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A theorem in convex programming

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  • William Karush

Abstract

An optimization problem which frequently arises in applications of mathematical programming is the following: Where fi are convex functions. In this paper, the function F is studied and shown to satisfy F(A, B) = M (A) + N(B), where M and N are increasing and decreasing convex functions, respectively. Also, the functional equation F (A, C) = F(A, B) + F(B, C) − F(B, B) is established. These results generalize to the continuous case F(A, B)=min ∫ OT f(t, x(t))dt, with x(t) increasing and A ≤ x (0) ≤ x (T) ≤ B. The results obtained in this paper are useful for reducing an optimization problem with many variables to one with fewer variables.

Suggested Citation

  • William Karush, 1959. "A theorem in convex programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 6(3), pages 245-260, September.
    • Handle: RePEc:wly:navlog:v:6:y:1959:i:3:p:245-260
    DOI: 10.1002/nav.3800060306
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    Cited by:

    1. Rossi, Roberto & Chen, Zhen & Tarim, S. Armagan, 2024. "On the stochastic inventory problem under order capacity constraints," European Journal of Operational Research, Elsevier, vol. 312(2), pages 541-555.
    2. Tong Wang & Xiting Gong & Sean X. Zhou, 2017. "Dynamic Inventory Management with Total Minimum Order Commitments and Two Supply Options," Operations Research, INFORMS, vol. 65(5), pages 1285-1302, October.

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