Optimization Algorithms & Methods

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Abstractions about solving optimization problems, including metaheuristics like ant colony and particle swarm methods, line-search and interval-reduction techniques, and problem classes such as bilinear, nonlinear and semi-infinite programming.

19 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Ant colony optimization algorithms — A population metaheuristic in which stochastic construction agents reinforce useful graph choices through shared, evaporating pheromone values.
• Bilinear program — A nonlinear optimization problem whose objective or constraints contain products that are linear in either variable block when the other is fixed.
• Chance-constrained portfolio selection — A portfolio optimization model that maximizes a return objective while limiting the probability that final wealth or another outcome falls below a declared safety threshold.
• Covector mapping principle — A compatibility principle giving conditions under which discretizing an optimal-control problem and then dualizing yields covectors corresponding to a discretization of the continuous Pontryagin adjoint system.
• Fuzzy finite element — A finite-element uncertainty method that represents imprecise parameters as fuzzy numbers or fields and propagates their membership levels to response bounds.
• Golden-section search — A derivative-free interval-reduction algorithm for optimizing a unimodal function by placing interior evaluations in the golden ratio so one point can be reused each iteration.
• Guillotine cutting — A rectangular stock-cutting constraint in which every cut must pass straight from one edge of the current rectangular piece to the opposite edge, recursively partitioning it into two rectangles.
• Local search (optimization) — A heuristic optimization method that repeatedly moves to neighboring candidate solutions using local objective information.
• Maximum satisfiability problem — The optimization problem of assigning Boolean variables to maximize the number or total weight of satisfied clauses in a conjunctive normal form formula.
• Multifit algorithm — An approximation algorithm for identical-machine makespan scheduling that repeatedly runs first-fit-decreasing bin packing while binary-searching a trial capacity.
• Nonlinear programming — Optimization of an objective subject to constraints when the objective or at least one constraint is nonlinear in the decision variables.
• Pareto front — The set of feasible objective vectors or solutions not dominated by any alternative in a multi-objective optimization problem.
• Particle swarm optimization — A population-based optimization method in which candidate positions move through a search space using their own best experience and information from a neighborhood or global best.
• Powell's method — A derivative-free local optimization algorithm that performs successive line minimizations along a changing set of directions and replaces a direction with the net displacement to build approximate conjugacy.
• Prune and search — An optimization technique that repeatedly discards a guaranteed constant fraction of candidate input while preserving at least one optimum, then recurses on the remainder.
• Relaxation (approximation) — The replacement of a difficult optimization problem by an easier problem with weakened constraints or simplified structure whose solution bounds or informs the original.
• Semi-infinite programming — Optimization with finitely many decision variables and infinitely many constraints, or dually infinitely many variables and finitely many constraints.
• Simulation-based optimization — Optimization in which candidate decisions are evaluated by a computational simulation—often noisy, expensive and derivative-free—rather than a closed-form objective or constraint model.
• Ternary search — An interval-reduction search for the extremum of a unimodal function that compares two interior points and discards the third of the domain that cannot contain the optimum.