Smooth and Flexible Dual Optimal Inequalities

We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOIs). We study two novel classes of DOIs, which are referred to as Flexible DOIs (F-DOIs) and Smooth-DOIs (S-DOIs), respectively (and jointly as SF-DOIs). F-DOIs provide rebates for covering items more than necessary. S-DOIs describe the payment of a penalty to permit the undercoverage of items in exchange for the overinclusion of other items. Unlike other classes of DOIs from the literature, the S-DOIs and F-DOIs rely on very little problem-specific knowledge and, as such, have the potential to be applied to a vast number of problem domains. In particular, we discuss the application of the new DOIs to three relevant problems: the single-source capacitated facility location problem, the capacitated p-median problem, and the capacitated vehicle-routing problem. We provide computational evidence of the strength of the new inequalities by embedding them within a column-generation solver for these problems. Substantial speedups can be observed as when compared with a nonstabilized variant of the same CG procedure to achieve the linear-relaxation lower bound on problems with dense columns and structured assignment costs.

Optimization on Submanifolds of $\delta$-Lorentzian trans-Sasakian Manifolds with Casorati Curvatures

The present research article is concerned about a couple of optimal inequalities for submanifolds of $\delta$-Lorentzian trans-Sasakian manifolds endowed with semi-symmetric metric connection (briefly says $SSM$). Some examples of $\delta$-Lorentzian trans-Sasakiam manifolds are also discussed here. This paper ends with some open problems.

Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds

In [An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23(3) (2012)], B.-Y. Chen introduced the CR δ-invariant for CR-submanifolds. Then, in [Two optimal inequalities for anti-holomorphic submanifolds and their applications, Taiwan. J. Math. 18 (2014), 199–217], F. R. Al-Solamy, B.-Y. Chen and S. Deshmukh proved two optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR δ-invariant. In this paper, we obtain optimal inequalities for this invariant for contact CR-submanifolds in almost contact metric manifolds.

Optimal and Nonoptimal Gronwall Lemmas

In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. A. Rus. This paper is also motivated by the fact that, in many inequalities, the upper bound is not an optimal one.

Accelerating Column Generation via Flexible Dual Optimal Inequalities with Application to Entity Resolution

In this paper, we introduce a new optimization approach to Entity Resolution. Traditional approaches tackle entity resolution with hierarchical clustering, which does not benefit from a formal optimization formulation. In contrast, we model entity resolution as correlation-clustering, which we treat as a weighted set-packing problem and write as an integer linear program (ILP). In this case, sources in the input data correspond to elements and entities in output data correspond to sets/clusters. We tackle optimization of weighted set packing by relaxing integrality in our ILP formulation. The set of potential sets/clusters can not be explicitly enumerated, thus motivating optimization via column generation. In addition to the novel formulation, we also introduce new dual optimal inequalities (DOI), that we call flexible dual optimal inequalities, which tightly lower-bound dual variables during optimization and accelerate column generation. We apply our formulation to entity resolution (also called de-duplication of records), and achieve state-of-the-art accuracy on two popular benchmark datasets. Our F-DOI can be extended to other weighted set-packing problems.