Markowitz's criterion aims to balance expected return and risk when optimizing the portfolio. The expected return level is usually fixed according to the risk appetite of an investor, then the risk is minimized at this fixed return level. However, the investor may not know which return level is suitable for her/him and the current financial circumstance. It motivates us to find a novel approach that adaptively optimizes this return level and the portfolio at the same time. It not only relieves the trouble of deciding the return level during an investment but also gets more adaptive to the ever-changing financial market than a subjective return level. In order to solve the new model, we propose an exact, convergent, and efficient Krasnoselskii-Mann Proximity Algorithm based on the proximity operator and Krasnoselskii-Mann momentum technique. Extensive experiments show that the proposed method achieves significant improvements over state-of-the-art methods in portfolio optimization. This finding may contribute a new perspective on the relationship between return and risk in portfolio optimization.

翻译：马科维茨准则旨在优化投资组合时平衡期望收益与风险。通常根据投资者的风险偏好固定期望收益水平，然后在此固定收益水平下最小化风险。然而，投资者可能并不清楚何种收益水平适合自身需求及当前金融环境。这促使我们寻找一种能够同时自适应优化收益水平与投资组合的新方法。该方法不仅可减轻投资过程中确定收益水平的困扰，相较于主观设定的收益水平，更能适应不断变化的金融市场。为解决这一新模型，我们基于邻近算子与Krasnoselskii-Mann动量技术，提出了一种精确、收敛且高效的Krasnoselskii-Mann邻近算法。大量实验表明，所提方法在投资组合优化中较现有先进方法取得了显著改进。这一发现可能为投资组合优化中收益与风险关系的研究提供新视角。