# Mathematical optimization
<!-- MICROSIMGEN:BEGIN v1.7 — generated by g08_place_microsims.py; three.js first (§15); do not hand-edit inside -->
## Microsims — three.js
### Mathematical optimization (three.js)
<div class="microsim-player">
<iframe src="https://wikitube-3d-microsims.netlify.app/Mathematical_optimization.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Mathematical optimization — three.js microsim"></iframe>
</div>
**Open it full-screen:** [Mathematical_optimization.html](https://wikitube-3d-microsims.netlify.app/Mathematical_optimization.html) · library `threejs` · route `microsim/threejs/`
### Related microsims
Live sims on neighbouring articles — 2 of them inside this article's own Wikipedia link tree:
- [[Reliability_engineering]] *(in tree)*
- [[Stanford_torus]] *(in tree)*
- [[Failure_mode_and_effects_analysis]]
- [[Queueing_theory]]
- [[Structural_engineering]]
- [[Tribology]]
*Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*
<!-- MICROSIMGEN:END -->
## MicroSim spec
### Parameters (tunable controls)
- `Step size` · 0.001–0.5 · gradient-descent learning rate
- `Start x` · -10–10 · initial guess on the objective landscape
- `Landscape ruggedness` · 0–5 · number of local minima in the surface
### What animates
A marker steps downhill across a contoured objective landscape toward a minimum.
### Learning objective
Show how gradient descent searches a landscape for a minimum.
## Links (Wikipedia order)
<!-- injected from _registry/childlinks/Mathematical_optimization.json (2026-07-30T02:09:12Z) -->
`Abstract_algebra` · `Active-set_method` · `Active_filter` · [[Aerospace_engineering]] · `Affine_geometry` · `Affine_scaling` · `Agent_(economics)` · `Albert_W._Tucker` · `Algebra` · `Algebraic_geometry` · `Algebraic_number_theory` · `Algebraic_topology` · [[Algorithm]] · `American_Economic_Review` · `An_Essay_on_the_Nature_and_Significance_of_Economic_Science` · `Analytic_geometry` · `Analytic_number_theory` · `Applied_mathematics` · `Approximation_algorithm` · `Arg_max` · `Argument_of_a_function` · `Arithmetic` · `Arithmetic_geometry` · `Arkadi_Nemirovski` · [[Arthur_David_Hall_III]] · [[Artificial_intelligence]] · `Asset_pricing` · `Augmented_Lagrangian_method` · `Automated_reasoning` · `Barrier_function` · `Bayesian_optimization` · `Bellman_equation` · `Bellman–Ford_algorithm` · [[Benjamin_S._Blanchard]] · `Bernard_Koopman` · `Berndt–Hall–Hall–Hausman_algorithm` · [[Biological_systems_engineering]] · `Borůvka's_algorithm` · `Bounded_set` · `Brachistochrone_curve` · `Branch_and_bound` · `Branch_and_cut` · `Broyden–Fletcher–Goldfarb–Shanno_algorithm` · [[Business_process]] · [[Calculus]] · [[Calculus_of_variations]] · `Carl_Friedrich_Gauss` · `Category_theory` · `Christodoulos_Floudas` · `Claude_Berge` · [[Cognitive_systems_engineering]] · [[Combinatorial_optimization]] · `Combinatorics` · `Commutative_algebra` · `Comparative_statics` · [[Comparison_of_optimization_software]] · `Complementarity_theory` · [[Complex_analysis]] · `Complex_geometry` · `Computational_complexity_theory` · `Computational_geometry` · `Computational_mathematics` · `Computer_algebra` · [[Computer_engineering]] · [[Computer_programming]] · [[Computer_science]] · `Concave_function` · [[Configuration_management]] · `Conjugate_gradient_method` · `Constrained_optimization` · `Constraint_(mathematics)` · `Constraint_programming` · `Constraint_satisfaction` · `Construction_management` · `Consumer` · `Continuous_optimization` · [[Control_engineering]] · [[Control_theory]] · `Convex_analysis` · `Convex_function` · `Convex_geometry` · `Convex_optimization` · `Convex_set` · `Coordinate_descent` · `Countable_set` · `Criss-cross_algorithm` · `Critical_point_(mathematics)` · `Curve_fitting