In this contribution we introduce the dual mixed-integer least-squares problem and study it in relation to its primal counterpart. The dual differs from the primal formulation in the order in which the integer ambiguity vector \(a \in {\mathbb {Z}}^{n}\) and baseline vector \(b \in {\mathbb {R}}^{p}\) are estimated. As not the ambiguities, but rather the entries of b are usually the parameters of interest, the attractiveness of the dual formulation stems from its direct computation of b. It is shown that this potential advantage relies on the ease with which an implicit integer least-squares problem of the dual can be solved. For the convoluted cases, we introduce two methods of simplifying approximations. To be able to describe their quality, we provide a complete distributional analysis of their estimators, thus allowing users to judge whether or not the approximations are acceptable for their application. It is shown that this approach implicitly introduces a new class of admissible integer estimators of which we also determine the pull-in regions. As the dual function is shown to lack convexity, special care is required to be able to compute its global minimizer \({\check{b}}\). Our proposed method, which has finite termination with a guaranteed \(\epsilon \)-tolerance, is constructed from combining the branch-and-bound principle, with a special convex-relaxation of the dual, to which the projected-gradient-descent method is applied to obtain the required bounds. Each of the method’s three constituents are described, whereby special emphasis is given to the construction of the required continuously differentiable, convex lower bounding function of the dual.

在这篇文章中，我们介绍了对偶混合整数最小二乘问题，并研究了它与其原始对应问题的关系。对偶与原始公式的不同之处在于整数模糊向量 \(a \in {\mathbb {Z}}^{n}\) 和基线向量 \(b \in {\mathbb {R}}^ {p}\) 是估计的。由于 b 的条目通常不是模糊性，而是感兴趣的参数，因此对偶公式的吸引力源于其对 b 的直接计算。结果表明，这种潜在优势依赖于解决对偶的隐式整数最小二乘问题的容易程度。对于复杂的情况，我们引入了两种简化近似的方法。为了能够描述它们的质量，我们提供了对其估计器的完整分布分析，从而允许用户判断近似值是否适合他们的应用。结果表明，这种方法隐式地引入了一类新的可接受的整数估计器，我们还确定了其中的拉入区域。由于对偶函数缺乏凸性，因此需要特别注意计算其全局最小值 \({\check{b}}\)。我们提出的方法具有有限终止和保证的 epsilon 容差，是通过将分支定界原理与对偶的特殊凸松弛相结合而构建的，其中投影梯度下降应用方法来获得所需的边界。描述了该方法的三个组成部分中的每一个，其中特别强调了所需的连续可微的对偶凸下界函数的构造。