1,992 research outputs found
On generalized semi-infinite optimization and bilevel optimization
The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems
Linear bilevel problems: Genericity results and an efficient method for computing local minima
The paper is concerned with linear bilevel problems. These nonconvex problems are known to be NP-complete. So, no efficient method for solving the global bilevel problem can be expected. In this paper we give a genericity analysis of linear bilevel problems and present a new algorithm for computing efficiently local minimizers. The method is based on the given structural analysis and combines ideas of the Simplex method with projected gradient steps
A New Approach to Electricity Market Clearing With Uniform Purchase Price and Curtailable Block Orders
The European market clearing problem is characterized by a set of
heterogeneous orders and rules that force the implementation of heuristic and
iterative solving methods. In particular, curtailable block orders and the
uniform purchase price (UPP) pose serious difficulties. A block is an order
that spans over multiple hours, and can be either fully accepted or fully
rejected. The UPP prescribes that all consumers pay a common price, i.e., the
UPP, in all the zones, while producers receive zonal prices, which can differ
from one zone to another.
The market clearing problem in the presence of both the UPP and block orders
is a major open issue in the European context. The UPP scheme leads to a
non-linear optimization problem involving both primal and dual variables,
whereas block orders introduce multi-temporal constraints and binary variables
into the problem. As a consequence, the market clearing problem in the presence
of both blocks and the UPP can be regarded as a non-linear integer programming
problem involving both primal and dual variables with complementary and
multi-temporal constraints.
The aim of this paper is to present a non-iterative and heuristic-free
approach for solving the market clearing problem in the presence of both
curtailable block orders and the UPP. The solution is exact, with no
approximation up to the level of resolution of current market data. By
resorting to an equivalent UPP formulation, the proposed approach results in a
mixed-integer linear program, which is built starting from a non-linear integer
bilevel programming problem. Numerical results using real market data are
reported to show the effectiveness of the proposed approach. The model has been
implemented in Python, and the code is freely available on a public repository.Comment: 15 pages, 7 figure
New models for the location of controversial facilities: A bilevel programming approach
Motivated by recent real-life applications in Location Theory in which the location decisions generate controversy, we propose a novel bilevel location
model in which, on the one hand, there is a leader that chooses among a number of fixed potential locations which ones to establish. Next, on the second hand, there is one or several followers that, once the leader location facilities have been set, chooses his location points in a continuous framework. The leader’s goal is to maximize some proxy to the weighted distance to the follower’s location points, while the follower(s) aim is to locate his location points as close as possible to the leader ones. We develop the bilevel location model for one follower and for any polyhedral distance, and we extend it for several followers and any ℓp-norm, p ∈ Q, p ≥ 1. We prove the NP-hardness of the problem and propose different mixed integer linear programming formulations. Moreover, we develop alternative Benders decomposition algorithms for the problem. Finally, we report some computational results comparing the formulations and the Benders decompositions on a set of instances.Fonds de la Recherche Scientique - FNRSMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona
A Consensus-ADMM Approach for Strategic Generation Investment in Electricity Markets
This paper addresses a multi-stage generation investment problem for a
strategic (price-maker) power producer in electricity markets. This problem is
exposed to different sources of uncertainty, including short-term operational
(e.g., rivals' offering strategies) and long-term macro (e.g., demand growth)
uncertainties. This problem is formulated as a stochastic bilevel optimization
problem, which eventually recasts as a large-scale stochastic mixed-integer
linear programming (MILP) problem with limited computational tractability. To
cope with computational issues, we propose a consensus version of alternating
direction method of multipliers (ADMM), which decomposes the original problem
by both short- and long-term scenarios. Although the convergence of ADMM to the
global solution cannot be generally guaranteed for MILP problems, we introduce
two bounds on the optimal solution, allowing for the evaluation of the solution
quality over iterations. Our numerical findings show that there is a trade-off
between computational time and solution quality
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