关键词: 逆问题, 二次规划问题, 非精确牛顿法

Abstract: A positive optimization problem is a problem in which all the parameters of a model are given, and the optimal solution is found from the feasible solutions of the model. In reality, there are many optimization problems where some of the parameters are unknown, and we only know their estimated values, but we can get the optimal solution for the problem through experience, observation or experimentation. The so-called inverse optimization problem is to adjust the values of unknown parameters to minimize the difference between these values and their estimated values on the premise of obtaining the optimal solution of the problem.
The inverse optimization problem was first studied on the shortest path inverse problem arising from seismic tomography. From this point on, the research enthusiasm for inverse optimization problems rapidly rose among operations researchers. At the same time, inverse problems were also widely applied in many fields of combinatorial optimization, such as inverse spanning tree problems, inverse portfolio optimization, inverse network flow problems and so on.
Professor ZHANG from City University of Hong Kong suggests that the study of inverse problems about continuous optimization is an important research direction, and under his leadership, he, his collaborators, and other scholars have achieved systematic results in this direction since the late 1990s.
For the quadratic programming problem, scholars have studied its inverse problems in different forms. Some are solved for the parameters in the objective function, others for both the objective function and the parameters in the constraints.
This paper is a further exploration of a class of inverse quadratic programming problem studied by Professor ZHANG. The problem is solved only for the parameters of the objective function, and the minimization problem is the square of the matrix F-norm plus the square of the vector Euclidean norm. The train of thought about the problem is to convert this inverse problem into a minimized convex problem with semi-positive definite cone constraints based on the duality theory. Then it is directly transformed into a generalized equation based on the first-order optimality condition of the convex problem and by means of the relation between the normal cone and the projection operator. Under a simple assumption, it is proved that the generalized Jacobi element of this equation at its solution point is non-singular. With this conclusion, further we employ Newton’s method to solve the generalized equation. An inexact Newton method with two-line search techniques, monotone line search and non-monotone line search, is used to solve this generalized equation. And the convergence theorem for the monotone line search Newton method is given without proof.
In the numerical experiments, the known parameter values in the inverse problem and the parameter values taken in each algorithm are given first. And for comparison, it is ensured that the two-line search algorithms have the same stopping criterion. Then the solution efficiency of this two-line search techniques is first compared and the results show that the non-monotonic Newton method is more efficient. Next, we compare the monotonic Newton method with the alternating direction method for solving the same inverse problem and the results show that the method in this paper has better efficiency.
This inverse optimization problem is solved only for the parameters in the objective function. Whereas in reality the parameters in the constraints may also be unknown. So, solving parameter values for both in the objective function and the constraints has a wider practical significance. There is already literature on this aspect of the problem, and we are prepared to study this aspect in the future.

Key words: inverse problem, quadratic programming problem, inexact Newton method