Bilevel Optimisation
A mathematical programme with another mathematical programme nested inside it as a constraint: the feasible set contains only solutions where the inner decision-maker is also optimising.
Read →Ambiguity Set
Instead of betting on one assumed distribution, collect all distributions consistent with what the data shows and optimise for the worst one in that set.
Read →Time-Space Network
A graph where nodes are (place, time) pairs and arcs encode movement or waiting — turning scheduling into network flow.
Read →Arc Consistency
Arc consistency removes every variable value that has no compatible partner in any constrained pair, permanently pruning it before the solver branches. One deadline on Machine B can cascade to eliminate start times from Machine A. Modern CP solvers enforce it continuously at every node.
Read →Conflict Analysis
When a MIP solver discovers an infeasible subproblem during branch-and-bound, conflict analysis extracts the reason and converts it into a reusable constraint. The solver never rediscovers the same dead region. Every modern solver runs it automatically — practitioners see the output as a conflict count in logs.
Read →Bellman Equation
The value of where you are equals the best immediate payoff you can get, plus the value of wherever that action leads. Applying that one equality consistently across every state is all of dynamic programming.
Read →Budgeted Uncertainty Set
A Γ-budget set caps how many uncertain parameters can simultaneously be adversarial. It sits between the deterministic nominal point and the full worst-case box, producing solutions robust to realistic disruptions without the over-conservatism of guarding against every parameter being bad at once.
Read →Second-Order Cone Programming
Optimises a linear objective over cone constraints, subsuming LP and convex QP, solvable to global optimality in polynomial time — the natural solver for portfolio CVaR bounds, power flow relaxations, and Wasserstein-ball distributionally robust optimisation.
Read →Distributionally Robust Optimisation
A hedge against being wrong about the probability distribution itself — minimise worst-case expected cost over a whole ambiguity ball of distributions around the one you estimated from data.
Read →Conditional Value at Risk
The average loss across the worst alpha fraction of outcomes — the mean of what lies beyond Value at Risk's threshold, not the threshold itself, which is what makes it coherent where Value at Risk is not.
Read →Augmented Lagrangian
A Lagrangian plus a quadratic stabiliser on constraint violation; the stabiliser lets a finite penalty weight do what would otherwise take a pure penalty heading to infinity.
Read →Chance Constraint
Requires a condition to hold with at least probability 1 − α rather than in every scenario, with α as the single most consequential number the modeller picks.
Read →Scenario Recourse Inequality
Allows each scenario to choose optimal recourse independently, then projects scenario-specific actions back onto robust first-stage decisions.
Read →Rolling Horizon Optimisation
Solve the near horizon, commit the first decisions, slide forward, re-optimise -- trading global optimality for tractability, at the cost of suboptimality near each window boundary.
Read →Large Neighbourhood Search
A metaheuristic that escapes local optima by repeatedly destroying a large portion of the current solution and rebuilding it using an exact solver.
Read →Predict-then-Optimize
A two-phase workflow where an ML model forecasts uncertain parameters passed to an optimisation solver, creating a risk of decision infeasibility if forecast accuracy metrics don't align with decision error.
Read →Non-Anticipativity
A structural constraint in multi-stage stochastic models that forces decisions to be made using only information available at the time — preventing inadvertent peeking at the future.
Read →Big-M Formulation
Use a large constant to linearise disjunctive logic in mixed-integer models by scaling constraints with binary variables, at the cost of numerical fragility.
Read →Lexicographic Optimisation
Rank multiple objectives in strict sequential order: optimise the first criterion absolutely, then fix its optimal value and optimise the second criterion within that constraint.
Read →Integrality Gap
The fractional gap between the optimal value of an LP relaxation and the optimal value of the integer problem — a measure of how weak the relaxation is.
Read →Column Generation
An algorithmic technique for solving large LPs and MIPs by iterating between a restricted master problem and a pricing subproblem that identifies the most improving variable to add.
Read →Two-Stage Robust Optimisation
Optimise a here-and-now decision against the worst-case realisation of uncertain parameters in a given uncertainty set, where a recourse action is allowed after uncertainty is revealed.
Read →Pareto Optimality
A solution is Pareto optimal when no objective can be improved without worsening at least one other objective. The Pareto front is the set of all such non-dominated solutions.
Read →Constraint Programming
Specify what constraints must hold over a set of decision variables; let a solver propagate those constraints and search for an assignment that satisfies all of them.
Read →Stochastic Programming
Minimises expected cost across a probability distribution of scenarios rather than a single forecast — producing decisions that are good on average rather than guaranteed for the worst case.
Read →Mixed Integer Programming
Optimisation where some decisions must be whole numbers — enabling yes/no choices, assignment decisions, and logical conditions that continuous LP cannot model.
Read →Branch and Bound
Solves integer programmes exactly by recursively splitting on fractional variables while pruning branches where the LP relaxation proves no improving integer solution can exist.
Read →Cutting Planes
Valid linear inequalities added to the LP relaxation that slice off fractional corners without removing any integer-feasible solution, steadily closing the gap toward the integer hull.
Read →Benders Decomposition
Solves large MIPs by fixing complicating integer variables, solving the resulting LP exactly, and feeding back optimality and feasibility cuts until the two subproblems converge.
Read →Lagrangian Relaxation
Moves hard coupling constraints into the objective with a penalty multiplier, creating a decomposable problem that provides a lower bound and often near-optimal solutions.
Read →Scenario Recourse Inequalities
Valid cuts for two-stage stochastic MIPs that encode optimal recourse policy choices as integer constraints, enabling exact solution with provable optimality certificates.
Read →Vehicle Routing Problem
Find the cheapest way to assign customers to a fleet of vehicles and sequence each vehicle's stops, subject to capacity and time constraints.
Read →Dual Decomposition
Duplicates shared variables across independently-solved subproblems and enforces agreement through iteratively-updated Lagrange multiplier prices; enables parallel solvers coordinated by a compact price vector.
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