10,206 research outputs found
Designing q-Unique DNA Sequences with Integer Linear Programs and Euler Tours in De Bruijn Graphs
DNA nanoarchitechtures require carefully designed oligonucleotides with certain non-hybridization guarantees, which can be formalized as the q-uniqueness property on the sequence level. We study the optimization problem of finding a longest q-unique DNA sequence. We first present a convenient formulation as an integer linear program on the underlying De Bruijn graph that allows to flexibly incorporate a variety of constraints; solution times for practically relevant values of q are short. We then provide additional insights into the problem structure using the quotient graph of the De Bruijn graph with respect to the equivalence relation induced by reverse complementarity. Specifically, for odd q the quotient graph is Eulerian, so finding a longest q-unique sequence is equivalent to finding an Euler tour and solved in linear time with respect to the output string length. For even q, self-complementary edges complicate the problem, and the graph has to be Eulerized by deleting a minimum number of edges. Two sub-cases arise, for one of which we present a complete solution, while the other one remains open
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Reformulations of mathematical programming problems as linear complementarity problems
A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are
(i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables,
(ii.) Minimum Linear Complementarity Problem (MLCP) which is an
LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized,
(iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value.
A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems
Strong Stationarity Conditions for Optimal Control of Hybrid Systems
We present necessary and sufficient optimality conditions for finite time
optimal control problems for a class of hybrid systems described by linear
complementarity models. Although these optimal control problems are difficult
in general due to the presence of complementarity constraints, we provide a set
of structural assumptions ensuring that the tangent cone of the constraints
possesses geometric regularity properties. These imply that the classical
Karush-Kuhn-Tucker conditions of nonlinear programming theory are both
necessary and sufficient for local optimality, which is not the case for
general mathematical programs with complementarity constraints. We also present
sufficient conditions for global optimality.
We proceed to show that the dynamics of every continuous piecewise affine
system can be written as the optimizer of a mathematical program which results
in a linear complementarity model satisfying our structural assumptions. Hence,
our stationarity results apply to a large class of hybrid systems with
piecewise affine dynamics. We present simulation results showing the
substantial benefits possible from using a nonlinear programming approach to
the optimal control problem with complementarity constraints instead of a more
traditional mixed-integer formulation.Comment: 30 pages, 4 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
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