Hubbry Logo
Karmarkar's algorithmKarmarkar's algorithmMain
Open search
Karmarkar's algorithm
Community hub
Karmarkar's algorithm
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Karmarkar's algorithm
Karmarkar's algorithm
Karmarkar's algorithm
View on Wikipedia
from Wikipedia

Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient algorithm that solves these problems in polynomial time. The ellipsoid method is also polynomial time but proved to be inefficient in practice.

Denoting by the number of variables, m the number of inequality constraints, and the number of bits of input to the algorithm, Karmarkar's algorithm requires operations on -digit numbers, as compared to such operations for the ellipsoid algorithm.[1] In "square" problems, when m is in O(n), Karmarkar's algorithm requires operations on -digit numbers, as compared to such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus using FFT-based multiplication (see Big O notation).

Karmarkar's algorithm falls within the class of interior-point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but moves through the interior of the feasible region, improving the approximation of the optimal solution by a definite fraction with every iteration and converging to an optimal solution with rational data.[2]

The algorithm

[edit]

Consider a linear programming problem in matrix form:

maximize cTx
subject to Ax ≤ b.

Karmarkar's algorithm determines the next feasible direction toward optimality and scales back by a factor 0 < γ ≤ 1. It is described in a number of sources.[3][4][5][6][7][8] Karmarkar also has extended the method[9][10][11][12] to solve problems with integer constraints and non-convex problems.[13] 

Algorithm Affine-Scaling

Since the actual algorithm is rather complicated, researchers looked for a more intuitive version of it, and in 1985 developed affine scaling, a version of Karmarkar's algorithm that uses affine transformations where Karmarkar used projective ones, only to realize four years later that they had rediscovered an algorithm published by Soviet mathematician I. I. Dikin in 1967.[14] The affine-scaling method can be described succinctly as follows.[15] While applicable to small scale problems, it is not a polynomial time algorithm.[14] 

Input:  A, b, c, , stopping criterion, γ.


do while stopping criterion not satisfied
    
    
    
    
    if  then
        return unbounded
    end if
    
    
    
end do
  • "←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
  • "return" terminates the algorithm and outputs the following value.

Example

[edit]
Example solution

Consider the linear program That is, there are 2 variables and 11 constraints associated with varying values of . This figure shows each iteration of the algorithm as red circle points. The constraints are shown as blue lines.

Patent controversy

[edit]

At the time he invented the algorithm, Karmarkar was employed by IBM as a postdoctoral fellow in the IBM San Jose Research Laboratory in California. On August 11, 1983 he gave a seminar at Stanford University explaining the algorithm, with his affiliation still listed as IBM. By the fall of 1983 Karmarkar started to work at AT&T and submitted his paper to the 1984 ACM Symposium on Theory of Computing (STOC, held April 30 - May 2, 1984) stating AT&T Bell Laboratories as his affiliation.[16] After applying the algorithm to optimizing AT&T's telephone network,[17] they realized that his invention could be of practical importance. In April 1985, AT&T promptly applied for a patent on his algorithm.

The patent became more fuel for the ongoing controversy over the issue of software patents.[18] This left many mathematicians uneasy, such as Ronald Rivest (himself one of the holders of the patent on the RSA algorithm), who expressed the opinion that research proceeded on the basis that algorithms should be free. Even before the patent was actually granted, it was argued that there might have been prior art that was applicable.[19] Mathematicians who specialized in numerical analysis, including Philip Gill and others, claimed that Karmarkar's algorithm is equivalent to a projected Newton barrier method with a logarithmic barrier function, if the parameters are chosen suitably.[20] Legal scholar Andrew Chin opines that Gill's argument was flawed, insofar as the method they describe does not constitute an "algorithm", since it requires choices of parameters that don't follow from the internal logic of the method, but rely on external guidance, essentially from Karmarkar's algorithm.[21] Furthermore, Karmarkar's contributions are considered far from obvious in light of all prior work, including Fiacco-McCormick, Gill and others cited by Saltzman.[21][22][23] The patent was granted in recognition of the essential originality of Karmarkar's work, as U.S. patent 4,744,028: "Methods and apparatus for efficient resource allocation" in May 1988.

