多目标条件梯度方法的紧致基于代理的自适应步长
A Tight Surrogate-Based Adaptive Step Size for the Multiobjective Conditional Gradient Method
- Chongqing University of Science and Technology（重庆科技大学）
- Chongqing Jiaotong University（重庆交通大学）
- Chongqing Normal University（重庆师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对多目标条件梯度方法中自适应步长因代理函数保守而偏小的问题，提出利用各目标梯度Lipschitz常数的更紧致二次代理，在有限候选集上评估得到新步长，保持下降性并改进复杂度，实验验证有效。
AI中文摘要：
多目标条件梯度（MCondG）方法是一种用于求解约束多目标优化问题的有效下降型算法。在该算法中，自适应步长通过最小化一个一维凸二次代理函数来推导，该函数保证了每次迭代中所有目标函数的充分下降。然而，我们观察到该代理函数相对保守，产生过小的步长，从而减慢了算法的收敛速度。这种保守性源于代理函数受所有目标函数梯度中最大Lipschitz常数支配的事实。受此观察启发，我们开发了一个更紧致的二次代理函数，该函数利用各个梯度的Lipschitz常数。基于此代理，我们提出了一种新的自适应步长策略，通过在有限候选集上评估该代理来获得。我们证明了所提出的步长保持了MCondG方法的下降性质，并且与原始自适应步长策略相比，具有改进的最坏情况复杂度界。在基准问题和实际问题上的数值实验证明了所提出策略在计算效率和解决方案质量方面的有效性。
英文摘要：
The multiobjective conditional gradient (MCondG) method is an effective descent-type algorithm for solving constrained multiobjective optimization problems. In this algorithm, the adaptive step size is derived by minimizing a one-dimensional convex quadratic surrogate function, which guarantees a sufficient decrease in all objective functions at each iteration. However, we observe that this surrogate function is relatively conservative, producing overly small step sizes and consequently slowing the convergence of the algorithm. This conservativeness stems from the fact that the surrogate function is dominated by the largest Lipschitz constant among the gradients of all objective functions. Motivated by this observation, we develop a tighter quadratic surrogate function that exploits the individual gradient Lipschitz constants. Based on this surrogate, we propose a new adaptive step size strategy obtained by evaluating this surrogate over a finite candidate set. We prove that the proposed step size preserves the descent property of the MCondG method, and has an improved worst-case complexity bound compared with the original adaptive step size strategy. Numerical experiments on benchmark and real-world problems demonstrate the effectiveness of the proposed strategy in terms of computational efficiency and solution quality.
