1,079 research outputs found
A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs
Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online
A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs
International audienceBilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online
Mathematical Programming Formulations for the Collapsed k-Core Problem
In social network analysis, the size of the k-core, i.e., the maximal induced
subgraph of the network with minimum degree at least k, is frequently adopted
as a typical metric to evaluate the cohesiveness of a community. We address the
Collapsed k-Core Problem, which seeks to find a subset of users, namely the
most critical users of the network, the removal of which results in the
smallest possible k-core. For the first time, both the problem of finding the
k-core of a network and the Collapsed k-Core Problem are formulated using
mathematical programming. On the one hand, we model the Collapsed k-Core
Problem as a natural deletion-round-indexed Integer Linear formulation. On the
other hand, we provide two bilevel programs for the problem, which differ in
the way in which the k-core identification problem is formulated at the lower
level. The first bilevel formulation is reformulated as a single-level sparse
model, exploiting a Benders-like decomposition approach. To derive the second
bilevel model, we provide a linear formulation for finding the k-core and use
it to state the lower-level problem. We then dualize the lower level and obtain
a compact Mixed-Integer Nonlinear single-level problem reformulation. We
additionally derive a combinatorial lower bound on the value of the optimal
solution and describe some pre-processing procedures and valid inequalities for
the three formulations. The performance of the proposed formulations is
compared on a set of benchmarking instances with the existing state-of-the-art
solver for mixed-integer bilevel problems proposed in (Fischetti et al., A New
General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs, Operations
Research 65(6), 2017)
Evaluating Resilience of Electricity Distribution Networks via A Modification of Generalized Benders Decomposition Method
This paper presents a computational approach to evaluate the resilience of
electricity Distribution Networks (DNs) to cyber-physical failures. In our
model, we consider an attacker who targets multiple DN components to maximize
the loss of the DN operator. We consider two types of operator response: (i)
Coordinated emergency response; (ii) Uncoordinated autonomous disconnects,
which may lead to cascading failures. To evaluate resilience under response
(i), we solve a Bilevel Mixed-Integer Second-Order Cone Program which is
computationally challenging due to mixed-integer variables in the inner problem
and non-convex constraints. Our solution approach is based on the Generalized
Benders Decomposition method, which achieves a reasonable tradeoff between
computational time and solution accuracy. Our approach involves modifying the
Benders cut based on structural insights on power flow over radial DNs. We
evaluate DN resilience under response (ii) by sequentially computing autonomous
component disconnects due to operating bound violations resulting from the
initial attack and the potential cascading failures. Our approach helps
estimate the gain in resilience under response (i), relative to (ii)
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
Complexity of fuzzy answer set programming under Łukasiewicz semantics
Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution
Inverse Optimization with Noisy Data
Inverse optimization refers to the inference of unknown parameters of an
optimization problem based on knowledge of its optimal solutions. This paper
considers inverse optimization in the setting where measurements of the optimal
solutions of a convex optimization problem are corrupted by noise. We first
provide a formulation for inverse optimization and prove it to be NP-hard. In
contrast to existing methods, we show that the parameter estimates produced by
our formulation are statistically consistent. Our approach involves combining a
new duality-based reformulation for bilevel programs with a regularization
scheme that smooths discontinuities in the formulation. Using epi-convergence
theory, we show the regularization parameter can be adjusted to approximate the
original inverse optimization problem to arbitrary accuracy, which we use to
prove our consistency results. Next, we propose two solution algorithms based
on our duality-based formulation. The first is an enumeration algorithm that is
applicable to settings where the dimensionality of the parameter space is
modest, and the second is a semiparametric approach that combines nonparametric
statistics with a modified version of our formulation. These numerical
algorithms are shown to maintain the statistical consistency of the underlying
formulation. Lastly, using both synthetic and real data, we demonstrate that
our approach performs competitively when compared with existing heuristics
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