New approaches are making it capable to solve large-scale non-convex optimization problems with reduced computational cost through a combination of machine learning, surrogate modeling, and hybrid methods. Surrogate-assisted optimization (including deep surrogate-assisted optimization), multi-fidelity and high-dimensional optimization methods, and gradient-free optimization with learned directionality and active-fidelity methods are all emerging methods that have shown to help reduce computational costs while maintaining accuracy. Other methods can accelerate solving large-scale non-convex optimization problems that imply fundamental physical laws, such as utilizing physics-informed neural networks (PINNs) or landscape smoothing to help navigate a complicated objective landscape. The high-dimensional Bayesian optimization methods and adaptive latent-space dimensionality reduction methods like neural guided optimizer strategies, including transformers, have yielded promising results for high-dimensional black-box optimization and sequence-directed optimization. Distributed, parallel, and federated optimization frameworks may have added scalability options with regard to solving optimization problems of real-world systems. If applied with strategic integration, these optimization innovations, can potentially help mitigate heuristics of local minima or provide smoother convergence over different problems and applications.

Many domains like machine learning, engineering design, and operational study have significant non-convex optimization problems that are hard because of high dimensionality and numerous local minima. Traditional optimization algorithms are unable to handle these problems due to poor scalability and convergence guarantees. However, recent research shows promise in finding ways to navigate these difficulties more effectively. One good approach could be utilizing stochastic optimization strategies that integrate adaptive learning rates and variance reduction. The recent gains in variance reduction methods, such as SVRG and SAGA, show concentrating the gradient noise leads to faster convergence, specifically in large datasets (Johnson & Zhang, 2013).

Such strategies can efficiently address vast data while sailing through  the difficult non-convex spaces. In a different exceptional approach, researchers utilize second-order data from quasi-Newton schemas and approximate Hessian methods specifically engineered for non-convex issues. For example, Limited-memory BFGS (L-BFGS) or trust-region methods estimate curvature effectively to speed up convergence without the computational burden of Hessian calculations (Nocedal, & Wright, 2006). Approaches hybridizing first and second-order data have shown potential in escaping saddle points, which are hurdles in non-convex optimization (Dauphin et al., 2014). New research also integrates machine learning — driven strategies from tactics such as meta-learning and reinforcement learning in customizing optimization approaches to adapt to the challenge. Such approaches can gain knowledge of optimization approaches from the problem instance, enhancing efficiency in large-scale issues (Li & Malik, 2017).

On top of these, distributed and parallel computing grounds, such as federated evaluation and consensus-based methods, break large issues into smaller simultaneous problems, enhancing the scalability limits (Boyd et al., 2011). Ultimately, taking into account problem-specific structures, including smoothness, low-rankness, or sparsity, through constraint or regularization improvements extend efficiency by minimally reducing the search space reach (Bach et al., 2012).  In summary, cutting-edge approaches to deal with vast-scale non-convex optimization blend hybrid optimization & machine learning-driven heuristics, distributed computing, structural adaptation, and stochastic variance reduction. These improvements are directing the complex optimization problem-solving endeavor fruitfully.

References:

Bach, F., Jenatton, R., Mairal, J., & Obozinski, G. (2012). Optimization with sparsity-inducing penalties. Foundations and Trends® in Machine Learning, 4(1), 1-106.

Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine Learning, 3(1), 1-122.

Dauphin, Y. N., Pascanu, R., Gulcehre, C., Cho, K., Ganguli, S., & Bengio, Y. (2014). Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In Advances in Neural Information Processing Systems (pp. 2933-2941).

Johnson, R., & Zhang, T. (2013). Accelerating stochastic gradient descent using predictive variance reduction. In Advances in Neural Information Processing Systems (pp. 315-323).

Li, K., & Malik, J. (2017). Learning to optimize. arXiv preprint arXiv:1606.01885.

Nocedal, J., & Wright, S. (2006). Numerical Optimization (2nd ed.). Springer.

In order to navigate complicated landscapes and avoid local minima, novel approaches to addressing large-scale non-convex optimization problems frequently combine tried-and-true methods with creative alternatives. Methods such as hybrid descent techniques and random perturbation of the conditional gradient method (RPCGB) have the potential to achieve global convergence. Other active research fields include investigating novel kinds of gradient approximation and utilizing the potential of parallel computing.