Particle Swarm Optimization is a population-based metaheuristic inspired by social behavior of bird flocking and fish schooling. Each particle represents a candidate solution that moves through the search space with velocity influenced by: (1) its own best position found so far (cognitive component); (2) the best position found by the entire swarm (social component); (3) inertia from previous velocity. Update equations: vᵢ = ωvᵢ + c₁r₁(pbestᵢ - xᵢ) + c₂r₂(gbest - xᵢ) and xᵢ = xᵢ + vᵢ. PSO is simple to implement, requires few parameters, and is effective for continuous optimization, neural network training, and engineering design problems.

PERT is a probabilistic project management technique developed by the U.S. Navy in the 1950s for the Polaris missile program. Unlike CPM which uses deterministic durations, PERT accounts for uncertainty using three time estimates for each activity: optimistic time (a), most likely time (m), and pessimistic time (b). Expected duration is calculated as: tₑ = (a + 4m + b)/6, with variance σ² = [(b-a)/6]². PERT enables probability analysis of project completion times using the Central Limit Theorem, assuming activity durations follow beta distributions.

Polyhedral Theory studies geometric properties of polyhedra (solution sets of linear inequality systems) and their application to optimization. A polyhedron P = {x : Ax ≤ b} can be described by vertices (extreme points) or facets (defining inequalities). Key results: (1) Minkowski-Weyl theorem - polyhedra have dual representation as convex hulls of points plus cones; (2) facets of conv(S) for integer set S provide strongest valid inequalities; (3) totally unimodular matrices ensure integer vertices. For integer programs, the convex hull of integer solutions gives the ideal formulation. Polyhedral approaches: study combinatorial structures (matchings, stable sets), derive strong cutting planes, and design efficient formulations. Applications span: combinatorial optimization, network design, and scheduling.