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) |
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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 - Mar 1992 |
All Science Journal Classification (ASJC) codes
- Software
- Management Science and Operations Research
- Industrial and Manufacturing Engineering
- Applied Mathematics
Keywords
- Karmarkar's algorithm
- homogeneous functions