A New Hybrid Evolutionary Algorithm for the Treatment of Equality Constrained MOPs

Abstract

Multi-objective evolutionary algorithms are widely used by researchers and practitioners to solve multi-objective optimization problems (MOPs), since they require minimal assumptions and are capable of computing a finite size approximation of the entire solution set in one run of the algorithm. So far, however, the adequate treatment of equality constraints has played a minor role. Equality constraints are particular since they typically reduce the dimension of the search space, which causes problems for stochastic search algorithms such as evolutionary strategies. In this paper, we show that multi-objective evolutionary algorithms hybridized with continuation-like techniques lead to fast and reliable numerical solvers. For this, we first propose three new problems with different characteristics that are indeed hard to solve by evolutionary algorithms. Next, we develop a variant of NSGA-II with a continuation method. We present numerical results on several equality-constrained MOPs to show that the resulting method is highly competitive to state-of-the-art evolutionary algorithms.

COARSE-EMOA: An indicator-based evolutionary algorithm for solving equality constrained multi-objective optimization problems

“…Introduced in [15], this indicator measures the percentage of feasible solutions in a population. This measure is trivial to compute and is defined as follows:…”

“…As a way to cope with constrained MOPs, many methods have been proposed ranging from the transformation of constraints into additional objectives to the incorporation of local search techniques and heuristics within EMOAs [10,11,12,13,14,15,16,17,18]. Nevertheless, many result in the inclusion of additional conditions for the problem to be solvable, by computing internal optimization problems or the calculation of gradients to obtain feasible solutions [14,19].…”

Pareto Explorer for Finding the Knee for Many Objective Optimization Problems

“…, k). Hence, via imposing the non-negativity for the β i 's in (12), the search for the "knee" solution is not based on the entire set of solutions of a given multi-(or many) objective optimization problem. We hence suggest to drop the non-negativity restriction leading to the following problem:…”

“…The only difference between problems (12) and (16) is that in the latter one the non-negativity conditions for β i , i = 1, . .…”

“…Such problems are also termed many objective optimization problems (MaOPs). Though MOPs and MaOPs are identical mathematically, the above described "curse of dimensionality" calls for different numerical procedures for their treatment: while it is, in many cases, possible to compute suitable finite-size approximations of the entire Pareto sets/fronts of MOPs (e.g., Reference [1,[6][7][8][9][10][11][12][13][14][15]), this becomes more challenging and even intractable with increasing number of objectives. There exist some evolutionary approaches that aim to compute finite size approximations of the entire solution sets for MaOPs.…”