` · `Cutting-plane_method` · `David_Luenberger` · `Davidon–Fletcher–Powell_formula` · [[Decision-making]] · `Definite_matrix` · [[Derek_Hitchins]] · [[Design_review]] · `Deterministic_global_optimization` · `Differential_calculus` · [[Differential_equation]] · `Differential_evolution` · `Differential_geometry` · `Differential_topology` · `Dijkstra's_algorithm` · `Dimension` · `Dimitri_Bertsekas` · `Dinic's_algorithm` · `Diophantine_geometry` · `Discrete_geometry` · `Discrete_mathematics` · `Discrete_optimization` · `Domain_of_a_function` · [[Dynamic_programming]] · `Dynamic_relaxation` · `Dynamic_stochastic_general_equilibrium` · [[Earth_systems_engineering_and_management]] · `Economics` · `Edmonds–Karp_algorithm` · [[Electrical_engineering]] · `Elementary_algebra` · `Ellipsoid_method` · [[Energy]] · [[Engineering]] · [[Engineering_mathematics]] · `Engineering_optimization` · [[Enterprise_systems_engineering]] · `Envelope_theorem` · `Euclidean_geometry` · `Euclidean_space` · `Evolutionary_algorithm` · `Expenditure_minimization_problem` · `Extreme_value_theorem` · [[Fault_tolerance]] · `Field_(mathematics)` · `Finite_difference` · `Finite_geometry` · `Flow_network` · `Floyd–Warshall_algorithm` · `Ford–Fulkerson_algorithm` · `Foundations_of_mathematics` · `Fractional_programming` · `Frank–Wolfe_algorithm` · `Fritz_John` · `Function_(mathematics)` · [[Function_model]] · `Function_of_a_real_variable` · `Functional_(mathematics)` · [[Functional_analysis]] · `Future_of_mathematics` · [[Game_theory]] · `Gauss–Newton_algorithm` · `General_topology` · `Geometric_programming` · `Geometric_topology` · [[Geometry]] · `Geophysics` · [[George_Dantzig]] · `Global_optimization` · `Glossary_of_mathematical_symbols` · `Goal_programming` · `Golden-section_search` · `Gradient` · `Gradient_descent` · `Graph_(discrete_mathematics)` · [[Graph_theory]] · `Greedy_algorithm` · `Group_theory` · `Harmonic_analysis` · [[Harold_Chestnut]] · [[Health_systems_engineering]] · `Hessian_matrix` · `Heuristic_(computer_science)` · `Hill_climbing` · `History_of_mathematics` · `Homological_algebra` · `Homotopy_theory` · `Hydrological_optimization` · [[IDEF]] · [[Industrial_engineering]] · `Infinite-dimensional_optimization` · `Infinity` · `Informal_mathematics` · `Information_geometry` · [[Information_theory]] · `Integer` · `Integer_programming` · `Interior-point_method` · `International_Journal_of_RF_and_Microwave_Computer-Aided_Engineering` · `International_trade_theory` · `Interpolation` · `Interval_(mathematics)` · [[Isaac_Newton]] · `Iterative_method` · [[James_S._Albus]] · `John_Geanakoplos` · [[John_N._Warfield]] · [[John_von_Neumann]] · `Jon_Lee_(mathematician)` · `Jorge_Nocedal` · `Joseph-Louis_Lagrange` · [[Joseph_Francis_Shea]] · `Journal_of_Economic_Literature` · `Julio_Rotemberg` · `Karl_Weierstrass` · `Karmarkar's_algorithm` · `Karush–Kuhn–Tucker_conditions` · [[Kathleen_Carley]] · [[Katia_Sycara]] · `Kruskal's_algorithm` · `Lagrange_multiplier` · `Lagrangian_relaxation` · `Large_numbers` · `Lawrence_E._Blume` · `Least_squares` · `Lemke's_algorithm` · `Leonid_Kantorovich` · `Leonid_Khachiyan` · `Lev_Pontryagin` · `Levenberg–Marquardt_algorithm` · `Limited-memory_BFGS` · `Line_search` · `Linear-fractional_programming` · [[Linear_algebra]] · `Linear_complementarity_problem` · `Linear_programming` · `Lionel_Robbins` · `Lipschitz_continuity` · `List_of_films_about_mathematicians` · `List_of_optimization_software` · `List_of_publications_in_mathematics` · `Lists_of_mathematics_topics` · `Local_convergence` · `Local_search_(optimization)` · [[Logistics]] · `Loss_function` · `László_Lovász` · [[Machine_learning]] · `Macroeconomics` · [[Manuela_M._Veloso]] · `Margaret_H._Wright` · `Martin_Grötschel` · `Mathematical_Optimization_Society` · `Mathematical_Programming` · `Mathematical_analysis` · `Mathematical_and_theoretical_biology` · `Mathematical_chemistry` · `Mathematical_economics` · `Mathematical_finance` · `Mathematical_linguistics` · `Mathematical_logic` · [[Mathematical_model]] · `Mathematical_object` · `Mathematical_physics` · `Mathematical_programming_with_equilibrium_constraints` · `Mathematical_psychology` · `Mathematical_sociology` · `Mathematical_statistics` · `Mathematics` · `Mathematics_and_art` · `Mathematics_education` · `Matrix_(mathematics)` · `Matroid` · `Maximum_theorem` · `Measure_(mathematics)` · `Memetic_algorithm` · `Metaheuristic` · `Michel_Bierlaire` · `Microwave` · `Mineral_physics` · `Minimum_spanning_tree` · `Mirror_descent` · [[Model_predictive_control]] · `Multi-objective_optimization` · `Multidisciplinary_design_optimization` · `Narendra_Karmarkar` · `Natalia_K._Nikolova` · `Naum_Z._Shor` · `Nelder–Mead_method` · `Newton's_method_in_optimization` · `Nonlinear_conjugate_gradient_method` · `Nonlinear_programming` · `Number` · `Number_theory` · `Numerical_analysis` · [[Operations_research]] · [[Optimal_control]] · `Optimization_(disambiguation)` · `Optimization_problem` · `Optimum_(disambiguation)` · [[Ordinary_differential_equation]] · `Parallel_metaheuristic` · `Parameter` · [[Particle_swarm_optimization]] · `Pattern_search_(optimization)` · `Penalty_method` · `Performance_engineering` · `Permutation` · `Philosophy_of_mathematics` · [[Physics]] · `Pierre_de_Fermat` · [[Polyhedron]] · [[Polytope]] · `Portfolio_(finance)` · `Powell's_dog_leg_method` · `Powell's_method` · `Prim's_algorithm` · `Probability_theory` · `Process_optimization` · `Profit_(economics)` · [[Project_management]] · [[Projective_geometry]] · `Push–relabel_maximum_flow_algorithm` · `Quadratic_programming` · `Quality_function_deployment` · `Quality_management` · `Quantum_optimization_algorithms` · `Quasi-Newton_method` · `Quasiconvex_function` · `R._Tyrrell_Rockafellar` · `Rademacher's_theorem` · [[Radhika_Nagpal]] · `Random_variable` · `Real_analysis` · `Real_number` · `Recreational_mathematics` · `Relaxation_(approximation)` · [[Reliability_engineering]] · [[Requirements_engineering]] · `Resource_leveling` · `Revised_simplex_method` · `Richard_Bellman` · `Rigid_body_dynamics` · `Risk_aversion` · `Risk_management` · `Robert_Dorfman` · `Robert_E._Machol` · `Robust_optimization` · `Roger_Fletcher_(mathematician)` · `Ronald_A._Howard` · `Ruzena_Bajcsy` · `Saddle_point` · [[Safety_engineering]] · `Search_theory` · `Second-order_cone_programming` · `Seismology` · `Semidefinite_programming` · `Sequential_quadratic_programming` · `Set_(mathematics)` · [[Set_theory]] · `Shortest_path_problem` · `Simon_Ramo` · `Simplex_algorithm` · `Simulated_annealing` · `Simulation-based_optimization` · `Simultaneous_perturbation_stochastic_approximation` · `Slack_variable` · [[Software_engineering]] · `Space_mapping` · `Spare_part` · `Spiral_optimization_algorithm` · [[Stanford_torus]] · `Stationary_point` · `Statistics` · `Stephen_P._Boyd` · `Stochastic_optimization` · `Stochastic_process` · `Stochastic_programming` · `Stochastic_tunneling` · `Subgradient_method` · `Subset` · `Successive_linear_programming` · `Successive_parabolic_interpolation` · `Surrogate_model` · `Symmetric_rank-one` · [[System]] · [[System_dynamics]] · `System_integration` · [[System_of_systems_engineering]] · `Systems_analysis` · `Systems_development_life_cycle` · [[Systems_engineering]] · `Systems_modeling` · [[Systems_science]] · `Tabu_search` · `Test_functions_for_optimization` · `The_New_Palgrave_Dictionary_of_Economics` · `Theory_of_computation` · `Thomas_J._Sargent` · `Timeline_of_mathematics` · `Topology` · `Traffic` · [[Transportation_engineering]] · `Truncated_Newton_method` · `Trust_region` · `Type_theory` · `Undefined_(mathematics)` · `United_States` · `Utility` · `Utility_maximization_problem` · `V-model` · `Value_(mathematics)` · `Variable_(mathematics)` · `Vector_optimization` · `Vehicle_routing_problem` · `Verification_and_validation` · [[Wayback_Machine]] · `Wernher_von_Braun` · `William_Karush` · `Wolfe_conditions` · `Wolt_Fabrycky` · [[Work_breakdown_structure]] · `Yurii_Nesterov`
