Does the method move only along edges?

No. Its core iterations use interior points and transformed coordinates.

Does polynomial time mean strongly polynomial?

No. The stated complexity depends on numerical input length as well as dimensions.

Can it directly enforce integer quantities?

The original algorithm solves linear programs over continuous variables; integrality requires additional machinery.

AI Glossary / Mathematics and optimization / Intermediate

Karmarkar's algorithm

Definition

Karmarkar's algorithm is a polynomial-time method for linear programming introduced by Narendra Karmarkar in 1984. It uses projective transformations to recenter a strictly positive feasible point, takes a controlled improvement step in transformed coordinates, and maps back. It is a landmark in interior-point optimization, while remaining distinct from the many later algorithms in that broader family.

What it means

Linear programming chooses continuous decision variables to optimize a linear objective under linear constraints. A feasible region can be viewed geometrically as a polyhedron. Karmarkar's approach changes the coordinate system around an interior point to make a useful improvement step easier to construct.
Polynomial time refers to a worst-case complexity bound in terms of the encoded problem size, including numerical input length. It does not mean every practical instance takes the same time, that the method is strongly polynomial, or that it always beats another solver on a particular workload.

How it works

The original formulation transforms the problem into a canonical form involving nonnegative coordinates summing to one, homogeneous linear equalities, and additional stated assumptions. At a strictly positive iterate, a diagonal projective transformation sends that point to the center of the simplex. The constraints are transformed as well.
The method constructs a direction within the transformed equality constraints, takes a controlled step that remains interior, and transforms back. A potential function supports the progress analysis because a projective transformation does not generally preserve a linear objective as a linear function. Initialization, stopping, and recovery of a solution are part of the complete algorithm, not optional details.

Three practical examples

An optimization course can use the method to show how changing coordinates supports interior movement through a linear-programming feasible region.
A solver-research study can compare projective scaling with other interior-point formulations on carefully specified test problems.
An operations-research team can use its historical and mathematical ideas to understand why linear programming need not be approached only by movement between vertices.

Think of it this way

Think of adjusting a map so that your current position sits near the center of the useful working area before planning the next move. Distances and directions on the adjusted map must be translated back consistently. The analogy is limited because a projective transformation changes mathematical coordinates and objective expressions, not the actual business constraints.

Worked example

Consider an original geometric illustration on the three-coordinate simplex x1+x2+x3=1, with all coordinates nonnegative. Let the current interior point be a=(0.2,0.3,0.5). Use the projective coordinate transformation y_i=(x_i/a_i) divided by the sum of all x_j/a_j.
At x=a, all three ratios x_i/a_i equal 1. Their sum is 3, giving y=(1/3,1/3,1/3). Thus the current point has become the center of the simplex in the new coordinates even though its original coordinates were unequal.
The inverse map is x_i=a_i y_i divided by the sum of all a_j y_j. To check the round trip, insert y=(1/3,1/3,1/3). The denominator is (0.2+0.3+0.5)/3=1/3, so the original a is recovered.
Now choose the illustrative transformed candidate y=(0.4,0.35,0.25). The unnormalized inverse coordinates are (0.08,0.105,0.125), totaling 0.31. Dividing by that total gives x=(16/62,21/62,25/62), approximately (0.258065,0.338710,0.403226). All three coordinates remain positive and sum to one.
For the objective f(x)=x2+3x3, the original point has value 0.3+3(0.5)=1.8. The mapped candidate has value (21+75)/62=48/31, approximately 1.548387. It improves this illustrative objective, but it is not optimal: the simplex vertex (1,0,0) gives value zero.
The objective expressed in y is (0.3y2+1.5y3)/(0.2y1+0.3y2+0.5y3). It is a ratio of linear expressions, not simply the original coefficient vector applied to y. This explains why changing coordinates requires more care than rescaling a drawing.
This example deliberately demonstrates the transformation, inverse, and objective evaluation only. The candidate y was chosen for transparent arithmetic; it was not produced by Karmarkar's prescribed search direction, inscribed-ball step, or potential-function rule. A full algorithm trace would also specify those steps and any extra equality constraints.

Illustrative case study

In a fictional optimization-literacy workshop, Stonebridge Logistics uses a small allocation model to compare several ways of thinking about linear programming. An analyst demonstrates projective recentering, then explains which additional ingredients are needed to turn the geometry into a complete solver.
For an actual planning run, the company uses a documented solver and checks feasibility, objective quality, and status messages. It does not assume that selecting a product described as an interior-point solver means it implements the original 1984 method. The workshop is illustrative and implies no measured operational saving.

Common misconceptions and mistakes

Do not call Karmarkar's algorithm the first polynomial-time algorithm for linear programming; the ellipsoid method preceded it. Do not treat all interior-point methods as synonyms for it. A simple diagonal rescaling without the required normalization and transformed constraints is not the complete projective algorithm.
Also distinguish continuous linear programming from integer programming. A fractional allocation can be optimal for an LP while failing a whole-unit business requirement. The polynomial-time result does not settle arbitrary mixed-integer optimization.

Business use

For business users, the lasting importance is that linear-programming algorithms can have both rigorous complexity analysis and useful geometric design principles. Solver selection still requires testing representative models, considering sparsity and numerical scaling, and checking solution diagnostics. Report the actual implementation and version when making performance claims rather than attributing every modern interior-point result to this historical algorithm.

Check your understanding

Why is the illustrative objective not simply y2+3y3 after transformation?

Answer: The inverse map contains both coordinate scaling and normalization, so substituting it produces a ratio of linear expressions in y.

Related terms

Linear programming
Interior-point method
Projective transformation
Simplex
Potential function
Feasible region
Polynomial-time algorithm
Affine scaling
Numerical conditioning

Learn next

Apply the transformation to x=(0.5,0.25,0.25) using the same a, then invert the result. Verify the objective in both coordinate systems. Explain why a zero component of a would prevent using the displayed division formula.

Sources

Primary research: Karmarkar, A New Polynomial-Time Algorithm for Linear Programming, original 1984 paper: https://www.stat.uchicago.edu/~lekheng/courses/302/classics/karmarkar.pdf . The simplex transformation example and fictional workshop are original.

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