Getting Feasible Variable Estimates From Infeasible Ones: MRF Local Polytope Study

This paper proposes a method for construction of approximate feasible primal solutions from dual ones for large-scale optimization problems possessing certain separability properties. Whereas infeasible primal estimates can typically be produced from (sub-)gradients of the dual function, it is often not easy to project them to the primal feasible set, since the projection itself has a complexity comparable to the complexity of the initial problem. We propose an alternative efficient method to obtain feasibility and show that its properties influencing the convergence to the optimum are similar to the properties of the Euclidean projection. We apply our method to the local polytope relaxation of inference problems for Markov Random Fields and demonstrate its superiority over existing methods.

💡 Research Summary

The paper addresses a fundamental obstacle in large‑scale convex optimization: converting infeasible primal estimates—readily obtained from sub‑gradients of the dual function—into feasible primal solutions without incurring the prohibitive cost of a full Euclidean projection. While a global projection onto the primal feasible set guarantees the smallest Euclidean distance to the infeasible estimate, its computational complexity is essentially the same as solving the original problem, rendering it impractical for high‑dimensional applications such as graphical model inference.

To overcome this, the authors introduce a localized projection framework that exploits separability in the constraint structure. The key idea is to decompose the global feasible set (\mathcal{P}) into a union of small, overlapping local polytopes ({\mathcal{P}_{C_k}}), each associated with a subset (or “cluster”) of variables. For each cluster (C_k) the algorithm solves a tiny Euclidean‑distance minimization problem

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