# Introduction

This is a short guide to writing new functions for the MOEADr package. This package provides a component-based framework for developing (and applying) Multiobjective Evolutionary Algorithms based on Decomposition (MOEA/D)1.

The modular implementation provided in this package provides control over the following aspects of the algorithm:

• decomp, the decomposition strategy
• aggfun, the scalar aggregation function
• neighbors, the neighborhood assignment strategy
• variation, the variation operators used
• update, the population update method
• constraint, the constraint-handling method
• scaling, the strategy used for objective scaling
• stopcrit, the stop criteria used by the algorithm

This document describes how to write functions implementing new variants for any of these modules. A general description of the algorithm and the component-based interpretation behind the MOEADr package is available in our paper2

# General guidelines

## Nomenclature

• Functions should be preferably defined in the form verb_object (e.g., generate_weights or evaluate_population)

• Please try to follow the policy one function, one file, same name (very short functions for general use can be exceptions - in this case they should be placed, e.g., in the utils.R file.

• When passing variables between functions, there are three main rules that should be observed:

1. Unless absolutely impossible, functions should always receive all variables used in the function body via formal arguments, plus whatever other variables may be used downstream using the ... input.

2. If it is absolutely necessary, a function can eventually access variables from parent frames using, for instance, parent.frame()$variable_name, but this is not encouraged. It is definitely not kosher for functions to directly modify variables in the parent environment by any means except by formal, explicit outputs. Previous (development) versions of the MOEADr package used to allow it, but the resulting confusion (particularly when writing unit tests or debugging) heavily outweighted the benefits. 3. Functions should, with few exceptions, be able to handle any number of “useless” variables that may be passed to them (using the ... formal argument). • Documentation should be complete. Please use roxygen2-style documentation in all functions. This version of the package uses roxygen2 version 6.0.1 (which means some degree of markdown support and other conveniences). • Also, please make liberal use of in-code comments to clarify any non-trivial operation. ## Important variables defined in the package • W: matrix of weights ( N x m ) • X: matrix of candidate solutions at a given iteration. Each row is a point in the space of variables. ( N x nv ) • Xt: matrix of incumbent solutions at a given iteration ( N x nv ) • Y: matrix of objective function values (corresponding to the rows of X). Each row is a point in the space of objectives. ( N x m ) • Yt: matrix of objective function values (corresponding to the rows of Xt) ( N x m ) • V: List object with information related to the constraint values of points in X. This list contains three objects: • matrix V$Cmatrix, containing the raw value of each constraint (including box constraints, if present) on each point;
• matrix V$Vmatrix, containing the respective violation value of each constraint on each point; • vector V$v, containing the total sum of violations for each point.
• Vt: List object equivalent to V, but related to the points in Xt
• B: matrix of neighborhoods ( N x T )
• P: matrix of selection probabilities (derived from B) ( N x N ).
• nfe: counter, number of solutions evaluated
• iter: counter, number of iterations
• keep.running: flag. TRUE if any stop criterion is met
• time.start: starting time (system clock) of the algorithm
• iter.times: vector of times (in seconds) spent on each iteration
• ls.args: list containing information related to local search operators (if present)
• normYs: List object containing matrices of normalized objective values (regulated by the scaling strategy), plus information on the estimated ideal and nadir points.
• bigZ: matrix of scalarized values for the neighborhood of each subproblem, plus one row containing the scalarized values of the incumbent solutions of each subproblem.
• sel.indx: matrix of selection ranks (lower = better) for each subproblem, calculated from bigZ.

# Contributing to the modules

## Decomposition strategies

To discover the available decomposition strategies, use get_decomposition_methods(). Decomposition functions are called from within generate_weights().

• INPUTS:
• the decomposition parameters are defined in the input list decomp (use ?moead and ?decomposition_sld to get details on the structure of decomp). Any other required inputs should be explicitly declared in the function definition.
• OUTPUTS:
• the function must output a N x m matrix of weights, with N the number of subproblems and m the number of objectives.

