### Abstract

Let φ be a convex homogeneous function of degree K > 0 defined over the positive points of a subspace W of R^{n}, n ≥ 2. Assume φ(x) > 0 for some 0 < ε{lunate} W. Let V be the set of nonnegative nonzero x ε{lunate} W such that φ(x)^{k} converges to zero, for some sequence 0 < x ^{k} ε{lunate} W converging to x. We prove V is nonempty if and only if for every 0 < ε{lunate} W satisfying φ(d) > 0, there exists 0 < x_{d} ε{lunate} W such that e^{T}d = e^{T}x_{d} and f(x_{d}) ≤ γf(d), where e is the vector of ones, f(x) = φ(x)/Π^{n}
_{i=1xi}
^{K/n})), and γ = 〈[K + 1)/K]^{K} exp(-1)〉^{1/n}. Karmarkar's algorithm proves the above for the special case where φ is linear.

Original language | English (US) |
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Pages (from-to) | 93-98 |

Number of pages | 6 |

Journal | Operations Research Letters |

Volume | 11 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1992 |

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### All Science Journal Classification (ASJC) codes

- Management Science and Operations Research
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics
- Modeling and Simulation

### Cite this

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*Operations Research Letters*, vol. 11, no. 2, pp. 93-98. https://doi.org/10.1016/0167-6377(92)90039-6

**Generalization of Karmarkar's algorithm to convex homogeneous functions.** / Kalantari, Bahman.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalization of Karmarkar's algorithm to convex homogeneous functions

AU - Kalantari, Bahman

PY - 1992/1/1

Y1 - 1992/1/1

N2 - Let φ be a convex homogeneous function of degree K > 0 defined over the positive points of a subspace W of Rn, n ≥ 2. Assume φ(x) > 0 for some 0 < ε{lunate} W. Let V be the set of nonnegative nonzero x ε{lunate} W such that φ(x)k converges to zero, for some sequence 0 < x k ε{lunate} W converging to x. We prove V is nonempty if and only if for every 0 < ε{lunate} W satisfying φ(d) > 0, there exists 0 < xd ε{lunate} W such that eTd = eTxd and f(xd) ≤ γf(d), where e is the vector of ones, f(x) = φ(x)/Πn i=1xi K/n)), and γ = 〈[K + 1)/K]K exp(-1)〉1/n. Karmarkar's algorithm proves the above for the special case where φ is linear.

AB - Let φ be a convex homogeneous function of degree K > 0 defined over the positive points of a subspace W of Rn, n ≥ 2. Assume φ(x) > 0 for some 0 < ε{lunate} W. Let V be the set of nonnegative nonzero x ε{lunate} W such that φ(x)k converges to zero, for some sequence 0 < x k ε{lunate} W converging to x. We prove V is nonempty if and only if for every 0 < ε{lunate} W satisfying φ(d) > 0, there exists 0 < xd ε{lunate} W such that eTd = eTxd and f(xd) ≤ γf(d), where e is the vector of ones, f(x) = φ(x)/Πn i=1xi K/n)), and γ = 〈[K + 1)/K]K exp(-1)〉1/n. Karmarkar's algorithm proves the above for the special case where φ is linear.

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U2 - 10.1016/0167-6377(92)90039-6

DO - 10.1016/0167-6377(92)90039-6

M3 - Article

VL - 11

SP - 93

EP - 98

JO - Operations Research Letters

JF - Operations Research Letters

SN - 0167-6377

IS - 2

ER -