16,806 research outputs found
Exact Algorithms for 0-1 Integer Programs with Linear Equality Constraints
In this paper, we show -time and -space exact
algorithms for 0-1 integer programs where constraints are linear equalities and
coefficients are arbitrary real numbers. Our algorithms are quadratically
faster than exhaustive search and almost quadratically faster than an algorithm
for an inequality version of the problem by Impagliazzo, Lovett, Paturi and
Schneider (arXiv:1401.5512), which motivated our work. Rather than improving
the time and space complexity, we advance to a simple direction as inclusion of
many NP-hard problems in terms of exact exponential algorithms. Specifically,
we extend our algorithms to linear optimization problems
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Optimal exact designs of experiments via Mixed Integer Nonlinear Programming
Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studyingtheir properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariablyapplicable to the particular problem only.We propose a systematic approach to construct optimal exact designs by incorporatingthe Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. Asexamples, we apply the methodology to find D- and A-optimal exact designs for linear and nonlinear models using global orlocal optimizers. Our examples include design problems with constraints on the locations or the number of replicates at theoptimal design points
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving
mixed-integer quadratic programs and works in two phases: the first phase
determines an equivalent quadratic formulation with a convex objective function
by solving a semidefinite problem , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving that substantially
speeds up the first phase. Moreover, this algorithm leads to a reformulated
problem of smaller size than the one obtained by the original MIQCR method
which results in a shorter time for solving the second phase.
We present extensive computational results to show the efficiency of our
algorithm
Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies
Patriksson (2008) provided a then up-to-date survey on the
continuous,separable, differentiable and convex resource allocation problem
with a single resource constraint. Since the publication of that paper the
interest in the problem has grown: several new applications have arisen where
the problem at hand constitutes a subproblem, and several new algorithms have
been developed for its efficient solution. This paper therefore serves three
purposes. First, it provides an up-to-date extension of the survey of the
literature of the field, complementing the survey in Patriksson (2008) with
more then 20 books and articles. Second, it contributes improvements of some of
these algorithms, in particular with an improvement of the pegging (that is,
variable fixing) process in the relaxation algorithm, and an improved means to
evaluate subsolutions. Third, it numerically evaluates several relaxation
(primal) and breakpoint (dual) algorithms, incorporating a variety of pegging
strategies, as well as a quasi-Newton method. Our conclusion is that our
modification of the relaxation algorithm performs the best. At least for
problem sizes up to 30 million variables the practical time complexity for the
breakpoint and relaxation algorithms is linear
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
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