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**Second-order cone programming.**
*(English)*
Zbl 1153.90522

From the introduction: 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 range of
fields from engineering, control and finance to robust optimization and combinatorial optimization.

On the other hand semidefinite programming (SDP) – that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices – includes SOCP as a special case. Therefore, SOCP falls between linear (LP) and quadratic (QP) programming and SDP. Like LP, QP and SDP problems, SOCP problems can be solved in polynomial time by interior point methods. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP problems but less than that required to solve SDP’s of similar size and structure. Because the set of feasible solutions for an SOCP problem is not polyhedral as it is for LP and QP problems, it is not readily apparent how to develop a simplex or simplex-like method for SOCP.

While SOCP problems can be solved as SDP problems, doing so is not advisable both on numerical grounds and computational complexity concerns. For instance, many of the problems presented in the survey paper of L. Vandenberghe and S. Boyd [SIAM Rev. 38, No. 1, 49–95 (1996; Zbl 0845.65023)] as examples of SDPs can in fact be formulated as SOCPs and should be solved as such.

In §2, 3 below we give SOCP formulations for four of these examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadratic functions such as those that arise in structural optimization, logarithmic Chebyshev approximation and the problem of finding the smallest ball containing a given set of ellipsoids. Thus because of its broad applicability and its computational tractability, SOCP deserves to be studied in its own right.

On the other hand semidefinite programming (SDP) – that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices – includes SOCP as a special case. Therefore, SOCP falls between linear (LP) and quadratic (QP) programming and SDP. Like LP, QP and SDP problems, SOCP problems can be solved in polynomial time by interior point methods. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP problems but less than that required to solve SDP’s of similar size and structure. Because the set of feasible solutions for an SOCP problem is not polyhedral as it is for LP and QP problems, it is not readily apparent how to develop a simplex or simplex-like method for SOCP.

While SOCP problems can be solved as SDP problems, doing so is not advisable both on numerical grounds and computational complexity concerns. For instance, many of the problems presented in the survey paper of L. Vandenberghe and S. Boyd [SIAM Rev. 38, No. 1, 49–95 (1996; Zbl 0845.65023)] as examples of SDPs can in fact be formulated as SOCPs and should be solved as such.

In §2, 3 below we give SOCP formulations for four of these examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadratic functions such as those that arise in structural optimization, logarithmic Chebyshev approximation and the problem of finding the smallest ball containing a given set of ellipsoids. Thus because of its broad applicability and its computational tractability, SOCP deserves to be studied in its own right.

### MSC:

90C25 | Convex programming |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90C22 | Semidefinite programming |

90C51 | Interior-point methods |