ABSTRACT

A class of direct search methods for locally minimizing a Lipschitz continuous black-box function f subject to locally Lipschitz constraints is presented. A sequence of smoothed ${\ell }_1$ penalty functions is used. Each smoothed penalty function is approximately minimized in turn. The smoothing is reduced after each minimization, exposing the ${\ell }_1$ exact penalty function in the limit. Convergence to a constrained Clarke stationary point is shown under appropriate regularity conditions. Convergence to one or more KKT points is shown under similar conditions when f and all active constraints are strictly differentiable at each limit point. An implementation of one method in this class is numerically tested and shown to be effective in practice. The implementation uses a discrete quasi-Newton step when possible. Otherwise a global direction search is used to locate a descent direction. Theoretical convergence properties are independent of the quasi-Newton step.

摘要

提出了一类在局部Lipschitz约束下局部最小化Lipschitz连续黑盒函数f的直接搜索方法。平滑序列${\ell }_1$使用惩罚函数。每个平滑惩罚函数依次近似最小化。每次最小化后平滑度都会降低，从而暴露出${\ell }_1$极限内的精确惩罚函数。在适当的规律性条件下显示收敛到受约束的克拉克驻点。当f和所有活动约束在每个极限点处严格可微时，在类似条件下显示收敛到一个或多个 KKT点。对此类中一种方法的实现进行了数值测试，并证明在实践中是有效的。该实现尽可能使用离散的准牛顿步。否则，使用全局方向搜索来定位下降方向。理论收敛特性与准牛顿步无关。