132 research outputs found
Global solution of non-convex quadratically constrained quadratic programs
International audienceThe class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QC-QPs with the following restriction : all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P *). This equivalent problem (P *) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y = xx T , where x are the initial variables of problem (P). We then propose to solve (P *) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO
Linear Programming Relaxations of Quadratically Constrained Quadratic Programs
We investigate the use of linear programming tools for solving semidefinite
programming relaxations of quadratically constrained quadratic problems.
Classes of valid linear inequalities are presented, including sparse PSD cuts,
and principal minors PSD cuts. Computational results based on instances from
the literature are presented.Comment: Published in IMA Volumes in Mathematics and its Applications, 2012,
Volume 15
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
Polyhedral Properties of RLT Relaxations of Nonconvex Quadratic Programs and Their Implications on Exact Relaxations
We study linear programming relaxations of nonconvex quadratic programs given
by the reformulation-linearization technique (RLT), referred to as RLT
relaxations. We investigate the relations between the polyhedral properties of
the feasible regions of a quadratic program and its RLT relaxation. We
establish various connections between recession directions, boundedness, and
vertices of the two feasible regions. Using these properties, we present
necessary and sufficient exactness conditions for RLT relaxations. We then give
a thorough discussion of how our results can be converted into simple
algorithmic procedures to construct instances of quadratic programs with exact,
inexact, or unbounded RLT relaxations.Comment: Technical Report, School of Mathematics, The University of Edinburgh,
Edinburgh, EH9 3FD, Scotland, United Kingdo
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