Departure question
Move sameness aroundPort 8 of 8Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
Optimize a linear objective over a region cut out by linear constraints. If a finite optimum exists and the feasible region has vertices, at least one optimal vertex exists. An entire edge can also be optimal, and a model may be infeasible or unbounded. Simplex and interior-point methods support resource allocation, flow, and production planning.
Scenario | Constraints | Objective | Optimum |
|---|---|---|---|
Factory resource allocation | Materials ≤ 100; labor ≤ 80 | Profit = 5x + 3y → max | One of the vertices |
Blending and mixing | Ingredients, nutrition, and demand | Cost → min | An LP when quantities are continuous |
Network flow | Capacity and flow conservation | Cost → min | Some routing and transport problems |
Linear approximation to power dispatch | Demand and capacity | Operating cost → min | Real systems add nonlinear and integer constraints |
Some linear-classification variants | Linear constraints | Linear loss or cost → min | Can be an LP; a standard SVM is usually a quadratic program |
The simplex algorithm moves among adjacent basic feasible solutions, while interior-point methods travel through the feasible region. Model quality also depends on how faithfully the objective and constraints represent reality.
Optimize a linear objective over a feasible region defined by linear constraints. Kantorovich’s 1939 production planning and Dantzig’s 1947 planning model and simplex method formed in different institutions; a model may be infeasible, unbounded, or have multiple optima.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
In Leningrad, linear inequalities and solution methods were applied to production allocation. The work remained little known in the West for years and is not treated as a direct transmission into the later American route.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearIn US Air Force planning, Dantzig developed a method that improves an objective by moving among vertices of a feasible polytope. Programming meant organizing plans more than writing code.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearThe ellipsoid method showed that linear programming is solvable in polynomial time in theory. Worst-case complexity and practical speed are different comparisons.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearKarmarkar opened a major era for interior-point methods, but his was not the first polynomial-time LP algorithm and it did not replace simplex in every setting.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearAirline and logistics planning, network routing, portfolio optimization, and support-vector machines.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
Open the number voyage
Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
Open the number voyage
Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Linear Programming
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
