Modern engineering systems generate models that are inherently high-dimensional, nonconvex and nonsmooth, due to physical and other constraints. The emergence of new optimisation models beyond the scope of existing convex methods calls for the development of novel and efficient numerical algorithms, capable of addressing nonconvexity and nonsmoothness. A novel approach, initiated in the 1980s, explored the class of difference-of-convex (DC) functions with the pioneering work of [Reference Tao, Souad and Hiriart-Urruty6]. The class of DC functions is broad enough to address many optimisation applications, while remaining close enough to convexity to use existing convex optimisation tools.
This doctoral thesis is dedicated to addressing practical challenges in nonconvex optimisation. Specifically, the possibility of developing algorithms for structural problems, where there is little or no convexity, is investigated, with a focus on generalising the DC problems. Novel proximal and splitting algorithms with guaranteed convergence, moving beyond traditional convex analysis by leveraging modern variational analysis, are developed. These methods are successfully applied to problems in power systems, demonstrating both theoretical novelty and practical utility. In particular, the following contributions have been published.
In [Reference Pham, Dao, Shah, Sultanova, Li and Islam4], a forward–backward (proximal subgradient) type algorithm for a generalised DC problem is developed. This work shows the possibility of solving the DC problem without requiring the convexity of the components. It also provides a proof-of-concept for solving nonconvex optimal power flow problems without convex relaxation.
In [Reference Pham, Dao, Eberhard and Sultanova3], we extend the famous Alternating Direction Method of Multipliers (ADMM) for solving the DC problem. This work is the first to extend the DC program into a multi-block structure in a nonconvex setting. The theoretical convergence analysis also requires weaker assumptions than the pioneering work in [Reference Wang, Yin and Zeng7].
In [Reference Pham, Shah, Amjady and Islam5], we propose a lightweight deep feature selection framework that uses the DC structure in training neural networks. By adding this, the framework can select useful features in the training process itself.
In [Reference Dao, Pham and Tung1, Reference Pham, Dao, Amjady and Shah2], we explore the possibility of extending Doughlas–Rachford splitting to efficiently solve the DC problem with a weak convexity assumption, from two different perspectives. In [Reference Dao, Pham and Tung1], the algorithm still uses the subgradient of one component, while in [Reference Pham, Dao, Amjady and Shah2], a new full proximal splitting strategy is proposed. Both algorithms in [Reference Dao, Pham and Tung1, Reference Pham, Dao, Amjady and Shah2] are the first in the literature to have guaranteed full sequential convergence without the common smooth assumption on the concave part in the DC structure. Both have shown superior performance on real datasets. The work in [Reference Pham, Dao, Amjady and Shah2] also provides a lightweight, fast and explainable signal reconstruction method for power system operators, and it has the potential to be developed further into plug-and-play algorithms.
