The Random-Facet algorithm is a randomized variant of the simplex method which is known to solve any linear program with n variables and m constraints using an expected number of pivot steps which is subexponential in both n and m. This is the theoretically fastest simplex algorithm known to date if m ≈ n; it provably beats most of the classical deterministic variants which require exp(Ω(n)) pi...

We describe an 88-year-old woman with various lesions in the right submammary region that had remained stable for more than 40 years. She reported rapid growth and ulceration of one of the lesions in the past year. The examination showed a firm tumor of diameter 7 cm below the right breast. The surface of the tumor was keratotic at the periphery and ulcerated in the middle with foul-smelling se...

In this paper, we present a simplex method for linear programs in standard form, or more precisely, linear optimization problems that have countably many non-negative variables and countably many equality constraints. Important special cases of these problems include infinite horizon deterministic dynamic programming problems and network flow problems with countably infinite nodes and arcs. Aft...

The simplex algorithm travels, on the underlying polyhedron, from vertex to vertex until reaching an optimal vertex. With the same simplex framework, the proposed algorithm generates a series of feasible points (which are not necessarily vertices). In particular, it is exactly an interior point algorithm if the initial point used is interior. Computational experiments show that the algorithm ar...

Recently, computational results demonstrated remarkable superiority of a so-called “largest-distance” rule and “nested pricing” rule to other major rules commonly used in practice, such as Dantzig’s original rule, the steepest-edge rule and Devex rule. Our computational experiments show that the simplex algorithm using a combination of these rules turned out to be even more efficient. Mathemati...

The SecureSCM project (www.securescm.org) aims to develop cryptographic solutions to the problem of data sharing in Supply Chain Optimization (SCO). The SCO problem has a precise mathematical structure. It is an instance of the general Linear Programming (LP) problem. However, standard techniques for LP problems are not suitable for this purpose because they require participants to reveal priva...

Simplex Adjacency graphs represent the possible Simplex pivot operations (the edges) between pairs of feasible bases (the nodes) of linear optimization models. These graphs are mainly studied so far in the context of degeneracy. The more general point of view in this paper leads to a number of new results, mainly concerning the connectivity and the feasibility-optimality duality in these graphs...

The simplex method is frequently the most efficient method of solving linear programming (LP) problems. This paper reviews previous attempts to parallelise the simplex method in relation to efficient serial simplex techniques and the nature of practical LP problems. For the major challenge of solving general large sparse LP problems, there has been no parallelisation of the simplex method that ...

We consider heterogeneous distributed storage systems (DSSs) having flexible reconstruction degree, where each node in the system has nonuniform repair bandwidth and capacity. In particular, a data collector can reconstruct file using some $$k$$ nodes and, for failure, be repaired by set of active nodes. Using min-cut bound, we investigate fundamental trade-off between costs our model DSS. Fur...

As it is well-known, since the discovery of the interior-point methods linear programming (LP) is no longer synonymous with the celebrated simplex method. The interior-point methods (IPMs) have not only a better complexity bound than the simplex method (polynomial vs. exponential) but also enjoy practical efficiency and can be considerably faster than the simplex method for many (but not for al...