The essential features of the branch-and-bound approach to constrained optimization are described, and several specific applications are reviewed. These include integer linear programming (Land-Doig and Balas methods), nonlinear programming (minimization of nonconvex objective functions), the traveling-salesman problem (Eastman and Little, et al. methods), and the quadratic assignment problem (Gilmore and Lawler methods). Computational considerations, including trade-offs between length of computation and storage requirements, are discussed and a comparison with dynamic programming is made. Various applications outside the domain of mathematical programming are also mentioned.
Abstract. For every g ∈ N 0 and ǫ > 0, we construct a smooth genus g surface embedded into the unit ball with area 8π and Willmore energy smaller than 8π + ǫ. From this we deduce that a minimising sequence for Willmore's energy in the class of genus g surfaces embedded in the unit ball with area 8π converges to a doubly covered sphere for all g ∈ N 0 . We obtain the same result for certain Canham-Helfrich energies with χ K ≤ 0 without genus constraint and show that Canham-Helfrich energies with χ K > 0 are not bounded from below in the class of smooth surfaces with area S embedded into a domain Ω ⋐ R 3 .Furthermore, we prove that the class of connected surfaces embedded in a domain Ω ⋐ R 3 with uniformly bounded Willmore energy and area is compact under varifold convergence.
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