Objectives, Decision Variables, Constraints, and Feasibility › Summary

A constrained optimization problem has four components: a decision vector whose entries are the quantities we control, a domain that says which values are admissible, an objective that scores each decision, and constraints that partition decisions into feasible and infeasible. Written compactly, we minimize $f(x)$ over the set $\mathcal{F} = \{x \in X : g_i(x) \leq 0 \text{ for all } i,\ h_j(x) = 0 \text{ for all } j\}$, and in the linear case this becomes the familiar $\min c^\top x$ subject to $Ax \leq b$ and $x \geq 0$. The formulation is a model of the decision, not the decision itself, and its quality depends on consistent units, coefficient provenance, faithful requirements, and a suitable solver.

Formulation translates a decision into variables, domains, an objective, and constraints. Optimization searches that model: LPs with a nonempty bounded feasible polyhedron admit an optimal vertex, mixed-integer solvers use bounds to limit the possible improvement of the incumbent objective, and nonconvex local methods provide no automatic global certificate. Validation independently recomputes objectives, residuals, and domain requirements. Relaxations and Lagrangian dual values supply bounds, while feasible decisions supply achievable values. Penalties price violations; they do not automatically enforce hard rules or integrality. The worked allocation example distinguishes a fractional LP bound, the exact enumerated integer optimum, and the solver's final gap, giving us a way to read the more specialized models that follow.