# Documentation

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# gamultiobj

Find minima of multiple functions using genetic algorithm

## Syntax

```X = gamultiobj(FITNESSFCN,NVARS)X = gamultiobj(FITNESSFCN,NVARS,A,b)X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq)X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB)X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon)X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options)X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon,options)X = gamultiobj(problem)[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...)[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...)[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...)[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...)[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN, ...)```

## Description

`gamultiobj` implements the genetic algorithm at the command line to minimize a multicomponent objective function.

`X = gamultiobj(FITNESSFCN,NVARS)` finds a local Pareto set `X` of the objective functions defined in `FITNESSFCN`. For details on writing `FITNESSFCN`, see Compute Objective Functions. `NVARS` is the dimension of the optimization problem (number of decision variables). `X` is a matrix with `NVARS` columns. The number of rows in `X` is the same as the number of Pareto solutions. All solutions in a Pareto set are equally optimal; it is up to the designer to select a solution in the Pareto set depending on the application.

`X = gamultiobj(FITNESSFCN,NVARS,A,b)` finds a local Pareto set `X` of the objective functions defined in `FITNESSFCN`, subject to the linear inequalities $A\ast x\le b$, see Linear Inequality Constraints. Linear constraints are supported only for the default `PopulationType` option (`'doubleVector'`). Other population types, e.g., `'bitString'` and `'custom'`, are not supported.

`X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq)` finds a local Pareto set `X` of the objective functions defined in `FITNESSFCN`, subject to the linear equalities $Aeq\ast x=beq$ as well as the linear inequalities $A\ast x\le b$, see Linear Equality Constraints. (Set `A=[]` and `b=[]` if no inequalities exist.) Linear constraints are supported only for the default `PopulationType` option (`'doubleVector'`). Other population types, e.g., `'bitString'` and `'custom'`, are not supported.

`X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB)` defines a set of lower and upper bounds on the design variables `X` so that a local Pareto set is found in the range $LB\le x\le UB$, see Bound Constraints. Use empty matrices for `LB` and `UB` if no bounds exist. Bound constraints are supported only for the default `PopulationType` option (`'doubleVector'`). Other population types, e.g., `'bitString'` and `'custom'`, are not supported.

`X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon)` subjects the minimization to the constraints defined in `nonlcon`. The function `nonlcon` accepts `x` and returns vectors `C` and `Ceq`, representing the nonlinear inequalities and equalities respectively. `gamultiobj` minimizes the `fitnessfcn` such that `C(x) `` 0` and `Ceq(x) = 0`. (Set `LB=[]` and `UB=[]` if no bounds exist.)

`X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options)` or ```X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon,options)``` finds a Pareto set `X` with the default optimization parameters replaced by values in the structure `options`. Create `options` using `optimoptions` (recommended) or `gaoptimset`.

`X = gamultiobj(problem)` finds the Pareto set for `problem`, where `problem` is a structure containing the following fields:

 `fitnessfcn` Fitness functions `nvars` Number of design variables `Aineq` `A` matrix for linear inequality constraints `bineq` `b` vector for linear inequality constraints `Aeq` `Aeq` matrix for linear equality constraints `beq` `beq` vector for linear equality constraints `lb` Lower bound on `x` `ub` Upper bound on `x` `nonlcon` Nonlinear constraint function (optional) `solver` `'gamultiobj'` `rngstate` Optional field to reset the state of the random number generator `options` Options created using `optimoptions` or `gaoptimset`, or exported from the Optimization app

Create the structure `problem` by exporting a problem from Optimization app, as described in Importing and Exporting Your Work in the Optimization Toolbox™ documentation.

`[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...)` returns a matrix `FVAL`, the value of all the objective functions defined in `FITNESSFCN` at all the solutions in `X`. `FVAL` has `numberOfObjectives` columns and same number of rows as does `X`.

`[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...)` returns `EXITFLAG`, which describes the exit condition of `gamultiobj`. Possible values of `EXITFLAG` and the corresponding exit conditions are listed in this table.

EXITFLAG ValueExit Condition
`1`

Average change in value of the spread over `options.MaxStallGenerations` generations less than `options.FunctionTolerance`, and the final spread is less than the average spread over the past `options.MaxStallGenerations` generations

`0`

Maximum number of generations exceeded

`-1`

Optimization terminated by an output function or plot function

`-2`

No feasible point found

`-5`

Time limit exceeded

