The use of Grossone in Mathematical Programming and Operations Research

The concepts of infinity and infinitesimal in mathematics date back to anciens Greek and have always attracted great attention. Very recently, a new methodology has been proposed by Sergeyev for performing calculations with infinite and infinitesimal quantities, by introducing an infinite unit of measure expressed by the numeral grossone. An important characteristic of this novel approach is its attention to numerical aspects. In this paper we will present some possible applications and use of grossone in Operations Research and Mathematical Programming. In particular, we will show how the use of grossone can be beneficial in anti--cycling procedure for the well-known simplex method for solving Linear Programming Problems and in defining exact differentiable Penalty Functions in Nonlinear Programming

An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles

We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi

A simplicial algorithm for computing proper Nash equilibria of finite games

Stationary Point;Game Theory;Nash Equilibrium

An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles

We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-order cone program(SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurob

Polyhedral Diameters and Applications to Optimization

The Simplex method is the most popular algorithm for solving linear programs (LPs). Geometrically, it moves from an initial vertex solution to an improving neighboring one, selected according to a pivot rule. Despite decades of study, it is still not known whether there exists a pivot rule that makes the Simplex method run in polynomial time. Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called the circuits, that is a superset of the edges of a polytope. The number of circuit augmentations has been of interest as a proxy for the number of steps in the Simplex method, and the circuit-diameter of polyhedra has been studied as a lower bound to the combinatorial diameter of polyhedra. We show that in the circuit-augmentation framework the Greatest Improvement and Dantzig pivot rules are NP-hard, even for 0/1 LPs. On the other hand, the Steepest Descent pivot rule can be carried out in polynomial time in the 0/1 setting, and the number of circuit augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1 LPs. We introduce a new rule in the circuit-augmentation framework which we call Asymmetric Steepest Descent. We show both that it can be computed in polynomial time and that it reaches an optimal solution of an LP in a polynomial number of augmentations. It was not previously known that such a rule was possible. We further show a weakly-polynomial bound on the circuit diameter of rational polyhedra. We next show that the circuit-augmentation framework can be exploited to make novel conclusions about the classical Simplex method itself: In particular, as a byproduct of our circuit results, we prove that (i) computing the shortest (monotone) path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for 0/1-LPs (i.e., those whose vertex solutions are in {0, 1}^n), a monotone path of strongly-polynomial length can be constructed using steepest improving edges. Inspired by this, we further examine the lengths of other monotone paths generated by some local decision rules – which we call edge rules – including two modifications of the classical Shadow Vertex pivot rule. We leverage the techniques we use for analyzing edge rules to devise pivot rules for the Simplex method on 0/1-LPs that generate the same monotone paths as their edge rule counterparts. In particular, this shows that there exist pivot rules for the Simplex method on 0/1 LPs that reach an optimal solution by performing only a polynomial number of non-degenerate pivots, answering an open question in the literature

The Complexity of Zadeh's Pivot Rule

The Simplex algorithm is one of the most important algorithms in discrete optimization, and is the most used algorithm for solving linear programs in practice. In the last 50 years, several pivot rules for this algorithm have been proposed and studied. For most deterministic pivot rules, exponential lower bounds were found, while a probabilistic pivot rule exists that guarantees termination in expected subexponential time. One deterministic pivot rule that is of special interest is Zadeh's pivot rule since it was the most promising candidate for a polynomial pivot rule for a long time. In 2011, Friedmann proved that this is not true by providing an example forcing the Simplex algorithm to perform at least a subexponential number of iterations in the worst case when using Zadeh's pivot rule. Still, it was not known whether Zadeh's pivot rule might achieve subexponential worst case running time. Next to analyzing Friedmann's construction in detail, we develop the first exponential lower bound for Zadeh's pivot rule. This closes a long-standing open problem by ruling out this pivot rule as a candidate for a deterministic, subexponential pivot rule in several areas of linear optimization and game theory