Polynomial Optimization And Its Applications To Epidemic Models

Optimization is an important branch of decision analysis. The results andmethods of optimization are applied to many areas, e.g. management decision,economic planning, network transportation, system engineering, military engineering,etc. As a great lot of nonlinear problems arise in management decision, economics,biology and system engineering, the nonlinear optimization provoke interest of thescholars in decision analysis and other subjects. Since a nonlinear function can beapproximated to a polynomial function by Taylor series, many problems of nonlinearoptimization can be approximately converted into those of polynomial optimization.So the polynomial optimization becomes an important part of nonlinear optimization,and is a hot topic investigated by many researchers in different manners.In views of the current situation of researching the polynomial optimization, thisdissertation aims to the accuracy of computation and the scope of application, andattempts to treat the polynomial optimization problems in some new ways. Thewell-known Wu’s method is utilized as a main tool to establish some theoretic resultsand effective algorithms.This dissertation is divided into ten chapters, and the respective contents of thesechapters are as follows.Chapter1is the introduction of this dissertation. In this chapter, we describe thebackground and current situation of the study on polynomial optimization, andexplains the work of this dissertation and its value.Chapter2presents some technical preliminaries. Here, we recall some basicconcepts and results, which involve the theory of ordered fields, real algebraicgeometry and the Wu’s method of decomposing polynomial systems into triangularchains. Moreover, the notions of strictly critical points and strongly critical points areintroduced, and an effective algorithm is provided for catching strongly critical pointsin a real algebraic set.The purpose of Chapter3is to present a new algorithm for computing therational univariate representations of zero-dimensional systems. This algorithm is based on the Wu’s method. As an application to polynomial optimization, we give aneffective method to compute the global minima of certain polynomials. With the aidof the computer algebraic system Maple and the software wsolve adaptive to the theWu’s method, our algorithm has been made into a general program named RUR-Wu.In Chapter4, an improved algorithm is presented for deciding the semi-definiteness of multivariate polynomials with coefficients in a computable orderedfield, which admits an effective method of finding an isolating set for every non-zerounivariate polynomial. The technique in this algorithm is to compute the triangulardecompositions of polynomial systems into regular chains, which are weaker thanirreducible chains. Moreover, such a polynomial as f+t is not involved in thetriangular decomposition, where f is a considered polynomial and t is anauxiliary variable. With the aid of the computer algebraic system Maple, ouralgorithm has been made into a general program named DecidePsd.Chapter5presents two algorithms for global minimization of multivariatepolynomials. For a multivariate polynomial f with real coefficients, we provide aneffective algorithm for deciding whether or not the global infimum of f is finite. Inthe case of f having a finite infimum, the infimum of f can be accurately codedas (h, a, b), where h is a real polynomial in one variable, a and b are tworational numbers with a b, and (h; a, b) stands for the only real root of h in theopen interval]a, b[. Moreover, another algorithm is provided to decide whether ornot the infimum of f is attained when the infimum of f is finite. With the aid ofthe computer algebraic system Maple and the software wsolve adaptive to the theWu’s method, our algorithm has been made into a general program namedGlobalOpt.Chapter6may be considered as a complement of Chapter5. In this chapter, weinvestigate the semi-algebraically connected components of minimum points of apolynomial function. For a given multivariate polynomial f with real coefficients,it is shown that the algorithm, presented in Chapter5, can find at least one point ineach semi-algebraically connected component of minimum points of f wheneverf has its global minimum. Moreover, an algorithm is presented for finding at leastone point in each semi-algebraically connected component of minimum points of f in the case when f has its global minimum.The purpose of Chapter7is to present a new algorithm for the global minimi-zation of rational functions. For two multivariate real polynomials f and g, we provide an effective algorithm for computing the global infimum of f/g. In the case of f/g having a finite global infimum, we adopt the Interval Representation of real roots to code the infimum. Totally different from the existing algorithms, the strategy of our algorithm is to decompose a finite set of polynomials into the so-called regular chains. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program named FindInf.Chapter8investigates the equality-constrained minimization of polynomial functions. Let f be a multivariate polynomial with real coefficients, and let H be a finite subset of multivariate polynomials with real coefficients, and denote by V(f:H) the set{f(α)|α∈Rn,且h(α)=0,(?)h∈H｝. In this chapter, we provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum of V(f:H) is a root of some polynomial in U whenever this constrained infimum is finite. Our strategy is computing the so-called revised resultants. With the aid of the computer algebraic system Maple and the software wsolve, our algorithm has been made into a general program, named ConstrainedInfimum, to treat the equality-constrained minimization of polynomials with rational coefficients.In Chapter9, based on the Lyapunov functions, the domain of attraction (DOA) about ceitain SIR epidemic model is computed with the employment of our proposed method. The computed result is able to help us to determine and forecast the development trend of this epidemic.Chapter10is the conclusion of this dissertation. Here, the main results in this dissertation are summarized, and some further jobs in the future are put forward.