```[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...)``` returns a structure `OUTPUT` with the following fields:

OUTPUT FieldMeaning
`problemtype`Type of problem:
• `'unconstrained'` — No constraints

• `'boundconstraints'` — Only bound constraints

• `'linearconstraints'` — Linear constraints, with or without bound constraints

`rngstate`

State of the MATLAB® random number generator, just before the algorithm started. You can use the values in `rngstate` to reproduce the output of `ga`. See Reproduce Results.

`generations`Total number of generations, excluding `HybridFcn` iterations
`funccount`Total number of function evaluations
`message``gamultiobj` termination message
`averagedistance`Average "distance," which by default is the standard deviation of the norm of the difference between Pareto front members and their mean
`spread`Combination of the "distance," and a measure of the movement of the points on the Pareto front between the final two iterations
`maxconstraint`Maximum constraint violation at the final Pareto set

```[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...)``` returns the final `POPULATION` at termination.

```[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN, ...)``` returns the `SCORE` of the final `POPULATION`.

## Examples

This example optimizes two objectives defined by Schaffer's second function, which has two objectives and a scalar input argument. The Pareto front is disconnected. Define this function in a file:

```function y = schaffer2(x) % y has two columns % Initialize y for two objectives and for all x y = zeros(length(x),2); % ready for vectorization % Evaluate first objective. % This objective is piecewise continuous. for i = 1:length(x) if x(i) <= 1 y(i,1) = -x(i); elseif x(i) <=3 y(i,1) = x(i) -2; elseif x(i) <=4 y(i,1) = 4 - x(i); else y(i,1) = x(i) - 4; end end % Evaluate second objective y(:,2) = (x -5).^2;```

First, plot the two objectives:

```x = -1:0.1:8; y = schaffer2(x); plot(x,y(:,1),'.r'); hold on plot(x,y(:,2),'.b');```

The two component functions compete in the range [1, 3] and [4, 5]. But the Pareto-optimal front consists of only two disconnected regions: [1, 2] and [4, 5]. This is because the region [2, 3] is inferior to [1, 2].

Next, impose a bound constraint on `x`, setting

```lb = -5; ub = 10;```

The best way to view the results of the genetic algorithm is to visualize the Pareto front directly using the `@gaplotpareto` option. To optimize Schaffer's function, a larger population size than the default (15) is needed, because of the disconnected front. This example uses 60. Set the optimization options as:

`options = optimoptions('gamultiobj','PopulationSize',60,'PlotFcn',@gaplotpareto);`

Now call `gamultiobj`, specifying one independent variable and only the bound constraints:

```[x,f,exitflag] = gamultiobj(@schaffer2,1,[],[],[],[],... lb,ub,options); Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance. exitflag exitflag = 1```

The vectors `x`, `f(:,1)`, and `f(:,2)` respectively contain the Pareto set and both objectives evaluated on the Pareto set.

### Examples Included in the Toolbox

The `gamultiobjfitness` example solves a simple problem with one decision variable and two objectives.

The `gamultiobjoptionsdemo` example shows how to set options for multiobjective optimization.

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### Algorithms

`gamultiobj` uses a controlled elitist genetic algorithm (a variant of NSGA-II [1]). An elitist GA always favors individuals with better fitness value (rank). A controlled elitist GA also favors individuals that can help increase the diversity of the population even if they have a lower fitness value. It is important to maintain the diversity of population for convergence to an optimal Pareto front. Diversity is maintained by controlling the elite members of the population as the algorithm progresses. Two options, `ParetoFraction` and `DistanceFcn`, control the elitism. `ParetoFraction` limits the number of individuals on the Pareto front (elite members). The distance function, selected by `DistanceFcn`, helps to maintain diversity on a front by favoring individuals that are relatively far away on the front. The algorithm stops if the spread, a measure of the movement of the Pareto front, is small.

## References

[1] Deb, Kalyanmoy. Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, 2001.