Lagrangian Dual Theory and SQP Method in Optimization Problem

Abstract: Lagrangian dual problem and SQP method are two important research subjects of optimization problem. In this paper we establish a class of Lagrangian dual problem which has zero duality gap property and develop a kind of SQP method. This paper is composed of three chapters.Chapter 1 is the introduction of this paper, which introduces the Lagrangian dual problem and SQP method, and the main results obtained in this paper.In chapter 2, we modify and improve the results in [9]. We establish the model of a class of Lagrangian dual problem under very ordinary conditions and obtain that this Lagrangian dual problem has zero duality gap property. We also get several important corollaries. We discuss detailedly the existence of zero duality gap for a class of augmented Lagrangian dual problem and several classes of nonlinear Lagrangian dual problems, and obtain important results. We also give some particular examples. Finally, we discuss the convergence of optimal paths. Compared with [9], our model overcomes the weakness of " punishment " in [9], and is more beneficial to apply.In chapter 3, we discuss how to ensure that QP subproblem is feasible in the SQP method. We develop a kind of SQP method by modifying the traditional QP subproblem and applying the Armijo-type linear search to a class of l_∞- penalty function which penalty parameters can be adjusted automatically. The idea of our method comes from [51]. Here we delete the quadratic item…
Key words: Duality gap; Lagrangian function; Convergence; SQP method; Optimal path

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