SIAM Journal on Computing
On generalized bisection of n-simplices
Mathematics of Computation
A Strip-Packing Algorithm with Absolute Performance Bound 2
SIAM Journal on Computing
A Simplicial Branch-and-Bound Method for Solving Nonconvex All-Quadratic Programs
Journal of Global Optimization
Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs
Optimization Methods & Software - GLOBAL OPTIMIZATION
Packing unit spheres into the smallest sphere using VNS and NLP
Computers and Operations Research
Stability and convergence analysis of a dynamics-based collective method for random sphere packing
Journal of Computational Physics
The discrete ellipsoid covering problem: A discrete geometric programming approach
Discrete Applied Mathematics
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The problem of the unequal sphere packing in a 3-dimensional polytope is analyzed. Given a set of unequal spheres and a polytope, the double goal is to assemble the spheres in such a way that (i) they do not overlap with each other and (ii) the sum of the volumes of the spheres packed in the polytope is maximized. This optimization has an application in automated radiosurgical treatment planning and can be formulated as a nonconvex optimization problem with quadratic constraints and a linear objective function. On the basis of the special structures associated with this problem, we propose a variety of algorithms which improve markedly the existing simplicial branch-and-bound algorithm for the general nonconvex quadratic program. Further, heuristic algorithms are incorporated to strengthen the efficiency of the algorithm. The computational study demonstrates that the proposed algorithm can obtain successfully the optimization up to a limiting size.