What Are Optimization Models?

Optimization models, also written optimisation models, are mathematical representations of a decision problem that state an objective to be maximized or minimized, a set of decision variables that can be chosen, and a set of constraints that any acceptable choice must satisfy. They are the formulation layer that sits between a real problem and the algorithms that solve it: the model says what is being decided and what counts as better, while a solver searches the space the model defines. The distinction matters because the same physical situation can be modeled in several ways, and the choice of formulation often determines whether a problem is solvable in seconds or is computationally hopeless.

Optimization modeling grew out of operations research during and after the Second World War, when George Dantzig introduced the simplex method for linear programs in 1947. It draws on linear algebra, convex analysis, graph theory, and numerical computation, and it now underpins large parts of control engineering, machine learning, and network design. Models are classified by the algebraic form of the objective and constraints and by whether the variables are continuous, discrete, or uncertain, since those properties determine which solution methods apply.

Formulation Structure

Every optimization model has the same three parts. Decision variables encode what the modeler is free to choose, such as production quantities, routing flows, or filter coefficients. The objective function assigns a scalar score to any assignment of those variables, and it may combine competing terms with weights or be replaced by a vector objective in multi-objective formulations. Constraints delimit the feasible region, and they come in two families: hard constraints that must hold, such as a conservation law or a capacity limit, and soft constraints implemented as penalty terms when a modest violation is preferable to declaring the model infeasible. Good practice is to keep the model as tight as possible, meaning that its continuous relaxation should bound the true optimum closely, since formulation tightness often affects solver performance more than raw machine speed does.

Linear and Integer Models

When both objective and constraints are linear in the decision variables, the model is a linear program, and reliable general-purpose methods find a provably optimal solution without reference to the problem's origin, a property that explains the wide reach of linear programming. Requiring some or all variables to take integer values produces integer and mixed-integer linear programs, which express indivisible decisions such as whether to build a facility, assign a task, or open a route. That expressiveness comes at a cost: integer programming problems are NP-hard in general and are attacked with branch-and-bound, cutting planes, and branch-and-cut rather than with a direct method. Combinatorial models such as assignment, set covering, knapsack, and vehicle routing are usually stated in this form.

Nonlinear, Stochastic, and Heuristic Formulations

Nonlinear models arise whenever physics, economics, or engineering relationships are not affine. Convex nonlinear programs retain the guarantee that a local optimum is global, which is why semidefinite and second-order cone formulations are preferred where they apply. Combining discrete variables with nonlinear functions produces mixed-integer nonlinear programming, the hardest of the standard model classes to solve to proven optimality. Stochastic and robust models account for uncertainty explicitly, either by optimizing an expectation over scenarios or by guaranteeing performance across an uncertainty set. Where exact methods are impractical, metaheuristics such as genetic algorithms, simulated annealing, and particle swarm methods trade optimality guarantees for speed, and a unified taxonomy of mixed optimization metaheuristics surveys how these approaches handle variable-size and mixed problems.

Applications

Optimization models have applications in a wide range of disciplines, including:

• Supply chain planning, production scheduling, and inventory control
• Power system unit commitment and economic dispatch
• Telecommunications network design and traffic engineering
• Portfolio selection and financial risk management
• Model predictive control in process and automotive engineering
• Machine learning training objectives and hyperparameter tuning