# constraint_vbr: "Violation-based Ranking" constraint handling method for... In MOEADr: Component-Wise MOEA/D Implementation

## Description

Uses the Violation-based Ranking handling method to generate a preference index for the MOEADr framework.

## Usage

 1 constraint_vbr(bigZ, bigV, type = c("ts", "sr", "vt"), pf = NULL, ...) 

## Arguments

 bigZ Matrix of scalarized objective values for each neighborhood and the incumbent solution (generated by scalarize_values()) bigV Matrix of violation values for each neighborhood and the incumbent solution (generated in order_neighborhood()) type type of c(x) function to use (see c(x) Criteria for details). pf probability parameter for type = "sr" (ignored in other modes). ... other parameters (unused, included for compatibility with generic call)

## Details

This function calculates the preference index of a set of neighborhoods based on the "violation-based ranking" (VBR) constraint handling method. Please see order_neighborhood() for more information on the preference index matrix.

The VBR strategy generalizes some well-known methods for handling constraints in population-based metaheuristics (see Section c(x) Criteria). This strategy essentially ranks points within for a given subproblem based on their aggregated function value (f^{agg}(x|w_i)) or their total constraint violation (v(x)). Specific variations of this strategy differ on the criteria for using one or the other.

The value used for ranking a given point x can be summarized as:

 Violation | c(x) criterion | Rank using: v(x) = 0 | c(x) = * | f^{agg}(x|w_i) v(x) > 0 | c(x) == TRUE | f^{agg}(x|w_i) v(x) > 0 | c(x) == FALSE | v(x)

Points compared according to their f^{agg}(x|w_i) values (i.e., feasible points and those for which c(x) = TRUE) are ranked first (i.e., receive ranks between 1 and n_{feas}, where n_{feas} is the number of feasible points in the i-th neighborhood), with points that are compared according to their v(x) values receiving ranks between (n_{feas} + 1) and T + 1 (T being the size of the neighborhood. The +1 comes from including the incumbent solution in the comparison).

## Value

[ N x (T+1) ] matrix of preference indices. Each row i contains a permutation of {1, 2, ..., (T+1)}, where 1,...,T correspond to the solutions contained in the neighborhood of the i-th subproblem, B[i, ], and T+1 corresponds to the incumbent solution for that subproblem. The order of the permutation is defined by the specific strategy defined by the input variable type).

## c(x) Criteria

Specific variations of the VBR differ on how the criterion c(x) is implemented. Three variants are currently implemented in the MOEADr package:

## References

[Deb2000] K. Deb, "An efficient constraint handling method for genetic algorithm", Computer Methods in Applied Mechanics and Engineering 186(2–4):311–338, 2000.

[Runarsson2000] T. Runarsson, X. Yao, "Stochastic ranking for constrained evolutionary optimization", IEEE Transactions on Evolutionary Computation4(3):284–294, 2000.

[Asafuddoula2014] M. Asafuddoula, T. Ray, R. Sarker, K. Alam, "An adaptive constraint handling approach embedded MOEA/D,” 2012 IEEE Congress on Evolutionary Computation (CEC).

[Campelo2017] F. Campelo, L.S. Batista, C. Aranha (2020): The MOEADr Package: A Component-Based Framework for Multiobjective Evolutionary Algorithms Based on Decomposition. Journal of Statistical Software doi: 10.18637/jss.v092.i06

MOEADr documentation built on Feb. 18, 2020, 1:07 a.m.