# Second-order cone programming

@article{Alizadeh2003SecondorderCP, title={Second-order cone programming}, author={Farid Alizadeh and Donald Goldfarb}, journal={Mathematical Programming}, year={2003}, volume={95}, pages={3-51} }

Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems, as can many other problems that do not fall into these three categories. These latter problems model applications from a broad…

## 1,407 Citations

Interior-point methods for large-scale cone programming

- Computer Science
- 2011

In this chapter, an overview of ad hoc techniques that can be used to exploit non-sparse structure in specific classes of applications are given.

Second-order Cone Programming (SOCP)

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- 2019

NAG introduces a new SOCP solver at Mark 27 of the NAG Library based on interior point method (IPM), which can be highly utilized in a broad range of fields including finance, engineering and control.

Quadratic optimization over one first-order cone

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- 2013

This paper studies the first-order cone constrained homogeneous quadratic programming problem.
For efficient computation, the problem is reformulated as a linear conic programming problem. A union…

Applications of Second Order Cone Programming

- Computer Science
- 2012

A significant special case of the problems which could be solved were those whose constraints were given by semidefinite cones, and these have a wide range of applications, some of which are discussed in Section 5, and can still be solved efficiently using interior point methods.

An unconstrained smooth minimization reformulation of the second-order cone complementarity problem

- MathematicsMath. Program.
- 2005

This paper extends this merit function and its analysis, including continuous differentiability, to the second-order cone complementarity problem (SOCCP).

Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming

- Computer Science, MathematicsArXiv
- 2018

This thesis provides the first theoretical framework for constructing converging hierarchies of lower bounds on POPs whose computation simply requires the ability to multiply certain fixed polynomials together and to check nonnegativity of the coefficients of their product.

Convex relaxations in nonconvex and applied optimization

- Computer Science
- 2010

It is proved that the sequential relaxation technique generates the convex hull of 0-1 solutions asymptotically and is equivalent to an existing SDP relaxation at the root node, and it is significantly stronger on the children nodes in a branch-and-bound setting.

The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones

- Mathematics
- 2014

The focus of this paper is on studying an inverse second-order cone quadratic programming problem, in which the parameters in the objective function need to be adjusted as little as possible so that…

A perturbation approach for an inverse linear second-order cone programming

- Mathematics
- 2012

A type of inverse linear second-order cone programming problems is
discussed, in which the parameters in both the objective function
and the constraint set of a given linear second-order cone …

Sum of Squares Basis Pursuit with Linear and Second Order Cone Programming

- Mathematics, Computer ScienceArXiv
- 2015

A scheme for solving an iterative sequence of linear programs (LPs) or second order cone programs (SOCPs) to approximate the optimal value of any semidefinite program (SDP) or sum of squares (SOS) program is devised and applied to two problems of discrete optimization.

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