An overview of mathematical programming solvers: theory, development, and recent advances

Mathematical programming solvers are software tools designed to solve real-world problems using mathematical programming algorithms. They are widely used in areas such as manufacturing, transportation, finance, energy, logistics, and supply chains, where they help improve decision-making in planning, assignment, and scheduling.

In this review article, the authors provide an overview of the theoretical foundations, historical development, and recent advances of mathematical programming solvers. The paper discusses classical methods such as the simplex algorithm, interior-point methods, and branch-and-bound techniques, and then turns to newer developments supported by parallel computing, GPU acceleration, and artificial intelligence.

The review also highlights the recent rise of solvers developed by institutions and companies headquartered in China, including COPT, MindOpt, OptVerse, TAYLOR, and other representative China-originated solvers. Some of these solvers have ranked among the top performers in Hans Mittelmann’s public benchmark tests. However, benchmark results may not fully capture the diversity of real-world optimization problems.