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Conic Linear Programming

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Linear and Nonlinear Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 228))

Abstract

Conic Linear Programming, hereafter CLP, is a natural extension of Linear programming (LP). In LP, the variables form a vector which is required to be componentwise nonnegative, while in CLP they are points in a pointed convex cone (see Appendix B.1) of an Euclidean space, such as vectors as well as matrices of finite dimensions. For example, Semidefinite programming (SDP) is a kind of CLP, where the variable points are symmetric matrices constrained to be positive semidefinite. Both types of problems may have linear equality constraints as well. Although CLPs have long been known to be convex optimization problems, no efficient solution algorithm was known until about two decades ago, when it was discovered that interior-point algorithms for LP discussed in Chap. 5, can be adapted to solve certain CLPs with both theoretical and practical efficiency. During the same period, it was discovered that CLP, especially SDP, is representative of a wide assortment of applications, including combinatorial optimization, statistical computation, robust optimization, Euclidean distance geometry, quantum computing, optimal control, etc. CLP is now widely recognized as a powerful mathematical computation model of general importance.

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Authors and Affiliations

  1. Department of Management Science and Engineering, Stanford University, Stanford, CA, USA

    David G. Luenberger & Yinyu Ye

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  1. David G. Luenberger
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  2. Yinyu Ye
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Luenberger, D.G., Ye, Y. (2016). Conic Linear Programming. In: Linear and Nonlinear Programming. International Series in Operations Research & Management Science, vol 228. Springer, Cham. https://doi.org/10.1007/978-3-319-18842-3_6

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