ESTUDIOS DE LA RENTABILIDAD

Covariance Matrix Adaptation (CMA) (described in Section 2.2.3) has been se- lected from the literature as the desired variation operator for the design of an Evolutionary Algorithm (EA) intended for fast convergence within few func- tion evaluations on real-world many-objective problems. The single-objective CMA driven optimiser, the Covariance Matrix Adaptation Evolutionary Strat- egy (CMA-ES) (described in Section 2.2.3), has been shown to perform extremely well across a broad range of problems, including the single-objective optimiser performance comparisons in [58, 54].

The Multi-Objective Covariance Matrix Adaptation Evolution Strategy (MO- CMA-ES) (described in Section 2.7.3) is an existing Evolutionary Multi-Objective Optimisation (EMO) algorithm which utilises the CMA variation operator and has been shown to perform well in a number of algorithm variations in [36, 89, 88]. However, MO-CMA-ES relies on the contributing hypervolume (described in Section 2.5.2) indicator as a second-level sorting criterion and therefore suffers

from computational infeasibility on multi-objective problems which consist of more than three problem objectives. Real-world problems are often complex and require the optimisation of many objectives, therefore the CMA operator for variance needs to be incorporated into a new optimisation algorithm if it is to satisfy this requirement and be capable of being utilised for problems consisting of four or more problem objectives. In order to design an EMO algorithm which is driven by the CMA operator and capable of optimisation in the presence of many objectives, subjecting the entire non-dominated population to the contributing hypervolume indicator at each generation of the optimisation life-cycle must be avoided.

This chapter is divided into three sections. First, the Covariance Matrix Adaptation Pareto Archived Evolution Strategy (CMA-PAES) is introduced in Section 3.1 as a fast EMO algorithm which offers comparable performance to MO-CMA-ES, without reliance on the hypervolume indicator. Section 3.2 intro- duces the Multi-tier Covariance Matrix Adaptation Pareto Archived Evolution Strategy (m-CMA-PAES), an EMO algorithm which uses a multi-tier AGA with a grid-level hypervolume indicator, which outperforms MO-CMA-ES on problems consisting of two and three objectives. The chapter concludes with a summary of the developed algorithms and their intended use in Section 3.3.

3.1

CMA-PAES

The Pareto Archived Evolution Strategy (PAES) (described in Section 2.7.2) is an EMO algorithm which both contains a unique method of diversity preservation in the form of an Adaptive Grid Algorithm (AGA), and an algorithm which is does not have a high computational cost due to its simplicity [150]. The simplicity of

3.1. CMA-PAES 79

PAES has inspired a base framework which can be used to intuitively incorporate the CMA operator, such that an AGA and bounded Pareto archiving scheme will be responsible for diversity preservation and selection for variation and survival, and CMA will be used as the variation operator.

An algorithm inspired by the PAES structure and the CMA scheme for vari- ation has been designed under the name CMA-PAES. With the aim to be light in computational cost (without considering the computational cost of objective function evaluation), simple in structure in order to allow for easy extensibil- ity as the algorithm matures and develops in further work, and the ability to produce approximation sets with performance initially similar to or better than MO-CMA-ES on a test suite for which comparison between CMA-PAES and MO-CMA-ES is feasible (three objectives or lower). The field of Evolutionary Computation (EC) is growing year by year, with many contributions including the introduction of new methods for selection, diversity preservation, variance, etc. The development of a simple and modular framework (CMA-PAES) would allow for easy incorporation of these new methods in any number of combina- tions, meaning that CMA-PAES can be extended to target specific problems or to incorporate state of the art techniques.

The algorithm execution order for CMA-PAES has been illustrated in Figure 3.1. CMA-PAES begins by initializing the algorithm variables and parameters, these include the number of grid divisions used in the AGA, the archive for storing Pareto-optimal solutions, the parent vector Y and the covariance matrix. An initial current solution is then generated at random, which is evaluated and then the first to be archived (without being subjected to the PAES archiving procedure). The generational loop then begins, the square root of the covariance

matrix is resolved using Cholsky decomposition (as recommended by [151]) which offers a less computationally demanding alternative to spectral decomposition. Theλcandidate solutions are then generated using copies of the current solution and the CMA-ES procedure for mutation before being evaluated. The archive is then merged with the newly generated offspring and subjected to Pareto ranking, this assigns a rank of zero to all non-dominated solutions, and a rank reflecting the number of solutions that dominate the inferior solutions. The population is then purged of the inferior solutions so that only non-dominated solutions remain before being fed into the PAES archiving procedure. After the candidate solutions have been subjected to the archiving procedure and the grid has been adapted to the new solution coverage of objective space, the archive is scanned to identify the grid location with the smallest population, this is considered the lowest density grid population (ldgp). The solutions from the lowest density grid population are then spliced onto the end of the first µ−ldgp of the Pareto rank ordered population to be included in the adaptation of the covariance matrix, with the aim to improve the diversity of the next generation by encouraging movement into the least dense area of the grid. After the covariance matrix is updated, the generational loop continues onto its next iteration until the termination criteria is satisfied (maximum number of generations).

CMA-PAES has been benchmarked against the Nondominated Sorting Ge- netic Algorithm II (NSGA-II) and PAES in [15] in a performance comparison on the ZDT synthetic test suite, using two performance metrics to compare per- formance in terms of proximity (using the generational distance metric) and di- versity (using the spread metric). CMA-PAES displayed superior performance (the significance of which was supported with randomisation testing) in return-

3.1. CMA-PAES 81

Generate random current solution Evaluate and add to archive

Terminate

search? Stop

Use CMA to mutate current solution to generate population of

candidate solutions Evaluate candidate solutions

Is candidate dominated by

current?

Compare candidate solution with archive members

Update archive

Select new current solution from candidate and current solutions Yes

Yes

No

Reduce the population to only non-dominated solutions using

Pareto ranking

Candidates remaining?

Yes No

Get the square root of covariance matrix using cholsky

decomposition No

Recombination of mu best from lambda offspring by mean value

calculation

Update covariance matrix

Figure 3.1: Execution life-cycle for the CMA-PAES algorithm.

ing an approximation set close to or on the true Pareto-optimal front as well as maintaining diversity amongst solutions in the set.

The ultimate aim of CMA-PAES development is to utilise the benefits of the CMA operator for variance in an EMO algorithm that is computationally feasible on many-objective problems, and comparable in performance to MO-CMA-ES.

3.1.1

Comparison Between CMA-PAES and