## 1 Introduction
Moreover, by the appropriate selection of the model parameters, one can freely influence the numerical conditioning of the considered model equation system [26]. To improve the required stability condition, the multiple-shooting approach can be applied. This method is often used in numerical simulations of the nonlinear, as well as multi-stage production systems [27]. The multiple-shooting method can be combined together with other modifications, to ensure a failure-free operation of the designed computational algorithms [28]. Recently, to parametrize the trajectory described by the dynamical equations, variability constraints have been proposed [29]. Therefore, the appropriate model of the parametrization procedures can be applied to rewrite the optimization task in a well-tractable form. Some of the most-often-used parametrization approaches result in large-scale nonlinear optimization tasks. Moreover, methodologies such as the direct transcription method [5] and collocation-based approaches [20] require experience in the numerical treatment of the differential-algebraic constraints. On the other hand, they are independent of the external computational procedures for the DAE system solution. Although the fully parametrization methods are able to efficiently co-operate with large-scale numerical optimization procedures, the computational experience indicates that the external DAE solvers can be applied for some classes of constraints and thus reduce the dimension of the optimization task significantly. It is worth noticing that medium-scale optimization problems can be successfully solved by personal computers. Therefore, the use of specialized computing stations seems not to be necessary. The problem considered in this article is to overcome the difficulties associated with solving a system of the nonlinear differential-algebraic equations in a sequential optimization approach. One of the most important issues discussed in this work is an improvement of the optimization task's tractability. This problem was considered in the context of a homotopy method, which was adjusted for solving optimization tasks with the DAE constraints [30]. The main contributions of this article are:  
* The design of the \(\alpha\)-model parametrization procedure as a combination of homotopy and the multiple shooting method;* The implementation of the \(\alpha\)DAE model optimization algorithm;
* The application of the \(\alpha\)DAE model optimization algorithm for the design of a counter-flow exchanger. In the presented solution method, a well-known homotopy-based approach was used. Additionally, the designed method has never been used before to solve systems of differential- algebraic equations. Therefore, it is a significant extension of the applicability area of homotopy in solving optimization tasks subject to highly nonlinear differential-algebraic constraints. Based on the possibilities given by the homotopy and multiple-shooting method, an \(\alpha\)DAE model optimization algorithm was designed and implemented. The detailed discussion of the considered optimization task, a formula of the model constraints, as well as the features of the new methodology are discussed in the next sections. The optimization problem taken into account is formulated in Section 2. The \(\alpha\)-model parametrization procedure is introduced in Section 3. Then, the main steps of the \(\alpha\)DAE model optimization algorithm are presented in Section 4. An application of the new approach to the design task of a heat and mass transfer process is described in Section 5. Finally, the presented considerations are concluded in Section 6.