AT&T designed a vector multi-processor computer system specifically to run Karmarkar's algorithm, calling the resulting combination of hardware and software KORBX,[24] and marketed this system at a price of US$8.9 million.[25][26] Its first customer was the Pentagon.[27][28]

Opponents of software patents have further argued that the patents ruined the positive interaction cycles that previously characterized the relationship between researchers in linear programming and industry, and specifically it isolated Karmarkar himself from the network of mathematical researchers in his field.[29]

The patent itself expired in April 2006, and the algorithm is presently in the public domain.

The United States Supreme Court has held that mathematics cannot be patented in Gottschalk v. Benson,[30] In that case, the Court first addressed whether computer algorithms could be patented and it held that they could not because the patent system does not protect ideas and similar abstractions. In Diamond v. Diehr,[31] the Supreme Court stated, "A mathematical formula as such is not accorded the protection of our patent laws, and this principle cannot be circumvented by attempting to limit the use of the formula to a particular technological environment.[32] In Mayo Collaborative Services v. Prometheus Labs., Inc.,[33] the Supreme Court explained further that "simply implementing a mathematical principle on a physical machine, namely a computer, [i]s not a patentable application of that principle."[34]

Applications

[edit]

Karmarkar's algorithm was used by the US Army for logistic planning during the Gulf War.[1]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Karmarkar's algorithm

View on Grokipedia
from Grokipedia
Karmarkar's algorithm is an for solving problems, developed by mathematician at Bell Laboratories and published in 1984, which achieves a proven polynomial-time complexity of O(n3.5 L) arithmetic operations, where n denotes the problem dimension and L the input precision in bits. The approach transforms the standard-form linear program—maximizing cTx subject to Ax = b and x ≥ 0—into a sequence of approximate optimization steps within a transformed , using projective rescaling to maintain strict interior feasibility and a logarithmic barrier potential to guide progress toward the optimum. This marked a pivotal theoretical and practical advance over prior polynomial-time methods like the ellipsoid algorithm, which, despite proving resolvability in polynomial time in 1979, suffered from poor constant factors rendering it inefficient for real-world use. The algorithm's core innovation lies in its potential-reduction framework, where each reduces a by a fixed factor (typically via parameter γ ≈ 0.99), ensuring convergence in O(L log(1/ε)) steps for ε-optimal solutions, combined with per- costs dominated by solving a Newtonian via Cholesky . Initial implementations and analyses suggested substantial speedups over the prevalent method for large-scale problems, prompting widespread interest and the evolution of interior-point techniques into cornerstone solvers in . While practical performance varied—often trailing revised on sparse instances due to denser linear algebra—its guaranteed worst-case efficiency underscored linear programming's membership in P and influenced broader paradigms.

Historical Context

Pre-Karmarkar Linear Programming

The simplex algorithm, developed by George Dantzig in 1947, served as the primary practical method for solving linear programming problems for nearly four decades. Originating from Dantzig's work on resource allocation for the U.S. Air Force during and after World War II, the algorithm iteratively moves along the edges of the feasible polytope toward an optimal vertex, leveraging a tableau representation to perform pivot operations. In practice, it exhibited average-case polynomial performance and enabled solutions to problems with thousands of variables on early computers, finding widespread adoption in operations research for applications such as production planning and transportation scheduling. Despite its empirical efficiency, the simplex method lacked a proven polynomial-time worst-case guarantee, with counterexamples like the Klee-Minty hypercube demonstrating exponential iteration counts in contrived instances. This theoretical limitation spurred efforts in the 1970s to develop algorithms with rigorous polynomial complexity, amid growing computational demands from industrial sectors. Linear programming models were increasingly applied to optimize logistics networks, supply chain distribution, and early telecommunications routing, where problem sizes often exceeded the reliable performance bounds of simplex implementations on available hardware. These applications highlighted the need for methods that could handle larger-scale, sparse constraint systems without degenerate pivoting delays. In 1979, introduced the , marking the first algorithm proven to solve in time, specifically with O(n²L) iterations where n is the number of variables and L relates to the input size in bits. The method constructs a sequence of shrinking ellipsoids containing the , using subgradient information from violated constraints to update the bounding set until convergence to an optimal point. However, its high constant factors, sensitivity to numerical precision due to ellipsoid volume computations, and dependence on a deep degree rendered it unsuitable for practical large-scale problems, outperforming only in very small or specially structured cases. Thus, remained the dominant solver, underscoring a gap between theoretical advances and deployable efficiency for real-world optimization in fields like and .

