Interval Robust Multi-Objective Evolutionary Algorithm

Abstract

Uncertainties are commonly present in optimization systems, and when they are considered in the design stage, the problem usually is called a robust optimization problem. Robust optimization problems can be treated as noisy optimization problems, as worst case minimization problems, or by considering the mean and standard deviation values of the objective and constraint functions. The worst case scenario is preferred when the effects of the uncertainties on the nominal solution are critical to the application under consideration. Based on this worst case scenario, we developed the [I]RMOEA (Interval Robust Multi-Objective Evolutionary Algorithm), a hybrid method that combines interval analysis techniques to deal with the uncertainties in a deterministic way and a multiobjective evolutionary algorithm. We introduce [I]RMOEA and illustrate it on three robust test functions based on the ZDT problems. The results show that [I]RMOEA is an adequate way of tackling robust optimization problems with evolutionary techniques taking advantage of the interval analysis framework.

Robust feeder reconfiguration in radial distribution networks

“…The following evolutionary algorithm parameters were held fixed: population size (60), maximum number of generations (30), and probabilities of crossover (0.98) and mutation (0.05).…”

“…The worst case Robust Multi-objective Optimization Problem (RMOP) can be seen as maximizing the effects of uncertainty on the objective functions, and subsequently minimizing all objective functions, as described in [30]. Mathematically, given that x 2 X # R Nw ; p 2 P # R Np and f ðx; pÞ : R Nw  R Np #R Nf the RMOP is defined by: …”

Comparing Solutions under Uncertainty in Multiobjective Optimization

“…In [9], a robust multiobjective evolutionary algorithm was developed for solving optimization problems in which solutions should be invariant to small input changes. The uncertain parameters are represented with intervals, which results in solution objectives also being represented with intervals.…”

Interval-based ranking in noisy evolutionary multi-objective optimization

“…noise in the input values, which is sometimes referred to as robust optimization and its aim is to find solutions with the highest sta bility in their objective values. Soares et al [17] optimized the worst noisy objective values of the solutions in a min-max formulation using interval analysis. To decrease the amount of uncertainty in the intervals they propose to recursively divide the inter vals into halves, resulting in a grid which is placed on the objective space and is used to compute the worst objective values of the solutions.…”

“…Moreover, the proposed method can deal with MOPs consisting of both singular and interval objectives. Teich [20] Hughes [19] Büche et al [27] Babbar et al [29] Bui et al [22] Fieldsend and Everson [26] Basseur and Zitzler [38] Goh and Tan [30] Mehnen et al [35] Wo z´ niak [36] Boonma and Suzuki [39] Bui et al [32] Eskandari and Geiger [31] Kaji et al [37] Soares et al [17] Syberfeldt et al [33] Park and Ryu [34] Probabilistic Dominance One way to deal with the noise in an MOP when the objective values are represented with intervals is to extend the Pareto dominance definition (Definition 1). In the following definitions we assume that the set of objectives T is partitioned into two disjoint subsets of objectives with singular values Ts, and noisy objectives with interval values Tj, such that Ts U Tj = T and Ts n Ti = 0.…”