Abstract

Convex Matrix Inequalities (CMIs) and Linear Matrix Inequalities (LMIs) are fundamental tools in control theory, optimization, and systems engineering, providing powerful frameworks for expressing and solving a broad class of problems. This article offers a detailed comparison between CMIs and LMIs, highlighting their theoretical foundations, computational complexities, and practical applications. LMIs, defined by linear constraints on symmetric matrix variables, are well-studied convex problems solvable efficiently using semidefinite programming. CMIs extend this framework by allowing convex but nonlinear matrix inequalities, thereby encompassing a wider set of problems but often at increased computational cost and complexity. The paper explores conditions under which CMIs can be approximated or converted into LMIs, discusses solution methods including interior-point algorithms and relaxation techniques, and examines trade-offs between expressiveness and tractability. Applications in robust control design, system stability analysis, and signal processing are reviewed to demonstrate the practical implications of choosing between CMIs and LMIs. This comparative study aims to guide researchers and practitioners in selecting appropriate matrix inequality formulations tailored to specific problem requirements and computational resources.