Solver

A solver is a program, method, or system designed to find solutions to problems expressed in a formal domain. In mathematics and computer science, solvers handle tasks such as solving equations, optimizing objectives, or determining the satisfiability of logical formulas.

Common types include numeric solvers, which compute roots or approximate solutions to equations; optimization solvers for linear, nonlinear, or integer programming; logic solvers for SAT and SMT problems; and constraint or constraint-programming solvers. Numeric solvers use methods such as Gaussian elimination, Newton-Raphson, or iterative schemes; optimization solvers employ simplex or interior-point methods, branch-and-bound, or heuristic search.

Applications span engineering, physics, economics, operations research, scheduling, verification, and automated theorem proving. Solvers require a formulation of the problem and produce a solution, a certificate of optimality or feasibility, or a statement of infeasibility.

Performance depends on problem structure, model size, and algorithm; many solvers use exact methods with guarantees, while others rely on approximations. Prominent examples include commercial and open-source tools such as Gurobi, CPLEX, and GLPK for optimization; Z3 and other SMT solvers for logical formulas; and LAPACK for linear algebra routines.

Limitations include completeness, time complexity, and numerical stability; some problems are computationally intractable in general. The field has evolved from foundational numerical methods and logic to sophisticated hybrid solvers used in science, engineering, and software verification.