. Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing a family of linear programs with 3 r + 1 inequalities in dimension 2 r for which the number of iterations performed is in \Omega (2 r ). The total curvature of the central path of these linear programs is also exponential in r , disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through ``logarithmic glasses."" This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature in a general setting.

We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimize a global, possibly nonconvex, cost while satisfying the agents' constraints, for instance, an energy operator in charge of the management of energy consumption flexibilities of many individual consumers. We provide a privacy-preserving algorithm that computes the optimal allocation of resources, and in which each agent's private information (constraints and individual solution profile) is never revealed either to the central operator or to a third party. Our method relies on an aggregation procedure: we compute iteratively a global allocation of resources, and gradually ensure existence of a disaggregation, that is, individual profiles satisfying agents' private constraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations. To obtain these cuts, we use an alternate projection method, which is implemented locally by each agent, preserving her privacy needs. We address especially the case in which the local and global constraints define a transportation polytope. Then, we provide theoretical convergence estimates together with numerical results, showing that the algorithm can be effectively used to solve the allocation problem in high dimension, while addressing privacy issues.

We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimize a global, possibly non-convex, cost while satisfying the agents' constraints. We focus on the practical case of the management of energy consumption flexibilities by the operator of a mi-crogrid. This paper provides a privacy-preserving algorithm that does compute the optimal allocation of resources, avoiding each agent to reveal her private information (constraints and individual solution profile) neither to the central operator nor to a third party. Our method relies on an aggregation procedure: we maintain a global allocation of resources, and gradually disaggregate this allocation to enforce the satisfaction of private constraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations (SMC). To obtain these cuts, we use an alternate projections method à la Von Neumann, which is implemented locally by each agent, preserving her privacy needs. Our theoretical and numerical results show that the method scales well as the number of agents gets large, and thus can be used to solve the allocation problem in high dimension, while addressing privacy issues.

Unit commitment problem on an electricity network consists in choosing the production plan of the plants (units) of a company in order to meet demand constraints. It is generally solved using a decomposition approach where demand constraints are relaxed, resulting in one pricing subproblem for each unit. In this paper we focus on the pricing subproblem for thermal units at EDF, a major French electricity producer. Our objective is to determine an optimal two-day production plan that minimizes the overall cost while respecting several non-linear operational constraints. The pricing problem is generally solved by dynamic programming. However, due to the curse of dimensionality, dynamic programming reaches its limits when extra-constraints have to be enforced. We model the subproblem as a resource constrained shortest path (RCSP) problem. Leveraging on RCSP algorithms recently introduced by the second author, we obtain an order of magnitude speed-up with respect to traditional RCSP algorithms.

We prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $\Omega(2^r)$. The total curvature of the central path of these linear programs is also exponential in $r$, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature, in a general setting.

A key application of Operation Research to the energy industry is the design of production plans for electricity producers. The production plan of a unit is the sequence of production levels at which it operates. Due to technical constraints, a unit can produce only at given power levels, and there are constraints on the changes of level that can be operated : for instance, a nuclear unit cannot be launched or stopped instantaneously. A typical production plan is illustrated on Figure 1. The revenue and the costs generated by a unit both depend on its production plan. Given electricity prices, the single unit commitment problem builds a profit maximizing production plan for one unit. Electricity producers are primarily interested in solving the unit commitment problem, which aims at building the production plans of all their units in order to meet a given demand at minimum cost. Tanahan et al. mention in their recent survey on unit commitment [5] that one of the state of the art solution method for this problem is Lagrangian relaxation. This if for instance the method used at EDF [2], the main French electricity producer. The idea of the method is to relax the linking constraints between the different units in order to obtain one single unit commitment subproblem for each unit. For large producers like EDF, hundreds of single unit commitment subproblems must be solved at each iteration of the Lagrangian relaxation, and the total time needed by the method must not exceed one hour. It is therefore practically crucial to be able to solve single unit commitment problems in fractions of a second. The structure of the single unit commitment problem depends on the technology of the unit, and there is a literature specialized on hydro, nuclear, or thermal unit commitment [5]. The present contribution is the result of a partnership with EDF, whose objective was to design more efficient algorithms for EDF thermal single unit commitment problem. The standard method to solve the single unit commitment problem is dynamic programming [5]. It is the one currently in use at EDF. However, due to the curse of dimensionality, the complexity of the dynamic programming algorithm is exponential in the number of constraints

We disprove a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with $3r+4$ inequalities in dimension $2r+2$ where the central path has a total curvature in $\Omega(2^r)$. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. The lower bound for the classical curvature is obtained by developing a combinatorial concept of a tropical angle.

We develop a tropical analogue of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in O(n(m + n)) time, where m is the number of constraints and n is the dimension.

