A k×n array with entries from the q-letter alphabet {0, 1, …, q − 1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n, t, q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n, t, q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible ‘words’ of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein–Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D'Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions (w p with 0 < p < 1) and their increasing concave relatives. We provide (i) a sufficient condition (which applies to functions more general than root functions) for our smoothing to be increasing and concave, (ii) a proof that when p = 1/q for integers q ≥ 2, our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers 2 ≤ q ≤ 10, 000) on the conjecture that our smoothing is a sharper bound on the root function than the natural and simpler "shifted root function", and (iv) for all root functions, a quantification of the superiority (in an average sense) of our smoothing versus the shifted root function near 0.

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We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction x ∈ {0} ∪ [l, u], where z is a binary indicator of x ∈ [l, u] (u > > 0), and y "captures" f (x), which is assumed to be convex on its domain [l, u], but otherwise y = 0 when x = 0. This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex.Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the "perspective reformulation" inequality y ≥ zf (x/z). We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when f (x) := x p , for p > 1, relaxations utilizing the inequality yz q ≥ x p , for q ∈ [0, p − 1], are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with f (x) := x 2 ) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.