Convexification and Global Optimization in Continuous And
Convexification and Global Optimization in Continuous And
Convex Optimization
Global optimization of mixed-integer nonlinear programs: A theoretical and computational study
Mathematical Programming: Series A and B
An ε-accurate model for optimal unequal-area block layout design
Computers and Operations Research
Deterministic Global Optimization: Theory, Methods and (NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 37) (Nonconvex Optimization and Its Applications)
Journal of Global Optimization
| Hi-index | 0.00 |
Most branch-and-bound algorithms in global optimization depend on convex underestimators to calculate lower bounds of a minimization objective function. The $$\alpha $$BB methodology produces such underestimators for sufficiently smooth functions by analyzing interval Hessian approximations. Several methods to rigorously determine the $$\alpha $$BB parameters have been proposed, varying in tightness and computational complexity. We present new polynomial-time methods and compare their properties to existing approaches. The new methods are based on classical eigenvalue bounds from linear algebra and a more recent result on interval matrices. We show how parameters can be optimized with respect to the average underestimation error, in addition to the maximum error commonly used in $$\alpha $$BB methods. Numerical comparisons are made, based on test functions and a set of randomly generated interval Hessians. The paper shows the relative strengths of the methods, and proves exact results where one method dominates another.