Global optimization using interval constraints
Date
2017-08-30
Authors
Chen, Huaimo
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Abstract
Global optimization methods can be classified into two non-overlapping classes with respect to accuracy: those with guaranteed accuracy and those without. The former are called bounding methods, the latter point methods. Bounding methods compute lower and upper bounds of function over a box and give a lower bound and an upper bound for the minimum. Point methods compute function values at points and output as the minimum the function value at a point.
R. E. Moore was the first to propose the bounding method using interval arithmetic for unconstrained global optimization. The first bounding method using interval arithmetic for constrained global optimization was due to E. R. Hansen and S. Sengupta. These methods are the well known bounding methods. Since these methods use interval arithmetic, we call them interval arithmetic methods. This dissertation studies the new bounding methods that use interval constraints, which is called interval constraint methods.
We prove that interval constraints is a generalization of interval arithmetic, computing an interval function in interval constraints gives the same result as in interval arithmetic. We propose a hypernarrowing algorithm using interval constraints. This algorithm produces a smaller interval result for the range of function f over a given domain than interval arithmetic. We present a generic Branch-and-Bound algorithm for unconstrained global optimization, prove the properties of the algorithm, and propose improvements on the algorithm. From this algorithm, we can obtain its interval arithmetic version and interval constraint version. We investigate the role of interval constraints in global optimization and discuss the performance and characteristics of interval arithmetic methods and interval constraint ones.
Based on the Branch-and-Bound algorithm for unconstrained global optimization, we present a generic Branch-and-Bound algorithm for constrained global optimization, study the effect of Fritz-John conditions as redundant constraints and compare the interval arithmetic method for constrained optimization with the interval constraint one.
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Mathematical optimization, Constraints (Artificial intelligence)