Abstract
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)/Πni=1xiK/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) |
|---|---|
| Pages (from-to) | 93-98 |
| Number of pages | 6 |
| Journal | Operations Research Letters |
| Volume | 11 |
| Issue number | 2 |
| DOIs | |
| State | Published - Mar 1992 |
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All Science Journal Classification (ASJC) codes
- Software
- Management Science and Operations Research
- Industrial and Manufacturing Engineering
- Applied Mathematics
Keywords
- Karmarkar's algorithm
- homogeneous functions
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Generalization of Karmarkar's algorithm to convex homogeneous functions. / Kalantari, Bahman.
In: Operations Research Letters, Vol. 11, No. 2, 03.1992, p. 93-98.Research output: Contribution to journal › Article
TY - JOUR
T1 - Generalization of Karmarkar's algorithm to convex homogeneous functions
AU - Kalantari, Bahman
PY - 1992/3
Y1 - 1992/3
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)/Πni=1xiK/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)/Πni=1xiK/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.
KW - Karmarkar's algorithm
KW - homogeneous functions
UR - http://www.scopus.com/inward/record.url?scp=0347974316&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0347974316&partnerID=8YFLogxK
U2 - 10.1016/0167-6377(92)90039-6
DO - 10.1016/0167-6377(92)90039-6
M3 - Article
AN - SCOPUS:0347974316
VL - 11
SP - 93
EP - 98
JO - Operations Research Letters
JF - Operations Research Letters
SN - 0167-6377
IS - 2
ER -