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We present a global optimization approach for solving the maximuma-posteriori (MAP) clustering problem under the Gaussian mixture model.Ourapproach can accommodate side constraints and it preserves the combinatorialstructure of the MAP clustering problem by formulating it asa mixed-integernonlinear optimization problem (MINLP). We approximate the MINLP through amixed-integer quadratic program (MIQP) transformation that improvescomputational aspects while guaranteeing $\epsilon$-global optimality. Animportant benefit of our approach is the explicit quantification of the degreeof suboptimality, via the optimality gap, en route to finding the globallyoptimal MAP clustering. Numerical experiments comparing our method to otherapproaches show that our method finds a better solution than standardclustering methods. Finally, we cluster a real breast cancer gene expressiondata set incorporating intrinsic subtype information; the induced constraintssubstantially improve the computational performance and produce more coherentand bio-logically meaningful clusters.

MAP Clustering under the Gaussian Mixture Model via Mixed Integer Nonlinear Optimization