Investor sentiment has a profound impact on financial market volatility; however, it is difficult to accurately capture the complex nonlinear relationships among sentiment proxies with the existing methods. In this study, we propose a novel investor sentiment indicator, S^GAN, which uses generative adversarial networks (GANs) to extract the nonlinear latent structure from eight sentiment proxies from February 2003 to September 2023 in the Chinese A-share market. Unlike traditional linear dimensionality reduction methods, GANs are able to capture complex market dynamics through adversarial training, effectively reducing noise and improving prediction accuracy. The empirical analyses show that S^GAN significantly outperforms existing methods in both in-sample and out-of-sample prediction capabilities. The GAN-based investment strategy achieves impressive annualized returns and provides a powerful tool for portfolio construction and risk management. Robustness tests across economic cycles, industries, and U.S. markets further validate the stability of S^GAN. These findings highlight the unique advantages of GANs as sentiment-driven financial forecasting tools, providing market participants with new ways to more accurately capture sentiment-shifting trends and develop effective investment strategies.

This study establishes a novel mathematical framework for stochastic maintenance optimization in production systems by integrating Markov decision processes (MDPs) with convex programming theory. We develop a Z-transformation-based dual-space decomposition method to reconstruct MDPs into a solvable linear programming form, resolving the inherent instability of traditional models caused by uncertain initial conditions and non-stationary state transitions. The proposed approach introduces three mathematical innovations: (i) a spectral clustering mechanism that reduces state-space dimensionality while preserving Markovian properties, (ii) a Lagrangian dual formulation with adaptive penalty functions to handle operational constraints, and (iii) a warm start algorithm accelerating convergence in high-dimensional convex optimization. Theoretical analysis proves that the derived policy achieves stability in probabilistic transitions through martingale convergence arguments, demonstrating structural invariance to initial distributions. Experimental validations on production processes reveal that our model reduces long-term maintenance costs by 36.17% compared to Monte Carlo simulations (1500 vs. 2350 average cost) and improves computational efficiency by 14.29% over Q-learning methods. Sensitivity analyses confirm robustness across Weibull-distributed failure regimes (shape parameter $\beta \in$ [1.2, 4.8]) and varying resource constraints.