Convergence rates analysis of a multiobjective proximal gradient method
Convergence rates analysis of a multiobjective proximal gradient method
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DOI：
10.1007/s11590-022-01877-7
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发表时间：
2020-10
影响因子：
1.6
通讯作者：
H. Tanabe;E. H. Fukuda;N. Yamashita
中科院分区：
文献类型：
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作者：
H. Tanabe;E. H. Fukuda;N. Yamashita
在过去的二十年中，已经开发了许多用于多目标优化的下降算法。田边等人。 (Comput Optim Appl 72(2):339–361, 2019)提出了一种多目标优化的近端梯度法，可以解决多目标问题，其目标函数是连续可微函数与闭、真、凸函数之和。在合理的假设下，已知该方法生成的序列的累加点是Pareto平稳的。然而，该论文并未确定收敛率。在这里，我们展示了多目标近端梯度法的全局收敛率，与标量优化中已知的相匹配。更具体地说，通过使用评价函数来衡量复杂性，我们给出了非凸 ()、凸 (O(1/k)) 和强凸（对于某些）问题的收敛率。我们还扩展了所谓的 Polyak-Łojasiewicz (PL) 不等式以实现多目标优化，并建立了满足此类不等式（对于某些）的多目标问题的线性收敛率。
Many descent algorithms for multiobjective optimization have been developed in the last two decades. Tanabe et al. (Comput Optim Appl 72(2):339–361, 2019) proposed a proximal gradient method for multiobjective optimization, which can solve multiobjective problems, whose objective function is the sum of a continuously differentiable function and a closed, proper, and convex one. Under reasonable assumptions, it is known that the accumulation points of the sequences generated by this method are Pareto stationary. However, the convergence rates were not established in that paper. Here, we show global convergence rates for the multiobjective proximal gradient method, matching what is known in scalar optimization. More specifically, by using merit functions to measure the complexity, we present the convergence rates for non-convex (), convex (O(1/k)), and strongly convex (for some) problems. We also extend the so-called Polyak-Łojasiewicz (PL) inequality for multiobjective optimization and establish the linear convergence rate for multiobjective problems that satisfy such inequalities (for some).