求不定二次规划问题全局解的单调化方法

A Monotonic Optimization Approach for Solving Globally Generalized Quadratic Programming

不定二次规划是全局优化的一类重要问题,在金融、统计、工程设计等实际问题中有广泛应用。但此类问题可能存在多个非全局最优的局部极值点,所以求其全局最优解变得十分困难。运用单调优化理论提出一种求不定二次规划问题全局最优解的新方法:通过引入新变量将问题等价转化为单调优化问题,然后利用问题的单调结构进行缩减、分割、辅助问题最优值的定界等过程获得近似全局最优解。该解不仅可行且能充分接近真实的全局最优解,数值结果表明方法可行有效。

The generalized quadratic programming （GQP） is an important class of global optimization problems with wide applications in the fields of financial management, statistics and design engineering, with multiple local optimal solutions differing from the global solution. Thus, it is very difficult to obtain a global optimal solution for the GQP. Many solution methods were developed for globally solving the GQPs in a special form and the general form. However, these approaches may sometimes provide an infeasible solution, or one far from the true optimum. To overcome these limitations, a monotonic optimization approach is proposed for the GQP. In the approach, the original problem is first converted into an equivalent monotonic optimization problem, whose objective function is just a simple univariate by exploiting the particular features of this problem. Then, a range division and compression approach is used to reduce the range of each variable. Tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using monotonic functions, in which such solution is adequately guaranteed to be feasible and to be close to the actual global optimal solution. At last, several numerical examples are given to illustrate the feasibility and efficiency of the present algorithm.