Abstract
Many decision making situations are characterized by a hierarchical structure where a lower-level (follower) optimization problem appears as a constraint of the upper-level (leader) one. Such kind of situations is usually modeled as a BLOP (Bi-Level Optimization Problem). The resolution of the latter usually has a heavy computational cost because the evaluation of a single upper-level solution requires finding its corresponding (near) optimal lower-level one. When several objectives are optimized in each level, the BLOP becomes a multi-objective task and more computationally costly as the optimum corresponds to a whole non-dominated solution set, called the PF (Pareto Front). Despite the considerable number of recent works in multi-objective evolutionary bi-level optimization, the number of methods that could be applied to the combinatorial (discrete) case is much reduced. Motivated by this observation, we propose in this paper an Indicator-Based version of our recently proposed Co-Evolutionary Migration-Based Algorithm (CEMBA), that we name IB-CEMBA, to solve combinatorial multi-objective BLOPs. The indicator-based search choice is justified by two arguments. On the one hand, it allows selecting the solution having the maximal marginal contribution in terms of the performance indicator from the lower-level PF. On the other hand, it encourages both convergence and diversity at the upper-level. The comparative experimental study reveals the outperformance of IB-CEMBA on a multi-objective bi-level production-distribution problem. From the effectiveness viewpoint, the upper-level hyper-volume values and inverted generational distance ones vary in the intervals [0.8500, 0.9710] and [0.0072, 0.2420], respectively. From the efficiency viewpoint, IB-CEMBA has a good reduction rate of the Number of Function Evaluations (NFEs), lying in the interval [30.13%, 54.09%]. To further show the versatility of our algorithm, we have developed a case study in machine learning, and more specifically we have addressed the bi-level multi-objective feature construction problem.
Highlights
Bi-level optimization, as the name reflects, deals with the minimization or the maximization of two interconnected hierarchical levels: (1) a leader, called the upper-level problem, and (2) a follower, called the lower-level problem
IB-Co-Evolutionary Migration-Based Algorithm (CEMBA) ALGORITHMIC SCHEME In this subsection, we present the flowchart of IB-CEMBA in Fig. 2, and we describe the IB-CEMBA working principle using a number of steps as follows: 1) Step 0: Upper and lower population initialization: Initialize the upper-level population (UP) and the lower-level population (LP) using DSDM (Discrete Space Decomposition Method) [41] two times in order to obtain, at each level, two initial populations (UP1, UP2), (LP1, LP2)
Due to the lack of works proposed for multi-objective bi-level problems, we have compared our proposed IB-CEMBA to the extensions of three recently proposed combinatorial bi-level algorithms and to a nested bi-level algorithm that uses NSGA-II at both levels
Summary
Bi-level optimization, as the name reflects, deals with the minimization or the maximization of two interconnected hierarchical levels: (1) a leader, called the upper-level problem, and (2) a follower, called the lower-level problem. In this case, existing approaches select a lower-level solution randomly from the follower Pareto-optimal set This solution will be used for the upper-level problem. This further motivates the need to an efficient bi-level algorithmic scheme that significantly reduces the NFEs as possible This algorithmic design choice allows IB-CEMBA to preserve the population decomposition and migration strategies of the baseline CEMBA [28], which makes it able to solve combinatorial MOBPs with the least possible NFEs. In details, IB-CEMBA uses two populations in each level where each leader population works with its corresponding follower one. We mention here that our proposed algorithm is implemented using a multithreading (pseudo-parallel) mechanism in which the two sub-populations work in a parallel manner by the use of multiple threads
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