The working principles of model-based GAs fall within the PAC framework: A mathematical theory of problem decomposition

The concepts of linkage, building blocks, and problem decomposition have long existed in the genetic algorithm (GA) field and have guided the development of model-based GAs for decades. However, their definitions are usually vague, making it difficult to develop theoretical support. This paper provides an algorithm-independent definition to describe the concept of linkage. With this definition, the paper proves that any problems with a bounded degree of linkage are decomposable and that proper problem decomposition is possible via linkage learning. The way of decomposition given in this paper also offers a new perspective on nearly decomposable problems with bounded difficulty and building blocks from the theoretical aspect. Finally, this paper relates problem decomposition to PAC learning and proves that the global optima of these problems and the minimum decomposition blocks are PAC learnable under certain conditions.

Optimization by Pairwise Linkage Detection, Incremental Linkage Set, and Restricted / Back Mixing: DSMGA-II

This paper proposes a new evolutionary algorithm, called DSMGA-II, to efficiently solve optimization problems via exploiting problem substructures. The proposed algorithm adopts pairwise linkage detection and stores the information in the form of dependency structure matrix (DSM). A new linkage model, called the incremental linkage set, is then constructed by using the DSM. Inspired by the idea of optimal mixing, the restricted mixing and the back mixing are proposed. The former aims at efficient exploration under certain constrains. The latter aims at exploitation by refining the DSM so as to reduce unnecessary evaluations. Experimental results show that DSMGA-II outperforms LT-GOMEA and hBOA in terms of number of function evaluations on the concatenated/folded/cyclic trap problems, NK-landscape problems with various degrees of overlapping, 2D Ising spin-glass problems, and MAX-SAT. The investigation of performance comparison with P3 is also included.

Theoretical Perspective of Convergence Complexity of Evolutionary Algorithms Adopting Optimal Mixing

The optimal mixing evolutionary algorithms (OMEAs) have recently drawn much attention for their robustness, small size of required population, and efficiency in terms of number of function evaluations (NFE). In this paper, the performances and behaviors of OMEAs are studied by investigating the mechanism of optimal mixing (OM), the variation operator in OMEAs, under two scenarios -- one-layer and two-layer masks. For the case of one-layer masks, the required population size is derived from the viewpoint of initial supply, while the convergence time is derived by analyzing the progress of sub-solution growth. NFE is then asymptotically bounded with rational probability by estimating the probability of performing evaluations. For the case of two-layer masks, empirical results indicate that the required population size is proportional to both the degree of cross competition and the results from the one-layer-mask case. The derived models also indicate that population sizing is decided by initial supply when disjoint masks are adopted, that the high selection pressure imposed by OM makes the composition of sub-problems impact little on NFE, and that the population size requirement for two-layer masks increases with the reverse-growth probability.