Convex Optimization by Stephen Boyd and Lieven Vandenberghe is a comprehensive guide to the application of convex optimization in various fields, offering both theoretical insights and practical algorithms.
The excellent theoretical analysis properties, high practical computability, and powerful modeling capabilities are the reasons why people choose convex modeling. It is crucial to note that I am referring specifically to convex modeling.
In scientific research, the first step involves abstracting real-world problems into mathematical models. This process offers a significant degree of freedom in choosing how to represent these issues mathematically. Although non-convex modeling can be more straightforward and expressive, it comes with theoretical challenges and computational unreliability.
Over the past decade, areas such as compressed sensing, sparse representation, and low-rank recovery have gained prominence largely due to advancements made possible through convex modeling. Researchers analyze the properties of convex problems to explain and understand real-world mechanisms. It is worth noting that many issues had non-convex formulations decades ago; it was only with convex modeling that they were revitalized.
Moreover, by deeply understanding convex models, researchers have begun analyzing specific non-convex problems using their unique structural features, leading to profound insights—such as neural networks converging to local optima rather than saddle points and random algorithms aiding in escaping from saddle points. However, the analysis of non-convex issues is often case-by-case and lacks a unified effective method, which contrasts sharply with convex analysis.
From this perspective, convex modeling and optimization are preferred approaches for studying practical problems.
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