In this paper we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESS's evaluated at some points of interest in the parameter space aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers which is based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function, that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, like in Bayesian, Kiefer-Weiss and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS, and other related. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss is proposed, and is applied for a particular case of three hypotheses in the Bernoulli model.

翻译：本文研究多重假设的序贯检验问题，主要目标是在约束误差概率条件下最小化期望样本量。我们以参数空间中某些感兴趣点处期望样本量的加权和作为最小化准则，旨在满足误差概率限制的前提下实现最小化。采用基于拉格朗日乘子法的变体方法，通过最小化辅助目标函数（称为拉格朗日函数），该函数将目标函数与约束条件结合，并引入称为乘子的常数。随后利用乘子使解满足约束条件。我们开发了一种面向计算机的拉格朗日函数最小化方法，根据参数点的具体选择，可提供贝叶斯设定、基弗-韦斯设定及其他具体场景下的最优检验。针对伯努利总体抽样的特例，我们设计了一套计算机算法，用于构建最小化拉格朗日函数的序贯检验，并数值评估检验特性（如误差概率、期望样本量）及其他相关指标。算法采用R编程语言实现，程序代码已公开于GitHub仓库。针对伯努利模型，我们以三个假设的特例为例，开展了关于序贯多重假设检验最优性的一系列计算机评估，并与矩阵序贯概率比检验进行数值对比。提出了一种多重假设基弗-韦斯问题的求解方法，并将其应用于伯努利模型中三个假设的特例。