A union-closed family F is a finite collection of distinct subsets of a finite set such that the union of two subsets in F belongs to F . Péter Frankl conjectured in 1979 that, for any such family, there exists an element that belongs to at least half of its sets. This conjecture is still unsolved. In this paper, we present some results and conjectures related to Frankl’s conjecture. The new conjectures are based on evidence provided by some experimental results.

We introduce a new variant of Chubanov's method for solving linear homogeneous systems with positive variables. In the Basic Procedure we use a recently introduced cut in combination with Nemirovski's Mirror-Prox method. We show that the cut requires at most (Formula presented.) time, just as Chubanov's cut. In an earlier paper it was shown that the new cuts are at least as sharp as those of Chubanov. Our Modified Main Algorithm is in essence the same as Chubanov's Main Algorithm, except that it uses the new Basic Procedure as a subroutine. The new method has (Formula presented.) time complexity, where (Formula presented.) is a suitably defined condition number. As we show, a simplified version of the new Basic Procedure competes well with the Smooth Perceptron Scheme of Peña and Soheili and, when combined with Rescaling, also with two commercial codes for linear optimization.

We consider nine elementary problems in optimization. We simply explore the conditions for optimality as known from the duality theory for convex optimization. This yields a quite straightforward solution method for each of these problems. The main contribution of this paper is that we show that even in the harder cases the solution needs only O(n) time.

We deal with a recently proposed method of Chubanov [A polynomial projection algorithm for linear feasibility problems. Math. Program. 153 (2015), pp. 687–713] for solving linear homogeneous systems with positive variables. Some improvements of Chubanov's method and its analysis are presented. We propose a new and simple cut criterion and show that the cuts defined by the new criterion are at least as sharp as in [1]. The new cut criterion reduces the iteration bound for his Basic Procedure by a factor 5, without changing the order of its strongly polynomial complexity. Our Modified Main Algorithm is in essence the same as Chubanov's Main Algorithm, except that it uses our Modified Basic Procedure as a subroutine. It is shown that it has (Formula presented.) time complexity, just as in [1]. Some promising computational results are presented, in comparison with the optimization package Gurobi.