Conic Linear Programming

@article{Luenberger2021ConicLP,
  title={Conic Linear Programming},
  author={David G. Luenberger and Yinyu Ye},
  journal={Linear and Nonlinear Programming},
  year={2021}
}
  • D. Luenberger, Y. Ye
  • Published 2021
  • Computer Science, Mathematics
  • Linear and Nonlinear Programming
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… 
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References

SHOWING 1-10 OF 368 REFERENCES
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
TLDR
It is argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion.
Solving Semide nite Programs via Nonlinear Programming
In a semidefinite programming (SDP) problem, a linear function of a symmetric matrix variable X is minimized subject to linear equality constraints on X and the essential constraint that X be
Second-order cone programming
TLDR
SOCP formulations are given for four examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadRatic functions, and many of the problems presented in the survey paper of Vandenberghe and Boyd as examples of SDPs can in fact be formulated as SOCPs and should be solved as such.
On Cones of Nonnegative Quadratic Functions
TLDR
It is shown that optimizing a general quadratic function over the intersection of an ellipsoid and a half-plane can be formulated as semidefinite programming (SDP), thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming.
Conic convex programming and self-dual embedding
TLDR
The self-dual embedding technique proposed by Ye, Todd and Mizuno is extended to solve general conic convex programming, including semidefinite programmng and numerous examples from semideFinite programming are provided to illustrate various possibilities which have no analogue in the linear programming case.
Semidefinite programming for assignment and partitioning problems
TLDR
An efficient "partial infeasible" primal-dual interior-point algorithm is developed by using a conjugate gradient method and by taking advantage of the special data structure of the SDP relaxation, which plays a significant role in simplifying the problem.
Geometric Optimization Problems Likely Not Contained in APX
Maximizing geometric functionals such as the classical lp-norms over poly- topes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
TLDR
An extension of semidefinite programming to complex space to solve the natural relaxation, and a natural extension of the random hyperplane technique introduced by the authors to obtain near-optimal solutions to the problems.
Solving Semidefinite Programs via Nonlinear Programming, Part II: Interior Point Methods for a Subclass of SDPs
TLDR
This paper develops interior point methods for solving a subclass of the transformable linear SDP problems where the diagonal of a matrix variable is given and global convergence of these methods is proved.
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