• The name of the function (and of the file) must have the format decomposition_XYZ, with XYZ being the moniker for the contributed method (which is going to be passed as decomp$name). • Please follow the one function, one file, same name policy strictly (otherwise get_decomposition_methods() won’t be able to correctly list the method). ### Example file Check decomposition_sld.R for a good example of decomposition routine (e.g., to use as a template). ## Scalar aggregation functions To discover the available decomposition strategies, use get_scalarization_methods(). Scalarization functions are called from within scalarize_values(). • INPUTS: • the scalarization parameters are defined in the input list aggfun (use ?moead and ?scalarization_pbi to get details on the structure of aggfun). Any other required inputs (e.g., W, Y, etc.) should be explicitly declared in the function definition. • OUTPUTS: • the function must output a numeric vector of size N, containing the scalarized values. ### Other guidelines and requirements: • The name of the function (and of the file) must have the format scalarization_XYZ, with XYZ being the moniker for the contributed method (which is going to be passed as aggfun$name).
• Please follow the one function, one file, same name policy strictly (otherwise get_scalarization_methods() won’t be able to correctly list the method.

### Example file

Check scalarize_pbi.R for a good example of decomposition routine (e.g., to use as a template).

## Neighborhood assignment options

The strategy for defining the neighborhood structure in the MOEADr package is essentially the same (use Euclidean distances and use the neighbors$T nearest subproblems as a neighborhood). The only difference is the space in which the distances are calculated, which has implications in the need for re-calculating the neighborhood structure. The neighborhoods are defined using an efficient C implementation of the k-nearest-neighbors algorithm available in function FNN::get.knn, which is the only reason why package MOEADr lists FNN in its Imports field (see DESCRIPTION). The neighborhood assignment function is define_neighborhood(), which is called directly from the main function moead(). • INPUTS: • the neighborhood parameters are defined in the input list neighbors (use ?moead and ?define_neighborhood to get details on the structure of aggfun). Any other required inputs should be explicitly declared in the function definition. • OUTPUTS: • the function must output a list object containing the following matrices: • B: each row represents the neighborhood of a subproblem as indices (first element is the subproblem index, and the following neighbors$T - 1 elements are the neighbor indices). This is a N x T matrix.
• P: matrix of probabilities of selection to be used in the sampling of solutions for variation operators. Each element represents the probability of using the solution associated with the j-th subproblem when performing a variation operator (e.g., recombination) for the i-th subproblem. This is an N x N matrix.
• fullB: same as B, but assuming that the neighborhood size is equal to the number of subproblems (i.e., resulting in an N x N matrix.
• fullP: same as P, but with all elements set as 1 / N.

### Other guidelines and requirements:

• Unlike the previous modules, the neighborhood assignment strategies are defined (in the current version) as options passed to a single function define_neighborhood. Other possibilities (e.g., to deal with adaptive weights, which would require periodic recalculation) can, at least in principle, use the same strategy. However, if an alternative assignment method becomes too different from the one currently implemented, it may be better to break the options and use the one function, one file, same name policy. In this case, the current options should be moved to independent functions starting with a common preffix (as is the case with other modules, e.g., decomposition).

### Example file

Check define_neighborhood.R for the current neighborhood assignment alternatives (e.g., to use as a template).

## Variation operators

To discover the available variation operators, use get_variation_operators(). Variation methods are called from within perform_variation().

• INPUTS:
• The variation operators are defined in the input list variation (use ?moead and ?perform_variation to get details on the structure of variation). Any other required inputs should be explicitly declared in the function definition.
• OUTPUTS: the function must output either a matrix X containing the (modified) points, or a list object containing the matrix X as well as a counter nfe, containing the number of additional function evaluations performed by that operator..

### Other guidelines and requirements:

• The name of the function (and of the file) must have the format variation_XYZ, with XYZ being the moniker for the contributed method.
• the only exceptions to that naming convention are local search operators, which are called from within variation_localsearch(). Local search methods should follow the naming convention ls_XYZ, and available methods are discovered using get_localsearch_methods(). See ?variation_localsearch and the Variation section of ?moead for details.
• Please follow the one function, one file, same name policy strictly (otherwise get_variation_operators() and get_localsearch_methods() won’t be able to correctly list the method.

