8,856 research outputs found
Hebey-Vaugon conjecture II
In this paper we consider the remaining cases of Hebey-Vaugon conjecture
A construction of conformal-harmonic maps
Conformal harmonic maps from a 4-dimensional conformal manifold to a
Riemannian manifold are maps satisfying a certain conformally invariant fourth
order equation. We prove a general existence result for conformal harmonic
maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses
a geometric flow and relies on results of Gursky-Viaclovsky and Lamm
Analytic varieties with finite volume amoebas are algebraic
In this paper, we study the amoeba volume of a given dimensional generic
analytic variety of the complex algebraic torus (\C^*)^n. When , we show that is algebraic if and only if the volume of its amoeba is
finite. In this precise case, we establish a comparison theorem for the volume
of the amoeba and the coamoeba. Examples and applications to the linear
spaces will be given.Comment: 13 pages, 2 figure
A MIP framework for non-convex uniform price day-ahead electricity auctions
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, uniform-price 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
Revisiting minimum profit conditions in uniform price day-ahead electricity auctions
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
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