# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 1 Introduction  
The problem of identifying the best potential combinations of assets has been present in the academic financial literature for a long time. Among the efforts made, the seminal framework introduced by [PERSON] [1; 2] has been especially influential. [PERSON] suggested analyzing sets of assets, also known as financial portfolios, in terms of risk and return. These two issues are contradictory in nature, since the desired higher returns usually come at a price of higher risks. The portfolio selection problem formulated by [PERSON] addresses the minimization of portfolio return variance subject to a minimum expected value of return. The best combinations of assets for certain degrees of risk define a line, the efficient frontier. Portfolios along this line have the best properties as, for each level or risk, there is no alternative that offers a higher return. Under some assumptions, these combinations can be identified by repeated use of quadratic programming. Unfortunately, simplistic assumptions of the classical [PERSON] model usually do not hold in the real world. Problems faced by practitioners might include complex constraints or definitions of risk that differ from the canonical portfolio standard deviation of returns, among others. For this reason, there has been a need to find alternative ways to handle the optimization process. Among the recent advances in optimization algorithms [3], evolutionary algorithms (EAs) have proven successful in overcoming difficulties with traditional optimization techniques either in their standard version, or hybridized [4; 5; 6]. EAs work with a set of solutions, called population. This feature is particularly suitable for solving multiobjective problems, as it enables approximating the efficient frontier in a single run. As a result, multiobjective evolutionary algorithms (MOEAs) have received growing attention to financial applications [7]. In fact, portfolio optimization was one of the first successful applications of MOEAs in economics and finance. The body of literature devoted to the topic has grown significantly over the last years [8]. The interest in the general subject, stronger than ever, can be illustrated by recent innovations on algorithms [9; 10; 11] and evolutionary operators [12]. However, the problem of robustness has been subject to a much lower amount of research. In the context of mean-variance portfolio optimization, the term of robustness can refer to different sources of uncertainty, such as the accuracy of future asset returns or the nature of the variance-covariance matrix [13; 14]. The portfolio performance will not be optimal if incorrect parameter values are used. The risk due to parameter uncertainty is commonly referred to as estimation risk. Another source of uncertainty is portfolio implementation risk. With the latter, we make reference to the possibility of temporary deviations form a target allocation that could result in undesired exposure. The aim of this paper is to show how to limit this problem by means of a robustness-based S-metric evolutionary multiobjective optimisation algorithm [15] (SMS-EMOA). As of today, most of the efforts made to mitigate uncertainty during the optimization process have been focused on parameters and the ability of the solutions to tolerate a certain degree of noise. Some examples of these efforts are [16; 17; 18]. Among the few works that deal with this issue using evolutionary algorithms, we can mention some that rely on the basic principle of picking solutions based on a number of future potential scenarios instead of a single prediction. For instance, in [19], profitable trading strategies are found by means of genetic programming. In [20], a portfolio modeling approach that considers uncertainty in returns of risky asset is presented and solved with a genetic algorithm (GA).