Development at Bell Labs

Narendra Karmarkar, an Indian mathematician who joined Bell Laboratories in 1983, developed the algorithm while employed there as part of the organization's mathematical research efforts. The work was carried out internally at the labs in , focusing on advancing computational methods for optimization problems relevant to telecommunications planning and . In late 1984, publicly announced the algorithm through press releases and media coverage, emphasizing its potential for dramatically accelerating solutions to large instances. The company claimed it achieved up to a 50-fold on benchmark test problems compared to the prevailing simplex method, particularly for problems with thousands of variables and constraints previously deemed computationally prohibitive. Karmarkar's initial formal , titled "A new polynomial-time for ," appeared in the December issue of Combinatorica, providing a rigorous proof of its polynomial-time complexity bounded by O(n3.5 L), where n denotes the number of variables and L the total of the input data. Empirical validations accompanying the announcement demonstrated the method's efficacy on substantial real-world-scale problems, such as those involving over 10,000 constraints, which the could not resolve in feasible timeframes on contemporary hardware.

Algorithm Fundamentals

Core Principles

Karmarkar's algorithm represents a paradigm shift in linear programming solvers by employing interior-point methods, which navigate through the interior of the feasible polytope rather than traversing its boundary vertices as in the simplex method. The simplex algorithm, developed by George Dantzig in 1947, pivots from one basic feasible solution (vertex) to adjacent vertices, potentially requiring exponentially many steps in the worst case for problems with nn variables and mm constraints. In contrast, Karmarkar's approach, introduced in 1984, initiates from a strictly feasible interior point and follows a trajectory that asymptotically approaches the optimal face without touching the boundary until convergence, enabling polynomial-time complexity guarantees. Central to the method is the projective scaling transformation, which homogenizes the and maps it into a standard unit , facilitating symmetric treatment of constraints and promoting movement toward the "center" of the transformed space. This transformation, applied iteratively, distorts the geometry to recenter the current iterate near the analytic center of the , allowing a deep feasible step that reduces while maintaining strict feasibility. The projective framework draws from geometric insights into polytopes, ensuring that each iteration scales variables inversely to residual slacks, akin to a barrier against constraint violations without explicit penalty terms. Progress is rigorously controlled via a primal potential function, typically of logarithmic form ϕ(x)=−∑i=1mlog⁡vik+ρ∥c∥Tx\phi(x) = - \sum_{i=1}^m \log v_i^k + \rho \|c\|^T x, where vkv^k denotes slack residuals and ρ\rho is a large parameter scaling the objective. Each iteration guarantees a constant fractional reduction in this potential—approximately 0.02 to 0.2 depending on step size α\alpha (often set near 0.25)—leading to logarithmic convergence in the number of steps, bounded by O(nLlog⁡(1/ϵ))O(n L \log(1/\epsilon)) to achieve ϵ\epsilon-optimality, with LL the bit length of input data. This potential embodies barrier-like principles, penalizing proximity to boundaries through logarithmic terms that grow unbounded as slacks approach zero, and its consistent decrease provides a Lyapunov-style measure of global convergence independent of local geometry. The algorithm's foundations prefigure broader interior-point theory, including self-concordant barrier functions for , where the negative log-barrier induces a central path of Newton steps tracing minimizers of perturbed objectives. Karmarkar's projective variant approximates this path-following via scaled Newton directions in transformed coordinates, establishing causal links between interior geometry, potential reduction, and polynomial solvability that underpin subsequent primal-dual and barrier methods.

