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Semidefinite programs (SDPs) can be solved in polynomial time by interiorpoint methods. However, when the dimension of the problem gets large, interiorpoint methods become impractical in terms of both computational time and memoryrequirements. Certain first-order methods, such as Alternating DirectionMethods of Multipliers (ADMMs), established as suitable algorithms to deal withlarge-scale SDPs and gained growing attention over the past decade. In thispaper, we focus on an ADMM designed for SDPs in standard form and extend it todeal with inequalities when solving SDPs in general form. Beside numericalresults on randomly generated instances, where we show that our method comparesfavorably with respect to the state-of-the-art solver SDPNAL+, we presentresults on instances from SDP relaxations of classical combinatorial problemssuch as the graph coloring problem and the maximum clique problem. Throughextensive numerical experiments, we show that even an inaccurate dual solution,obtained at a generic iteration of our proposed ADMM, can represent anefficiently recovered valid bound on the optimal solution of the combinatorialproblems considered, as long as an appropriate post-processing procedure isapplied.

Dealing with inequality constraints in large-scale semidefinite relaxations for graph coloring and maximum clique problems