A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on certaincombinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solvemean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existenceof such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games.Mean payoff games are known to be in ${NP} \cap {co-NP}$; whether they can be solved in polynomial time isan open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field ofHahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrarymean payoff game instance into a nondegenerate linear program over Hahn series.

We introduce an algorithm which solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a flip invariance property on the set of actions associated with every state. The algorithm is a tropical analogue of the shadow-vertex simplex algorithm, which solves mean payoff games via linear feasibility problems over the tropical semiring (ℝ ∪ { − ∞ }, max , + ). The key ingredient in our approach is that the shadow-vertex pivoting rule can be transferred to tropical polyhedra, and that its computation reduces to optimal assignment problems through Plücker relations.

Cet expose presente un analogue de la methode du simplexe pour la programmation lineaire tropicale, c'est a dire les problemes d'optimisation decrits par des inegalites (max,+)-lineaires. Tropicalement, les operations de pivotage et de calcul des couts reduits sont purement combinatoires. Pivoter s'interprete comme une succession de graphes acycliques. Le calcul des couts reduits s'effectue via un probleme couplage de poids maximum et un probleme de plus court chemin. La sequence de bases obtenue par methode du simplexe tropical correspond exactement a la sequence de bases donnee par la methode du simplexe classique appliquee sur le corps des series de Puiseux.

We study the weighted circuit constraint in the context of constraint programming. It appears as a substructure in many practical applications, particularly routing problems. We propose a domain filtering algorithm for the weighted circuit constraint that is based on the 1-tree relaxation of Held and Karp. In addition, we study domain filtering based on an additive bounding procedure that combines the 1-tree relaxation with the assignment problem relaxation. Experimental results on Traveling Salesman Problem instances demonstrate that our filtering algorithms can dramatically reduce the problem size. In particular, the search tree size and solving time can be reduced by several orders of magnitude, compared to existing constraint programming approaches. Moreover, for medium-size problem instances, our method is competitive with the state-of-the-art special-purpose TSP solver Concorde.

This paper is motivated by operating self service transport systems that flourish nowadays. In cities where such systems have been set up with bikes, trucks travel to maintain a suitable number of bikes per station. It is natural to study a version of the C-delivery TSP defined by Chalasani and Motwani in which, unlike their definition, C is part of the input: each vertex v of a graph G = (V, E) has a certain amount x(v) of a commodity and wishes to have an amount equal to y(v) (we assume that Sigma(v is an element of V) x(v) = Sigma(v is an element of V) y(v) and all quantities are assumed to be integers); given a vehicle of capacity C, find a minimal route that balances all vertices, that is, that allows to have an amount y(v) of the commodity on each vertex v. This paper presents among other things complexity results, lower bounds, approximation algorithms, and a polynomial algorithm when G is a tree.

Dynamic constraint aggregation (DCA) and dual variable stabilization (DVS) are two methods that can reduce the negative impact of degeneracy when solving linear programs. The first uses a projection to reduce the primal space whereas the second acts in the dual space. In this paper, we develop a new method, called stabilized dynamic constraint aggregation (SDCA), that combines DCA and DVS for solving set partitioning problems. It allows to fight degeneracy from both primal and dual perspectives simultaneously. To assess the effectiveness of SDCA, we report computational results obtained for highly degenerate multi-depot vehicle scheduling problem instances solved by column generation. These results indicate that SDCA can reduce the average computational time of the master problem by a factor of up to 7 with respect to the best of the two combined methods. Furthermore, they show that its performance is robust with regard to increasing levels of degeneracy in test problems.

Held and Karp have proposed, in the early 1970s, a relaxation for the Traveling Salesman Problem (TSP) as well as a branch-and-bound procedure that can solve small to modest-size instances to optimality [4, 5]. It has been shown that the Held-Karp relaxation produces very tight bounds in practice, and this relaxation is therefore applied in TSP solvers such as Concorde [1]. In this short paper we show that the Held-Karp approach can benefit from well-known techniques in Constraint Programming (CP) such as domain filtering and constraint propagation. Namely, we show that filtering algorithms developed for the weighted spanning tree constraint [3, 8] can be adapted to the context of the Held and Karp procedure. In addition to the adaptation of existing algorithms, we introduce a special-purpose filtering algorithm based on the underlying mechanisms used in Prim’s algorithm [7]. Finally, we explored two different branching schemes to close the integrality gap. Our initial experimental results indicate that the addition of the CP techniques to the Held-Karp method can be very effective. The paper is organized as follows: section 2 describes the Held-Karp approach while section 3 gives some insights on the Constraint Programming techniques and branching scheme used. In section 4 we demonstrate, through preliminary experiments, the impact of using CP in combination with Held and Karp based branch-and-bound on small to modest-size instances from the TSPlib.