Projective Scaling and Potential Reduction

Projective rescaling in Karmarkar's algorithm applies a nonlinear transformation to the current feasible point, mapping it to the barycenter of the transformed —typically the uniform point (1/n,…,1/n)(1/n, \dots, 1/n) in a homogenized space where variables sum to unity—thus centering the iterate and symmetrizing the constraints geometrically. This step exploits to rescale the problem instance, ensuring subsequent computations occur relative to a canonical position that equalizes variable scales and facilitates isotropic analysis of the . The algorithm then minimizes a logarithmic barrier potential function, ϕ(x)=nlog⁡(cTx)−∑j=1nlog⁡xj\phi(x) = n \log(c^T x) - \sum_{j=1}^n \log x_j, which encodes objective progress via the first term while the barrier term −∑log⁡xj-\sum \log x_j enforces strict interiority by diverging as any xjx_j approaches zero. In the rescaled space, a Newton direction is computed as the projected minimizing this potential subject to linearized constraints, approximating the central path and yielding a search direction that reduces while respecting feasibility. A , often around 0.1 to 0.5, scales the step to guarantee a fixed relative decrease in ϕ\phi, such as at least 0.25 per in analyzed variants. This interior-point strategy causally evades degeneracy by maintaining iterates away from boundaries via the barrier's penalization, preventing the algorithm from stalling on low-dimensional faces or requiring pivots through ill-conditioned vertices as in boundary-following methods; instead, the smooth on the potential traces a approximating the analytic , inherently robust to constraint rank deficiencies. Iteration complexity derives from the potential's bounded decrease: starting from an ϕ(x0)\phi(x_0) scaling with input precision LL (the of coefficients), and terminating when cTxk<2−2LcTx0c^T x_k < 2^{-2L} c^T x_0, yields at most O(nL)O(n L) steps, where the factor nn arises from the potential's dimension dependence and LL captures the problem's condition number through logarithmic precision requirements.

Detailed Procedure

Mathematical Formulation

Karmarkar's algorithm solves linear programming problems, typically formulated in inequality standard form as maximizing cTx\mathbf{c}^T \mathbf{x} subject to Ax≤bA \mathbf{x} \leq \mathbf{b} and x≥0\mathbf{x} \geq \mathbf{0}, where AA is an m×nm \times n matrix with m<nm < n, c∈Rn\mathbf{c} \in \mathbb{R}^n, and b∈R+m\mathbf{b} \in \mathbb{R}^m_+. To convert inequalities to equalities, nonnegative slack variables s≥0\mathbf{s} \geq \mathbf{0} are introduced, yielding Ax+Is=bA \mathbf{x} + I \mathbf{s} = \mathbf{b} with x,s≥0\mathbf{x}, \mathbf{s} \geq \mathbf{0}. The algorithm requires transforming the problem into a canonical form suitable for projective transformations, where the feasible region lies within the standard simplex {y≥0∣eTy=1}\{\mathbf{y} \geq \mathbf{0} \mid \mathbf{e}^T \mathbf{y} = 1\}, with e\mathbf{e} the all-ones vector. This is achieved through homogenization by adjoining a scaling variable t≥0t \geq 0: the original problem is equivalent to maximizing tt subject to A(x/t)≤b/tA (\mathbf{x}/t) \leq \mathbf{b}/t, eT(x/t)=1\mathbf{e}^T (\mathbf{x}/t) = 1, and x/t≥0\mathbf{x}/t \geq \mathbf{0}, assuming the original optimum is bounded. Scaling ensures the right-hand side becomes the all-ones vector, embedding the scaled feasible set {z≥0∣A′z≤1,eTz=1}\{\mathbf{z} \geq \mathbf{0} \mid A' \mathbf{z} \leq \mathbf{1}, \mathbf{e}^T \mathbf{z} = 1\} inside the simplex, where A′A' incorporates the adjusted constraints. Theoretical guarantees rely on strict feasibility, meaning there exists x0>0\mathbf{x}^0 > \mathbf{0} such that Ax0<bA \mathbf{x}^0 < \mathbf{b} in the original form (or equivalently A′z0<1A' \mathbf{z}^0 < \mathbf{1}, z0>0\mathbf{z}^0 > \mathbf{0}, eTz0=1\mathbf{e}^T \mathbf{z}^0 = 1 in the ), ensuring the relative interior of the is nonempty and allowing barrier-like potential functions to drive convergence. Boundedness of the optimal objective value is also assumed, preventing unbounded rays that could hinder polynomial-time termination. Affine scaling methods, which linearly transform the to center iterates at the origin while preserving affine structure, conceptually preceded projective approaches by emphasizing interior-point trajectories but differ in using linear rather than nonlinear (projective) maps.

