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In this paper we study lattice rules which are cubature formulae toapproximate integrands over the unit cube $[0,1]^s$ from a weighted reproducingkernel Hilbert space. We assume that the weights are independent randomvariables with a given mean and variance for two reasons stemming frompractical applications: (i) It is usually not known in practice how to choosethe weights. Thus by assuming that the weights are random variables, we obtainrobust constructions (with respect to the weights) of lattice rules. This, tosome extend, removes the necessity to carefully choose the weights. (ii) Inpractice it is convenient to use the same lattice rule for many differentintegrands. The best choice of weights for each integrand may vary to somedegree, hence considering the weights random variables does justice to howlattice rules are used in applications.  We also study a generalized version which uses $r$ constraints which we callthe cbc$r$c (component-by-component with $r$ constraints) algorithm. We showthat lattice rules generated by the cbc$r$c algorithm simultaneously work wellfor all weights in a subspace spanned by the chosen weights$\boldsymbol{\gamma}^{(1)},..., \boldsymbol{\gamma}^{(r)}$. Thus, inapplications, instead of finding one set of weights, it is enough to find an$r$ dimensional convex polytope in which the optimal weights lie. The price forthis method is a factor $r$ in the upper bound on the error and in theconstruction cost of the lattice rule. Thus the burden of determining one setof weights very precisely can be shifted to the construction of good latticerules.

Random weights, robust lattice rules and the geometry of the cbc$r$c algorithm