### Example files

Check variation_sbx.R for a good example of a non-local search variation operator, and variation_localsearch.R and ls_dvls.R for local search methods (e.g., to use as a template).

## Update strategies

To discover the available decomposition strategies, use get_update_methods(). Update functions are called from within update_population().

• INPUTS:
• the update parameters are defined in the input list update (use ?moead and ?update_population to get details on the structure of update). Any other required inputs should be explicitly declared in the function definition.
• OUTPUTS:
• the function must output a list object containing the updated matrices X, Y, and the updated list V.

• The name of the function (and of the file) must have the format updt_XYZ, with XYZ being the moniker for the contributed method (which is going to be passed as update$name). • Please follow the one function, one file, same name policy strictly (otherwise get_update_methods() won’t be able to correctly list the method. ### Example file Check update_standard.R for a good example of update routine (e.g., to use as a template). ## Constraint handling methods To discover the available constraint handling strategies, use get_constraint_methods(). Constraint handling methods are called from within order_neighborhood(). • INPUTS: • the constraint handling parameters are defined in the input list constraint (use ?moead to get details on the structure of constraint). Any other required inputs should be explicitly declared in the function definition. • OUTPUTS: • the function must output a matrix of preference indices, indicating the selection preference relations between solutions (see the Value section of ?constraint_vbr for details). ### Other guidelines and requirements: • The name of the function (and of the file) must have the format constraint_XYZ, with XYZ being the moniker for the contributed method (which is going to be passed as constraint$name).
• Please follow the one function, one file, same name policy strictly (otherwise get_constraint_methods() won’t be able to correctly list the method.

### Example file

Check constraint_penalty.R for a good example of constraint handling routine (e.g., to use as a template).

## Objective scaling

The strategies for objective scaling currently available in the MOEADr package are essentially “none” (i.e., no scaling) and “simple” (simple linear scaling to the interval [0,1]).The scaling function is scale_objectives().

• INPUTS:
• the scaling parameters are defined in the input list scaling (use ?moead and ?scale_objectives to get details on the structure of scaling). Any other required inputs should be explicitly declared in the function definition.
• OUTPUTS:
• the function must output a list object containing the matrices Y and Yt (corresponding to the normalized values of the candidate and incumbent objective function matrices, respectively); as well as two vectors, minP and maxP, containing the estimates of the ideal and nadir points for the normalized matrices (i.e., a vector of 0s and a vector of 1s, if any scaling different from “none” is used).

### Other guidelines and requirements:

• Unlike the previous modules, the scaling strategies are defined (in the current version) as options passed to a single function scale_objectives(). Other possibilities can, at least in principle, use the same strategy. However, if an alternative assignment method becomes too different from the one currently implemented, it may be better to break the options and use the one function, one file, same name policy. In this case, the current options should be moved to independent functions starting with a common preffix (as is the case with other modules, e.g., decomposition).

## Termination Criteria

To discover the available termination criteria, use get_stop_criteria(). Termination methods are called from within check_stop_criteria().

• INPUTS:
• the stop criteria parameters are defined in the input list stopcrit (use ?moead and ?get_stop_criteria to get details on the structure of stopcrit). Any other required inputs should be explicitly declared in the function definition.
• OUTPUTS:
• the function must output a logical value indicating whether the termination criterion has been reached (TRUE) or not (FALSE).

### Other guidelines and requirements:

• The name of the function (and of the file) must have the format stop_XYZ, with XYZ being the moniker for the contributed method.
• Please follow the one function, one file, same name policy strictly (otherwise get_stop_criteria() won’t be able to correctly list the method.

### Example file

Check stop_maxiter.R for a good example of constraint handling routine (e.g., to use as a template).

1. Q. Zhang and H. Li, “MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition”, IEEE Trans. Evol. Comp. 11(6): 712-731, 2007.

2. F. Campelo, L.S. Batista and C. Aranha, “A Component-Wise Perspective on Multiobjective Evolutionary Algorithms based on Decomposition”, in preparation.