An efficient nonlinear interval uncertain optimization method using Legendre polynomial chaos expansion

Abstract

Many specialized numerical techniques have been developed on interval uncertainty optimization. However, the conventional nested optimization strategy is time-consuming. The improved Taylor-based interval optimization avoids the laborious inner optimization, nonetheless, it is intrusive and presents significant computational challenges in the large-level uncertain optimization problems. To end these, this paper proposes a Legendre polynomial expansion method combined with the subinterval technique to evaluate the range enclosure of an interval function, where the expansion coefficients are computed through the collocation method. The detailed comparisons demonstrate that such Legendre-based polynomial expansion provides much more confidence in interval bounds evaluation than the Taylor expansion, and is capable of maintaining very sharp solution enclosures even when the uncertainty level is increased. More exciting, it is highly efficient. Subsequent application of the Legendre polynomial expansion in the interval bound estimation of the uncertain objective function and constraints, and combining an appropriate outer optimizer, result in a novel nonlinear interval uncertain optimization method for complicated engineering systems, whereby the time-consuming optimization nesting is avoided. A beam design example and an electromagnetic buffer optimization example illustrate that the proposed interval optimization method has fine computational precision and high computational efficiency, especially is capable of attacking the increase of uncertainty level. Moreover, the new method does not require the derivative information of the uncertain objective and constraints, such that applies to any engineering uncertain optimization problems, including the complex engineering problems which are usually referred to as “black-box”.