Proximal Operator of Quotient Functions with Application to a Feasibility Problem in Query Optimization

Guido Moerkotte (1) , Martin Montag (2) , Audrey Repetti (3) , Gabriele Steidl (2)

1 Database Research Group [Mannheim]
2 University of Kaiserslautern [Kaiserslautern]
3 LIGM - Laboratoire d'Informatique Gaspard-Monge

In this paper we determine the proximity functions of the sum and the maximum of componentwise (reciprocal) quotients of positive vectors. For the sum of quotients, denoted by $Q_1$, the proximity function is just a componentwise shrinkage function which we call q-shrinkage. This is similar to the proximity function of the ℓ1-norm which is given by componentwise soft shrinkage. For the maximum of quotients $Q_∞$, the proximal function can be computed by first order primal dual methods involving epigraphical projections. The proximity functions of $Q_ν$ , $ν = 1,∞$ are applied to solve convex problems of the form $argmin_x Q _ν ( Ax/b )$ subject to $x ≥ 0$, $1^\top x ≤ 1$. Such problems are of interest in selectivity estimation for cost-based query optimizers in database management systems.

Domaines: Optimisation et contrôle [math.OC]

Guido Moerkotte, Martin Montag, Audrey Repetti, Gabriele Steidl. Proximal Operator of Quotient Functions with Application to a Feasibility Problem in Query Optimization. Journal of Computational and Applied Mathematics, 2015, 285, pp.243-255. ⟨10.1016/j.cam.2015.02.030⟩. ⟨hal-00942453v2⟩