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

摘要： 本文求解了一类二次规划的逆问题。可描述为在保证一个可行的解是原二次规划问题的最优解的前提下,使目标函数中的参数与它们的估计值的距离最小。根据对偶理论,先将该逆问题转换为具有半正定锥约束的最小化凸问题,然后再根据凸问题的一阶最优性条件以及法锥与投影算子的关系,直接将上述凸优化问题转化为广义方程,并在简单的假设条件下证明该方程在其解点处的广义雅克比元素是非奇异的。进一步文中分别采用了单调线搜索和非单调线搜索两种技术的非精确牛顿法求解该广义方程。在数值实验中,先比较了两种线搜索技术对求解本文逆问题的求解效率,结果表明非单调线搜索效果更好;同时还将非单调线搜索牛顿法与交替方向法进行比较,结果表明本文的方法具有更优的求解效率。

关键词: 逆问题, 二次规划问题, 非精确牛顿法