No abstract

No abstract

We analyze the polyhedral structure of the sets PCMIX = {(s, r, z) ∈ R × R+n × Zn ∣ s + rj + zj ≥ fj, j = 1, …, n} and P+CMIX = PCMIX ∩ {s ≥ 0}. The set P+CMIX is a natural generalization of the mixing set studied by Pochet and Wolsey [15, 16] and Günlük and Pochet [8] and recently has been introduced by Miller and Wolsey [12]. We introduce a new class of valid inequalities that has proven to be sufficient for describing conv(PCMIX). We give an extended formulation of size O(n) × O(n2) variables and constraints and indicate how to separate over conv(PCMIX) in O(n3) time. Finally, we show how the mixed integer rounding (MIR) inequalities of Nemhauser and Wolsey [14] and the mixing inequalities of Günlük and Pochet [8] constitute special cases of the cycle inequalities.

Abstract. We investigate the problem of solving traditional combinatorial graph problems using secure multi-party computation techniques, focusing on the shortest path and the maximum flow problems. To the best of our knowledge, this is the first time these problems have been addressed in a general multi-party computation setting. Our study highlights several complexity gaps and suggests the exploration of various trade-offs, while also offering protocols that are efficient enough to solve real-world problems.

We examine the problem of clearing day-ahead electricity market auctions where each bidder, whether a producer or consumer, can specify a minimum profit or maximum payment condition constraining the acceptance of a set of bid curves spanning multiple time periods in locations connected through a transmission network with linear constraints. Such types of conditions are for example considered in the Spanish and Portuguese day-ahead markets. This helps describing the recovery of start-up costs of a power plant, or analogously for a large consumer, utility reduced by a constant term. A new market model is proposed with a corresponding MILP formulation for uniform locational price day-ahead auctions, handling bids with a minimum profit or maximum payment condition in a uniform and computationally-efficient way. An exact decomposition procedure with sparse strengthened Benders cuts derived from the MILP formulation is also proposed. The MILP formulation and the decomposition procedure are similar to computationally-efficient approaches previously proposed to handle so-called block bids according to European market rules, though the clearing conditions could appear different at first sight. Both solving approaches are also valid to deal with both kinds of bids simultaneously, as block bids with a minimum acceptance ratio, generalizing fully indivisible block bids, are but a special case of the MP bids introduced here. We argue in favour of the MP bids by comparing them to previous models for minimum profit conditions proposed in the academic literature, and to the model for minimum income conditions used by the Spanish power exchange OMIE.

The main result of this paper is an O(n3) algorithm for the single-item lot-sizing problem with constant batch size and backlogging. We consider a general number of installable batches, i.e., in each time period t we may produce up to mt batches, where the mt are given and time-dependent. This generalizes earlier results as we consider backlogging and a general number of maximum batches. We also give faster algorithms for three special cases of this general problem. When backlogging is not allowed and the costs satisfy the Wagner-Whitin property, the problem is solvable in O(n2 log n) time. When the production in each period is required to be either zero or equal to the installed capacity, it is possible to solve the problem with and without backlogging in O(n2) and O(n log n) time, respectively.

It is well-known that a market equilibrium with uniform prices often does not exist in non-convex day-ahead electricity auctions. We consider the case of the non-convex, uniformprice Pan-European day-ahead electricity market "PCR" (Price Coupling of Regions), with non-convexities arising from so-called complex and block orders. Extending previous results, we propose a new primal-dual framework for these auctions, which has applications in both economic analysis and algorithm design. The contribution here is threefold. First, from the algorithmic point of view, we give a non-trivial exact (i.e. not approximate) linearization of a non-convex 'minimum income condition' that must hold for complex orders arising from the Spanish market, avoiding the introduction of any auxiliary variables, and allowing us to solve market clearing instances involving most of the bidding products proposed in PCR using off-the-shelf MIP solvers. Second, from the economic analysis point of view, we give the first MILP formulations of optimization problems such as the maximization of the traded volume, or the minimization of opportunity costs of paradoxically rejected block bids. We first show on a toy example that these two objectives are distinct from maximizing welfare. We also recover directly a previously noted property of an alternative market model. Third, we provide numerical experiments on realistic large-scale instances. They illustrate the efficiency of the approach, as well as the economics trade-offs that may occur in practice.An extensive literature now exists on non-convex day-ahead electricity markets or electricity pools, dealing in particular with market equilibrium issues in the presence of indivisibilities, see e.g. [3,4,5,11,10,13,14,15] and references therein. Research on the topic has been fostered by the ongoing liberalization and integration of electricity markets around the world during the past two decades. As electricity cannot be efficiently stored, non-convexities of production sets cannot be neglected, and bids introducing non-convexities in the mathematical formulation of the market clearing problem have been proposed for many years by most of power exchanges or electricity pools, allowing participants to reflect more accurately their operational constraints and cost structure.It is now well known that due to these non-convexities, a market equilibrium with uniform prices may fail to exist (a single price per market area and time slot, no transfer payments, no losses incurred, and no excess demand nor excess supply for the given uniform market prices). To deal with this issue, almost all ideas proposed revolve around reaching back, or getting close to a convex situation where strong duality holds and shadow prices exist. For example, a now classic proposition in [11] is to fix integer variables to optimal values for a welfare maximizing primal program whose constraints describe physically feasible dispatches of electricity, and compute multipart equilibrium prices using dual variables of these fixing constra...

No abstract