Recently, numerous multiobjective evolutionary algorithms (MOEAs) have been proposed to solve the multiobjective optimization problems (MOPs). One of the most widely studied MOEAs is that based on decomposition (MOEA/D), which decomposes an MOP into a series of scalar optimization subproblems, via a set of uniformly distributed weight vectors. MOEA/D shows excellent performance on most mild MOPs, but may face difficulties on ill MOPs, with complex Pareto fronts, which are pointed, long tailed, disconnected, or degenerate. That is because the weight vectors used in decomposition are all preset and invariant. To overcome it, a new MOEA based on hierarchical decomposition (MOEA/HD) is proposed in this paper. In MOEA/HD, subproblems are layered into different hierarchies, and the search directions of lower-hierarchy subproblems are adaptively adjusted, according to the higher-hierarchy search results. In the experiments, MOEA/HD is compared with four state-of-the-art MOEAs, in terms of two widely used performance metrics. According to the empirical results, MOEA/HD shows promising performance on all the test problems.

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The core size of the porphyrin macrocycles was closely related to their stability of the different electron structure in the central metal ion. Cobalt(II) ions can undergo a conversion in electron configurations upon N4 core contraction of 0.05 Å in nonplanar porphyrins, and these ions still maintain low spin forms after and before conversion. The structural fine-tuning can induce the appearance of a cross-hybrid stage [d(x(2)-y(2))sp(2) ↔ d(z(2))sp(2)] based on quadrilateral coordination of the planar core. The results indicate that the configuration conversion plays a key role in electron transfer in redox catalysis involving cobalt complexes. The electronic properties of six monostrapped cobalt(II) porphyrins were investigated by spectral, paramagnetic, and electrochemical methods. The macrocyclic deformations and size parameters of Co-containing model compounds were directly obtained from their crystal structures.

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