## From the Real GENERATIVE library
*Mathematical optimization — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Computation and Cybersecurity room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Max_paraboloid.svg).*
*Animated: Mathematical optimization — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Computation and Cybersecurity room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Nelder-Mead_Simionescu.gif).*
> Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives.[1][2] It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines f ([Wikipedia](https://en.wikipedia.org/wiki/Mathematical_optimization))
<!-- REAL-GENERATIVE-MEDIA:END -->
## Media (PD/CC)
<!-- MEDIA-DEPLOY:Mathematical_optimization/Nelder-Mead_Simionescu.gif -->
!Gif Library/Mathematical optimization/Nelder-Mead Simionescu.gif
*Nelder-Mead_Simionescu.gif · nicoguaro · CC BY-SA 4.0 · [source](https://commons.wikimedia.org/wiki/File:Nelder-Mead_Simionescu.gif)*
<!-- /MEDIA-DEPLOY -->
> p5.js MicroSim stub · part of Systems Engineering · [Wikipedia source](https://en.wikipedia.org/wiki/Mathematical_optimization)
## Concept summary
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization.
## Build checklist
- [ ] Claim it: set `status: in-progress` + `lease` + `leased_at`
- [ ] Write the child page explaining the concept (tie it back to Systems Engineering)
- [ ] Finalize parameter ranges and defaults
- [ ] Implement the p5.js sketch (drawing + control regions)
- [ ] Add caption + the learning objective on the page
- [ ] Set `status: done`
## p5.js sketch
```javascript
// MicroSim: Mathematical optimization
const ARTICLE = "Mathematical_optimization";
let p1, p2, p3;
function setup() {
createCanvas(720, 520);
pixelDensity(2);
// optimization-search sim — create the three sliders described above:
// Step size (0.001–0.5), Start x (-10–10), Landscape ruggedness (0–5).
// TODO: build labelled sliders in the control region (below the drawing region).
}
function draw() {
background(248);
// TODO: read p1/p2/p3; render the optimization-search in the drawing region and respond live.
// TODO: HUD — title + en.wikitube.io/wiki/Mathematical_optimization + control hints + live readouts.
}
```
---
Back to Systems Engineering
---
<!-- SEMIOTIC-PROFILE:START -->
## Semiotic profile
> *The semiotic universals this article invokes, machine-derived from the crossref — **unverified** (born so). Populated 2026-07-06 for the Systems room.*
**Universals (5):** 🟢 optimization (25) · 🟡 system (9) · 🟢 gradient (2) · 🟢 flow (1) · 🟢 discretization (1)
**Enter by sign:** Systems Semiotic Gateway · Alphabetum · Icon Registry · ← Systems Portal
<!-- SEMIOTIC-PROFILE:END -->
<!-- REAL-GENERATIVE-MEDIA:START -->
## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Mathematical_optimization) : [Wikitube](https://en.wikitube.io/wiki/Mathematical_optimization)
## Previous hub tags
Tree parents: [[Control_theory]] · [[Decision_theory]] · [[Game_theory]] · [[Graph_theory]] · [[Information_theory]] · [[Monte_Carlo_method]] · [[Operations_research]] · [[Reliability_engineering]] · [[Self-organization]] · [[System_dynamics]] · [[Systems_engineering]].
Legacy hubs: none.
---
*Sources: 2 legacy notes. Minted wave 1, 2026-07-30 (v1.6 order).*