A unified framework of exact and heuristic methods for minimizing sums of pointwise minima of convex functions, including a compact big-M mixed-integer formulation and relaxed alternating minimization with convergence guarantees.
The Boosted DC Algorithm for Clustering with Constraints
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abstract
This paper aims to investigate the effectiveness of the recently proposed Boosted Difference of Convex functions Algorithm (BDCA) when applied to clustering with constraints and set clustering with constraints problems. This is the first paper to apply BDCA to a problem with nonlinear constraints. We present the mathematical basis for the BDCA and Difference of Convex functions Algorithm (DCA), along with a penalty method based on distance functions. We then develop algorithms for solving these problems and computationally implement them, with publicly available implementations. We compare old examples and provide new experiments to test the algorithms. We find that the BDCA method converges in fewer iterations than the corresponding DCA-based method. In addition, BDCA yields faster CPU running-times in all tested problems.
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Global and local approaches for the minimization of a sum of pointwise minima of convex functions
A unified framework of exact and heuristic methods for minimizing sums of pointwise minima of convex functions, including a compact big-M mixed-integer formulation and relaxed alternating minimization with convergence guarantees.