Iterative Steps

The iterative procedure of Karmarkar's algorithm requires an initial strictly feasible point x0≥0x^0 \geq 0 such that Ax0<bAx^0 < b, ensuring positive slacks v0=b−Ax0>0v^0 = b - Ax^0 > 0, typically found via a preliminary phase I solver or of the problem instance. This point is mapped via a projective transformation to the interior of a standard {z∣eTz=1,z>0}\{z \mid e^T z = 1, z > 0\}, with the image under transformation centered at e/n=(1/n,…,1/n)e/n = (1/n, \dots, 1/n). The iteration index is initialized as k←0k \leftarrow 0. Each iteration computes an improving direction within the transformed space using a scaled Newton-like step that approximates the steepest descent for reducing a logarithmic potential function Φ(x)=−∑log⁡vi(x)−Rlog⁡(cTx+δ)\Phi(x) = -\sum \log v_i(x) - R \log(c^T x + \delta), where R>nR > n controls the trade-off and δ\delta offsets for boundedness. Specifically, the residual slacks are updated as vk←b−Axkv^k \leftarrow b - A x^k, followed by forming the diagonal scaling matrix Dv←\diag(v1k,…,vmk)D_v \leftarrow \diag(v_1^k, \dots, v_m^k). The primal direction solves the positive definite system hx←(ATDv−2A)−1ch_x \leftarrow (A^T D_v^{-2} A)^{-1} c for maximizing local objective progress under the metric induced by the barrier Hessian Dv−2D_v^{-2}. The corresponding slack direction is then hv←−Ahxh_v \leftarrow -A h_x. If hv≥0h_v \geq 0 componentwise, no improving feasible direction exists, indicating optimality. Otherwise, a line search determines the maximum step preserving interiority: α←γ⋅min⁡{−vik/(hv)i∣(hv)i<0, i=1,…,m}\alpha \leftarrow \gamma \cdot \min \{ -v_i^k / (h_v)_i \mid (h_v)_i < 0, \, i=1,\dots,m \}, with damping factor γ≈0.1\gamma \approx 0.1 to 0.250.25 ensuring potential reduction by at least 1−1/(2R)1 - 1/(2R) per step and avoiding numerical issues. The update follows as xk+1←xk+αhxx^{k+1} \leftarrow x^k + \alpha h_x, after which the point is remapped via the inverse projective transformation T−1(z)=Dz/(eTDz)T^{-1}(z) = D z / (e^T D z) with D=\diag(z)D = \diag(z), and k←k+1k \leftarrow k+1. Termination occurs when the computed α\alpha exceeds a large threshold (indicating near-optimality), the duality gap cTxk+bTyk−(Axk)Tykc^T x^k + b^T y^k - (A x^k)^T y^k falls below a precision ϵ>0\epsilon > 0, or potential stagnation is detected after O(nlog⁡(1/ϵ))O(n \log(1/\epsilon)) steps. In implementations, heuristics such as row/column equilibration of AA before solving the direction system, with rank-one updates for , or adaptive γ\gamma based on trial steps enhance stability against ill-conditioning from disparate slack magnitudes.

