Revisiting implicit variables in mathematical optimization: simplified modeling and a numerical evidence
本文通过一个更简单的模型问题，重新审视了将隐式变量视为显式变量的常见做法，指出该做法理论上会导致人工驻点和更强的约束规格，并通过基数约束投资组合优化问题展示了其数值上的劣势。
Abstract Implicit variables of an optimization problem are used to model variationally challenging feasibility conditions in a tractable way while not entering the objective function. Hence, it is a standard approach to treat implicit variables as explicit ones. Recently, it has been shown in terms of a comparatively complex model problem that this approach, generally, is theoretically disadvantageous as the surrogate problem typically suffers from the presence of artificial stationary points and the need for stronger constraint qualifications. The purpose of the present paper is twofold. First, it introduces a much simpler and easier accessible model problem which can be used to recapitulate and even broaden the aforementioned findings. Indeed, we will extend the analysis to two more classes of stationary points and the associated constraint qualifications. These theoretical results are accompanied by illustrative examples from cardinality-constrained, vanishing-constrained, and bilevel optimization. Second, the present paper illustrates, in terms of cardinality-constrained portfolio optimization problems, that treating implicit variables as explicit ones may also be disadvantageous from a numerical point of view.