Solver

A solver is the software category that reads a mathematical optimization model — typically through a file format such as LP, MPS, NL, OPL, or CNF, or through a modelling-language/API binding — and algorithmically produces an optimum (or near-optimum within tolerance) solution, primal and dual values, reduced costs, sensitivity information, and a feasibility report. In the OR ecosystem solvers split by problem class: linear programming (LP), mixed-integer linear programming (MILP), quadratic programming (QP), mixed-integer quadratic programming (MIQP), second-order cone programming (SOCP), semidefinite programming (SDP), general nonlinear programming (NLP), constraint programming (CP), and global optimization. The algorithmic engines split into families: for LP, the simplex method of Dantzig (1947) and the interior-point family of Karmarkar (1984) with barrier-predictor-corrector variants; for MILP, branch-and-cut (Padberg and Rinaldi 1991) built on branch-and-bound (Land and Doig 1960), branch-and-price (Barnhart et al. 1998), presolving, feasibility heuristics, gomory and other cut-generation layers; for NLP, sequential quadratic programming, interior-point, and trust-region variants. Solvers fall into two large groups: **commercial solvers** — licensed high-performance products with broad enterprise support and aggressive heuristic layers; **open-source solvers** — freely usable projects that are typically sufficient for academic and SMB workloads and run 2-10x slower than commercial peers on some problem classes. Models are passed via standard file formats (LP, MPS, NL, and similar) or via algebraic modelling languages and the mathematical-modelling packages of scientific-computing languages. Comparative performance is tracked in independent academic benchmark series. Bixby (2002) and Achterberg (2007) document the last thirty years of MILP solver speedup — not just hardware, but presolving, dual simplex, cut generation, and heuristics deliver a roughly 10⁶ total speedup.