针对只含有不等式约束的优化问题,本文首先给出了其Karush-Kuhn-Tucker (KKT)条件,并利用光滑互补函数将KKT系统转化为一类光滑的方程组问题;其次,将光滑方程组问题转化为无约束优化问题;最后,本文提出一类二阶微分方程系统求解无约束优化问题,并讨论了二阶微分方程系统的解的稳定性及收敛速度。For optimization problems with only inequality constraints, this paper first presents their Karush-Kuhn-Tucker (KKT) conditions, and uses smooth complementarity functions to transform the KKT system into a class of smooth system of equations problems. Secondly, this article transforms the problem of smooth equation systems into an unconstrained optimization problem. Finally, this article proposes a class of second-order differential equation systems for solving unconstrained optimization problems, and discusses the stability and convergence speed of the solutions of second-order differential equation systems.
微积分的一个重要应用就是微分方程,在工科院校的高等数学的课程当中,对于二阶常微分方程会花非常大的力气在二阶常系数线性常微分方程的求解中,对于其余的二阶常微分方程尤其是非线性的常微分方程很少涉及。本文通过对Kepler三大行星运动定律的数学推导,来说明非线性常微分方程才是实际当中碰到的大多数,以及微积分作为人类历史上的一项伟大发现的重要意义。An important application of calculus is differential equations. In the advanced mathematics courses of engineering colleges, a lot of effort is spent on solving second-order linear differential equations with constant coefficients, while other second-order differential equations, especially nonlinear differential equations, are rarely involved. This paper uses the mathematical derivation of Kepler’s three laws of planetary motion to illustrate that nonlinear differential equations are the majority of ordinary differential equations encountered in practice, and the importance of calculus as a great discovery in human history.
本文运用光滑化的自然残差函数建立了具有不等式约束条件的变分不等式问题的光滑化KKT方程组,并建立了与其等价的无约束优化问题。建立了具有阻尼惯性参数和时间尺度参数的二阶微分方程系统来求解该无约束优化问题,并证明了该二阶微分方程系统的稳定性,从而得到了具有不等式约束的变分不等式问题的KKT点的收敛性。并将二阶微分方程方法与已有的一阶微分方程方法进行了理论条件和数值结果的对比。在理论条件的要求上,二阶微分方程方法的条件要更容易实现,而在数值结果上,一阶微分方程方法的收敛速度要快,但是两种方法的差距可以忽略不计。In this paper, a smoothing KKT equation system with inequality constraints is established by using the smoothing natural residual function, and the unconstrained optimization problem is established. A second order differential equation system with damping coefficient and time scale coefficient is established to solve the unconstrained optimization problem, and the stability of the second-order differential equation system is proved, then the convergence of the KKT points of the variational inequality problem with inequality constraints is obtained. The theoretical conditions and numerical results of the second-order differential equation method are compared with the existing first order differential equation method. In terms of theoretical conditions, the conditions of the second-order differential equation method are easier to implement, while in the numerical results, the convergence speed of the first order differential equation method is faster, but the difference between the two methods is negligible.
