Cutting plane methods provide the means to solve large scale semidefinite programs (SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise from various perspectives, and include techniques based on interior point cutting plane approaches, non-differentiable optimization, and finally an approach which mimics the simplex method for linear programming (LP).We present an accessible introduction to various cutting plane approaches that have appeared in the literature. We place these methods in a unifying framework which illustrates how each approach arises as a natural enhancement of a primordial LP cutting plane scheme based on a semi-infinite formulation of the SDP.

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Abstract. We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of the algorithm. Cutting planes based on the polyhedral theory of the maxcut problem are then added to the primal problem in order to improve the SDP relaxation; this is the cutting phase of the algorithm. We provide computational results, and compare these results with a standard SDP cutting plane scheme.

Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primal-dual interior-point methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to non-convex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include non-convex potential reduction methods, interior point cutting plane methods, primal-dual IPMs and first-order algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.