10,742 research outputs found
A solvable class of quadratic 0–1 programming
AbstractWe show that the minimum of the pseudo-Boolean quadratic function Æ’(x) = xTQx + cTx can be found in linear time when the graph defined by Q is transformable into a combinatorial circuit of AND, OR, NAND, NOR or NOT logic gates. A novel modeling technique is used to transform the graph defined by Q into a logic circuit. A consistent labeling of the signals in the logic circuit from the set {0, 1} corresponds to the global minimum of Æ’ and the labeling is determined through logic simulation of the circuit. Our approach establishes a direct and constructive relationship between pseudo-Boolean functions and logic circuits.In the restricted case when all the elements of Q are nonpositive, the minimum of Æ’ can be obtained in polynomial time [15]. We show that the problem of finding the minimum of Æ’, even in the special case when all the elements of Q are positive, is NP-complete
Resolutions for unit groups of orders
We present a general algorithm for constructing a free resolution for unit
groups of orders in semisimple rational algebras. The approach is based on
computing a contractible -complex employing the theory of minimal classes of
quadratic forms and Opgenorth's theory of dual cones. The information from the
complex is then used together with Wall's perturbation lemma to obtain the
resolution
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
Convex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far reaching
generalization of the standard linear combinatorial optimization problem. We
show that it is strongly polynomial time solvable over any edge-guaranteed
family, and discuss several applications
OSQP: An Operator Splitting Solver for Quadratic Programs
We present a general-purpose solver for convex quadratic programs based on
the alternating direction method of multipliers, employing a novel operator
splitting technique that requires the solution of a quasi-definite linear
system with the same coefficient matrix at almost every iteration. Our
algorithm is very robust, placing no requirements on the problem data such as
positive definiteness of the objective function or linear independence of the
constraint functions. It can be configured to be division-free once an initial
matrix factorization is carried out, making it suitable for real-time
applications in embedded systems. In addition, our technique is the first
operator splitting method for quadratic programs able to reliably detect primal
and dual infeasible problems from the algorithm iterates. The method also
supports factorization caching and warm starting, making it particularly
efficient when solving parametrized problems arising in finance, control, and
machine learning. Our open-source C implementation OSQP has a small footprint,
is library-free, and has been extensively tested on many problem instances from
a wide variety of application areas. It is typically ten times faster than
competing interior-point methods, and sometimes much more when factorization
caching or warm start is used. OSQP has already shown a large impact with tens
of thousands of users both in academia and in large corporations
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