Read e-book online A branch and cut algorithm for nonconvex quadratically PDF

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 restricted quadratic programming challenge. the assumption is to estimate all quadratic phrases by way of successive linearizations inside of a branching tree utilizing Reformulation-Linearization ideas (RLT). to take action, 4 periods of linearizations (cuts), counting on one to 3 parameters, are distinct. for every type, we exhibit tips to choose the easiest member with recognize to an exact criterion. The cuts brought at any node of the tree are legitimate within the complete tree, and never simply in the subtree rooted at that node. as a way 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 mentioned that come with general try out difficulties taken from the literature. a few of these difficulties are solved for the 1st time with an explanation of world optimality.

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

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In C matters are simple: e± 2 π i/n is a primitive n-th root of unity for arbitrary n. e2 π i/21 is a 21-th root of unity. r = e2 π i/3 is also 21-th root of unity but not a primitive root, because r3 = 1. A primitive n-th root of 1 in Z/mZ is also called an element of order n. The ‘cyclic’ property of the elements r of order n lies in the heart of all FFT algorithms: rn+k = rk . In Z/mZ things are not that simple since primitive roots of unity do not exist for arbitrary n, they exist for some maximal order R only.

For the reason given above the computation of the column FFTs should not be done in place. One can insert additional transpositions in the algorithm to have the columns lie in contiguous memory when they are worked upon. The easy way is to use an additional scratch space for the column FFTs, then only the copying from and to the scratch space will be slow. If one interleaves the copying back with the exp()multiplications (to let the CPU do some work during the wait for the memory access) the performance should be ok.

X with n zeros appended). ω) B {ω} B is the cc. of C {ω2 } C and therefore every B {} B-term is the cc. of the C {} C-term in the same line. Is there a nice and general scheme for real valued convolutions based on the MFA? Read on for the positive answer. 6 s, d lower half plus/minus higher half of x CHAPTER 2. 6 46 Convolution of real valued data using the MFA For row 0 (which is real after the column FFTs) one needs to compute the (usual) cyclic convolution; for row R/2 (also real after the column FFTs) a negacyclic convolution is needed7 , the code for that task is given on page 62.

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

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