# minmaxPareto: Pareto minimum and maximum values In spsann: Optimization of Sample Configurations using Spatial Simulated Annealing

## Description

Compute the minimum and maximum attainable values of the objective functions that compose a multi-objective combinatorial optimization problem.

## Usage

 `1` ```minmaxPareto(osc, candi, covars) ```

## Arguments

 `osc` A list of objects of class `OptimizedSampleConfiguration` (OSC). Each OSC of the list must be named after the objective function with which it has been optimized. For example, `osc = list(CORR = osc_corr, DIST = osc_dist)`. `candi` Data frame or matrix with the candidate locations for the jittered points. `candi` must have two columns in the following order: `[, "x"]`, the projected x-coordinates, and `[, "y"]`, the projected y-coordinates. `covars` Data frame or matrix with the covariates in the columns.

## Details

#### Multi-objective combinatorial optimization problems

A method of solving a multi-objective combinatorial optimization problem (MOCOP) is to aggregate the objective functions into a single utility function. In spsann, the aggregation is performed using the weighted sum method, which incorporates in the weights the preferences of the user regarding the relative importance of each objective function.

The weighted sum method is affected by the relative magnitude of the different function values. The objective functions implemented in spsann have different units and orders of magnitude. The consequence is that the objective function with the largest values may have a numerical dominance during the optimization. In other words, the weights may not express the true preferences of the user, resulting that the meaning of the utility function becomes unclear because the optimization will favour the objective function which is numerically dominant.

A reasonable solution to avoid the numerical dominance of any objective function is to scale the objective functions so that they are constrained to the same approximate range of values. Several function-transformation methods can be used for this end and spsann has four of them available.

The upper-lower-bound approach requires the user to inform the maximum (nadir point) and minimum (utopia point) absolute function values. The resulting function values will always range between 0 and 1.

The upper-bound approach requires the user to inform only the nadir point, while the utopia point is set to zero. The upper-bound approach for transformation aims at equalizing only the upper bounds of the objective functions. The resulting function values will always be smaller than or equal to 1.

In most cases, the absolute maximum and minimum values of an objective function cannot be calculated exactly. If the user is uncomfortable with guessing the nadir and utopia points, there an option for using numerical simulations. It consists of computing the function value for many random system configurations. The mean function value obtained over multiple simulations is used to set the nadir point, while the the utopia point is set to zero. This approach is similar to the upper-bound approach, but the function values will have the same orders of magnitude only at the starting point of the optimization. Function values larger than one are likely to occur during the optimization. We recommend the user to avoid this approach whenever possible because the effect of the starting configuration on the optimization as a whole usually is insignificant or arbitrary.

The upper-lower-bound approach with the minimum and maximum attainable values of the objective functions that compose the MOCOP, also known as the Pareto minimum and maximum values, is the most elegant solution to scale the objective functions. However, it is the most time consuming. It works as follows:

1. Optimize a sample configuration with respect to each objective function that composes the MOCOP;

2. Compute the function value of every objective function that composes the MOCOP for every optimized sample configuration;

3. Record the minimum and maximum absolute function values attained for each objective function that composes the MOCOP – these are the Pareto minimum and maximum.

For example, consider ACDC, a MOCOP composed of two objective functions: CORR and DIST. The minimum absolute attainable value of CORR is obtained when the sample configuration is optimized with respect to only CORR, i.e. when the evaluator and generator objective functions are the same (see the intersection between the second line and second column in the table below). This is the Pareto minimum of CORR. It follows that the maximum absolute attainable value of CORR is obtained when the sample configuration is optimized with regard to only DIST, i.e. when the evaluator function is difference from the generator function (see the intersection between the first row and the second column in the table below). This is the Pareto maximum of CORR. The same logic applies for finding the Pareto minimum and maximum of DIST.

 Evaluator Generator DIST CORR DIST 0.5 8.6 CORR 6.4 0.3

## Value

A data frame with the Pareto minimum and maximum values.

## Author(s)

Alessandro Samuel-Rosa alessandrosamuelrosa@gmail.com

## References

Arora, J. Introduction to optimum design. Waltham: Academic Press, p. 896, 2011.

Marler, R. T.; Arora, J. S. Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization, v. 26, p. 369-395, 2004.

Marler, R. T.; Arora, J. S. Function-transformation methods for multi-objective optimization. Engineering Optimization, v. 37, p. 551-570, 2005.

Marler, R. T.; Arora, J. S. The weighted sum method for multi-objective optimization: new insights. Structural and Multidisciplinary Optimization, v. 41, p. 853-862, 2009.

`optimACDC`, `SPAN`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```## Not run: # This example takes more than 5 seconds require(sp) data(meuse.grid) candi <- meuse.grid[, 1:2] covars <- meuse.grid[, c(1, 2)] # CORR schedule <- scheduleSPSANN(initial.acceptance = 0.1, chains = 1, x.max = 1540, y.max = 2060, x.min = 0, y.min = 0, cellsize = 40) set.seed(2001) osc_corr <- optimCORR(points = 10, candi = candi, covars = covars, schedule = schedule) # DIST set.seed(2001) osc_dist <- optimDIST(points = 10, candi = candi, covars = covars, schedule = schedule) # PPL set.seed(2001) osc_ppl <- optimPPL(points = 10, candi = candi, schedule = schedule) # MSSD set.seed(2001) osc_mssd <- optimMSSD(points = 10, candi = candi, schedule = schedule) # Pareto pareto <- minmaxPareto(osc = list(DIST = osc_dist, CORR = osc_corr, PPL = osc_ppl, MSSD = osc_mssd), candi = candi, covars = covars) pareto ## End(Not run) ```