see the behavior of this constraints handling technique.