By Charles Audet, Pierre Hansen, Brigitte Jaumard

We current a department and minimize set of rules that yields in finite time, a globally ☼-optimal resolution (with recognize to feasibility and optimality) of the nonconvex quadratically limited quadratic programming challenge. the assumption is to estimate all quadratic phrases through successive linearizations inside of a branching tree utilizing Reformulation-Linearization innovations (RLT). to take action, 4 periods of linearizations (cuts), counting on one to 3 parameters, are exact. for every type, we exhibit how one can decide on the easiest member with admire to an exact criterion. The cuts brought at any node of the tree are legitimate within the complete tree, and never purely in the subtree rooted at that node. so as to improve the computational velocity, the constitution created at any node of the tree is versatile adequate for use at different nodes. Computational effects are stated that come with ordinary try difficulties taken from the literature. a few of these difficulties are solved for the 1st time with an evidence of world optimality.

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**Additional info for A branch and cut algorithm for nonconvex quadratically constrained quadratic programming**

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The advantage of Eq. 72 over Eq. 69 is that it can be applied to a constrained rigid body. Inverse inertia is a symmetric dyadic tensor that maps from F6 to M6 . 5. Another useful property is that the range of an inverse inertia is the subspace of M6 within which the body it applies to is free to move. This, in turn, implies that the rank of an inverse inertia equals the degree of motion freedom of the body to which it applies. Thus, if the body has fewer than six degrees of freedom, then Φ will be singular, but I will not exist.

On comparing these equations with Eq. 1, it is clear that FD and ID must evaluate to H −1 (τ − C) and H q¨ + C, respectively. However, the algorithms that implement them need not necessarily work by calculating C or H or H −1 . Thus, the useful property of Eqs. 4 is that they show clearly the inputs and outputs of each calculation, but do not imply any particular method of calculation. 1 assumes that the position variables are the integrals of the velocity variables. This is not always the case.

Lim δt→0 δt This accounts for the first two entries on the first line in the first table. 5(b) shows a coordinate frame that is moving with a velocity equal to dx , which means that it is translating with unit linear velocity in the x direction. Over a period of δt time units, this motion has no effect on the line Ox, but it causes Oy to shift by δt length units in the x direction. 10. 3: Properties of the spatial cross operators dOx (t + δt) = dOx (t), implying that d˙ Ox = 0, but dOy (t + δt) = dOy (t) + δt dz , implying that d˙ Oy = dz .

### A branch and cut algorithm for nonconvex quadratically constrained quadratic programming by Charles Audet, Pierre Hansen, Brigitte Jaumard

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