Illustrative Example

Consider the problem of maximizing z=3x1+2x2z = 3x_1 + 2x_2 subject to 2x1+x2≤1002x_1 + x_2 \leq 100, x1+x2≤80x_1 + x_2 \leq 80, x1≤40x_1 \leq 40, and x1,x2≥0x_1, x_2 \geq 0. This forms a bounded in the first quadrant, with vertices including (0,0), (40,0), (40,20), (20,60), and (0,80). Karmarkar's algorithm transforms the problem into a standard form with non-negative variables summing to a constant under linear equalities, starting from an interior point like the normalized all-ones vector scaled appropriately. In each iteration, a projective transformation centers the current point at the origin of the transformed space, preserving the feasible set's structure. The search direction is then found by minimizing a quadratic approximation of the objective over an inscribed within the transformed , yielding a descent direction hh that reduces the Karmarkar potential function Φ(x)=∑log⁡xi−μlog⁡(−∇f(x))\Phi(x) = \sum \log x_i - \mu \log(-\nabla f(x)), where μ\mu controls barrier strength. The step length α\alpha is selected to keep the new point interior, typically α=1R+1\alpha = \frac{1}{R+1} where RR is the transformed polytope's "radius," ensuring a constant potential decrease per . Updating xk+1=xk+αhx^{k+1} = x^k + \alpha h, the process repeats, converging in O(nLlog⁡(1/ϵ))O(n L \log(1/\epsilon)) iterations to an ϵ\epsilon-optimal solution, where nn is variables, mm constraints, and LL bit length. For the example problem, after slack variable introduction and transformation, iterations move interior points toward the optimum at approximately (20,60), improving the objective from an initial ~150 to near 200.

Performance Analysis

Theoretical Complexity

Karmarkar's algorithm solves problems in time, with a total arithmetic of O(n3.5[L](/page/L′))O(n^{3.5} [L](/page/L')), where [n](/page/N+)[n](/page/N+) is the number of variables and [L](/page/L′)[L](/page/L') is the total encoding the input coefficients, right-hand sides, and objective vector. This bound arises from O(nL)O(n L) iterations, each requiring O(n2.5L)O(n^{2.5} L) operations primarily for computing the Newton search direction via of the projected ATDv−2AA^T D_v^{-2} A, where DvD_v is a diagonal scaling matrix derived from the current residual. The iteration count follows from the potential reduction mechanism, which employs a projective potential function Φ(x)=−∑i=1mlog⁡vik+nlog⁡(n+1)\Phi(x) = - \sum_{i=1}^m \log v_i^k + n \log(n+1), decreasing by at least a fixed δ≈0.02\delta \approx 0.02 (or improved to ≈0.25\approx 0.25 with α=0.5\alpha = 0.5) per step through affine scaling and a damped Newton update with step length αγ\alpha \gamma, where γ<1/4\gamma < 1/4 ensures descent while staying within the feasible region. Starting from a scaled interior point with initial potential O(nlog⁡R)O(n \log R), where RR bounds the polytope size, the algorithm requires O(nlog⁡(R/ϵ))O(n \log(R / \epsilon)) steps to drive the duality gap below ϵ\epsilon, with log⁡(R/ϵ)=O(L)\log(R / \epsilon) = O(L) under standard input assumptions that coefficients and bounds are polynomially representable. The per-iteration cost depends on problem data through LL, which captures the Lipschitz-like conditioning via bit precision for matrix inversions and eigenvalue bounds in the projective , necessitating higher arithmetic precision to avoid numerical in the barrier term. While the framework leverages the geometry of the standard for bounded , akin to a self-concordant function with domain scaling as O(n)O(n), the resulting linear dependence on nn in iterations yields larger constants than later path-following interior-point methods achieving O(nlog⁡(1/ϵ))O(\sqrt{n} \log(1/\epsilon))
Add your contribution
Related Hubs
User Avatar
No comments yet.