Smoothing Methods For The Solution Of Systems Of Equalities And Inequalities

As a basic mathematical structure, systems of equalities and inequalitiesare widely used in various areas. Thus, it's pretty significant for theoreticalstudy and utilization to find out e?ective method for solving systems of equal-ities and inequalities. The study for solving systems of linear equalities andinequalities is almost mature now. But about systems of nonlinear equalitiesand inequalities, the existing results need improvement and there are manyproblems to be solved. It's one of focuses in the optimization community now.Smoothing algorithms have been proposed for solving various optimiza-tion problems successfully except for systems of equalities and inequalities.Whether or not and how to apply it to solve systems of equalities and inequal-ities, this paper will show some results on this area. We will focus on theanalysis of smoothing algorithm for solving systems of equalities and inequal-ities in this paper.Firstly, a system of nonlinear inequalities is originally reformed as a sys-tem of non-smooth equalities by using projection function in this paper. Basedon the idea of smoothing reformulation and the use of a special smoothingfunction, the problem is approximated via a family of parameterized smoothequations, and some smoothing algorithm is applied to solve them iterativelyso that a solution of the problem concerned is found. We show this algo-rithm is well-defined and globally convergent under two suitable assumptions.These assumptions in this paper are su?cient to be applied to many typesof functions. We obtain the solution of original problem by solving the re-sulting system. The proposed algorithm is locally super-linearly(or quadrat-ically)convergent. Moreover, we propose a more simple and e?ective methodfor solving a system of inequalities in which the number of variable is di?erentfrom that of inequalities. We also report some preliminary numerical resultsof the algorithm. Secondly, we generalize the smoothing method to solve a mixed system ofequalities and inequalities and propose a new smoothing algorithm. Compar-ing with the existing methods, the proposed method can solve more generalizedproblems. What's more, based on the results given before, we also improve thenumerical experiments and results of the algorithm by using a non-monotoneline search technique. We prove that without more conditions, the smoothingmethod can be used to solve a mixed system of equalities and inequalities andit's well-defined and globally convergent. We obtain the solution of the orig-inal system by solving new system using smoothing method. The proposedalgorithm is locally super-linearly (or quadratically) convergent. Since thenon-monotone degree of this algorithm can be controlled by changing the pa-rameter, this algorithm is more e?ective and useful. The proposed algorithmonly needs to solve at most one system of linear equations and execute onelinear search at each iteration. The numerical result demonstrates that ourtheoretical findings coincide with the practical results.Finally, we improve the former works from a new sight. We build a newclass of smoothing functions by p-norm and show its properties. Then wereform a system of inequalities by this function. Specially, for the system ofinequalities with P0 function, we prove the algorithm is well-defined withoutany conditions. Meanwhile, we show that the proposed smoothing algorithmis globally convergent and locally super-linearly (or quadratically ) convergentunder an assumption that the solution set of the system of inequalities con-cerned is nonempty and bounded, which is a weaker condition in the literature.We test the algorithm on some problems, and the experimental results showthat the proposed smooth function help to improve the computational results.