Abstract
We propose methods for improving the relaxations obtained by the normalized multiparametric disaggregation technique (NMDT). These relaxations constitute a key component for some methods for solving nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems. It is shown that these relaxations can be more efficiently formulated by significantly reducing the number of auxiliary variables (in particular, binary variables) and constraints. Moreover, a novel algorithm for solving MIQCQP problems is proposed. It can be applied using either its original NMDT or the proposed reformulation. Computational experiments are performed using both benchmark instances from the literature and randomly generated instances. The numerical results suggest that the proposed techniques can improve the quality of the relaxations.
Highlights
In this study, the following general nonconvex quadratically constrained quadratic programming ((MI)QCQP) problems with box constraints are considered.min x T Q0x + f0(x, y) (1) s.t.:x T Qr x + fr (x, y) ≤ 0, ∀r ∈ I1,m (2) xi ∈ [ X L i U i
It is known that mixed-integer quadratically constrained quadratic programming (MIQCQP) problems are equivalent to QCQP problems, as any integer variable can be defined as a sum of binary variables, and the constraint y = y2 can be added to represent the integrality condition y ∈ {0, 1}
We provide a formal proof that normalized multiparametric disaggregation technique (NMDT) can be used for generating arbitrarily tight relaxations for (MI)QCQP problems
Summary
The following general nonconvex (mixed-integer) quadratically constrained quadratic programming ((MI)QCQP) problems with box constraints are considered. It is known that MIQCQP problems are equivalent to QCQP problems, as any integer variable can be defined as a sum of binary variables, and the constraint y = y2 can be added to represent the integrality condition y ∈ {0, 1} This transformation is possible, it is usually undesirable because it generally results in more computationally difficult nonconvex problems. The resulting relaxed problem is still nonlinear, possibly with integer variables, but its continuous relaxation is convex This can be achieved by adding convex terms with sufficiently large coefficients [3] or by decomposing the quadratic matrices into a sum of positive and negative matrices and linearizing the second term only.
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