# Conic Linear Programming

@article{Luenberger2021ConicLP, title={Conic Linear Programming}, author={David G. Luenberger and Yinyu Ye}, journal={Linear and Nonlinear Programming}, year={2021} }

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…

## 6 Citations

Two Optimal Value Functions in Parametric Conic Linear Programming

- MathematicsJ. Optim. Theory Appl.
- 2022

We consider the conic linear program given by a closed convex cone in an Euclidean space and a matrix, where vector on the right-hand side of the inequality constraint and the vector defining the…

A New Use of Douglas-Rachford Splitting and ADMM for Identifying Infeasible, Unbounded, and Pathological Conic Programs

- Computer ScienceArXiv
- 2017

A method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM, which is simple to implement and has low per-iteration cost.

Light robustness in the optimization of Markov decision processes with uncertain parameters

- Computer ScienceComput. Oper. Res.
- 2019

On ambiguity function shaping for broadband constant-modulus signals

- Computer ScienceSignal Process.
- 2020

Adaptive Model Predictive Control for Cruise Control of High-Speed Trains with Time-Varying Parameters

- EngineeringJournal of Advanced Transportation
- 2019

It is proved theoretically that, with the proposed adaptive MPC, the high-speed trains track the desired speed with ultimately bounded tracking errors, while the estimated parameters are bounded and the relative spring displacement between the two neighboring cars is stable at the equilibrium state.

Practical measurement-device-independent entanglement quantification

- PhysicsPhysical Review A
- 2018

The robust estimation of entanglement is key to the validation of implementations of quantum systems. On the one hand, the evaluation of standard entanglement measures, either using quantum…

## References

SHOWING 1-10 OF 368 REFERENCES

Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

- MathematicsSIAM J. Optim.
- 1995

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

- Mathematics
- 1999

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

- Computer ScienceMath. Program.
- 2003

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

- MathematicsMath. Oper. Res.
- 2003

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

- Computer Science
- 1998

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

- Computer Science
- 1998

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

- Mathematics
- 2002

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

- Computer ScienceSTOC '01
- 2001

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

- Mathematics
- 1999

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.