JuMP is an open-source modeling language for mathematical optimization, designed to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. It leverages the advanced features of the Julia programming language to offer unique functionality while achieving performance comparable to commercial modeling tools for standard tasks. This paper provides benchmarks, discusses novel implementation aspects, and explores how JuMP can be extended to new problem classes and integrated with state-of-the-art tools for visualization and interactivity. Key features of JuMP include: - **Algebraic Modeling**: Users can express optimization problems in a natural, algebraic form, similar to the original mathematical expressions. - **Performance**: JuMP offers competitive performance with commercial systems, thanks to its integration with Julia's advanced features. - **Extensibility**: JuMP supports extensions for new problem classes and specialized solution approaches, making it suitable for modern scientific workflows. - **Automatic Differentiation**: JuMP can automatically differentiate user-defined functions, enhancing its utility in nonlinear optimization. - **Code Generation**: JuMP uses Julia's code generation capabilities to efficiently process large-scale problems, particularly in linear and conic-quadratic optimization. The paper also highlights the importance of modeling languages in the field of mathematical optimization, emphasizing their role in simplifying the translation of complex mathematical formulations into solver input. JuMP's design and implementation are discussed in detail, including its use of syntactic macros and code generation to achieve efficient performance. Benchmarks comparing JuMP with other modeling languages (AMPL, Pyomo, GAMS, CVX, YALMIP) demonstrate its competitive or superior performance in various optimization tasks. Finally, the paper explores extensions to JuMP, such as handling parallel multistage stochastic programming, robust optimization, and chance constraints, which further enhance its applicability in real-world optimization problems.JuMP is an open-source modeling language for mathematical optimization, designed to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. It leverages the advanced features of the Julia programming language to offer unique functionality while achieving performance comparable to commercial modeling tools for standard tasks. This paper provides benchmarks, discusses novel implementation aspects, and explores how JuMP can be extended to new problem classes and integrated with state-of-the-art tools for visualization and interactivity. Key features of JuMP include: - **Algebraic Modeling**: Users can express optimization problems in a natural, algebraic form, similar to the original mathematical expressions. - **Performance**: JuMP offers competitive performance with commercial systems, thanks to its integration with Julia's advanced features. - **Extensibility**: JuMP supports extensions for new problem classes and specialized solution approaches, making it suitable for modern scientific workflows. - **Automatic Differentiation**: JuMP can automatically differentiate user-defined functions, enhancing its utility in nonlinear optimization. - **Code Generation**: JuMP uses Julia's code generation capabilities to efficiently process large-scale problems, particularly in linear and conic-quadratic optimization. The paper also highlights the importance of modeling languages in the field of mathematical optimization, emphasizing their role in simplifying the translation of complex mathematical formulations into solver input. JuMP's design and implementation are discussed in detail, including its use of syntactic macros and code generation to achieve efficient performance. Benchmarks comparing JuMP with other modeling languages (AMPL, Pyomo, GAMS, CVX, YALMIP) demonstrate its competitive or superior performance in various optimization tasks. Finally, the paper explores extensions to JuMP, such as handling parallel multistage stochastic programming, robust optimization, and chance constraints, which further enhance its applicability in real-world optimization problems.