【作者】 全靖；

【导师】 吴至友；

【作者基本信息】 重庆师范大学 ， 运筹学与控制论， 2006， 硕士

【摘要】 本文主要研究几类广义凸函数的性质及其在极值问题、对偶问题等数学规划问题中的一些应用和单调优化规划问题的凸化、凹化方法. 第一类广义凸函数是预不变拟凸函数和半严格预不变拟凸函数。广义凸函数在优化理论中有较广泛的应用,预不变拟凸函数是一类重要的广义凸函数,它是拟凸函数和不变凸函数的推广,因此对预不变拟凸函数的研究有较强的理论和实际意义。Yang证明了实值函数在下半连续和满足条件D的条件下预不变拟凸函数的充分必要条件。本文利用集合的(弱)近似凸性,在较弱的条件下获得了预不变拟凸函数的一些等价条件,即通过检查集合的闭性来判断函数的预不变拟凸性。Yang建立了预不变拟凸函数和半严格预不变拟凸函数的性质,本文减弱条件得到了半严格预不变拟凸函数的一个充分条件和其它基本性质。 本文研究的第二类广义凸函数是B预不变凸函数。Bector和Singh介绍了B凸函数的概念,Sujea介绍了B预不变凸函数,从而统一了B凸函数和预不变凸函数。本文讨论B预不变凸函数的另外一个充分条件和新的性质,进而得到了关于B预不变凸函数的非线性规划问题的充分最优性条件和“Mond-Weir”型弱强对偶结果。 (ν,F,ρ,θ)凸函数是本文考虑的第三类广义凸函数.近年来,多目标分式规划问题的最优性条件和对偶已被很多人所研究。Bector推导了不可微凸多目标分式规划的Fritz John和KKT必要充分最优条件,并建立了对偶理论。Produ介绍了(F,ρ)凸函数的概念,Liu得到了关于(F,ρ)凸函数的非光滑多目标分式规划的最优性条件和对偶。H.Kuk等定义了(ν,ρ)-不变凸函数,并证明了在(ν,ρ)-不变凸性的条件下非光滑多目标规划问题的广义KKT充分最优条件和弱强对偶更多还原

【Abstract】 In this thesis, the properties of several classes of generalized convex functions and their applications in optimization problems such as extremum problems and dual problems etc. and a convexification, concavification method of monotone optimization problem are resarched.The first kind of generalized convex function is prequasi-invex and semistrictly prequas-invex funtions. Convex and generalized convex functions play a central role in optimization theories. Prequas-invex is an important generalized convex fuction, it is a generalization of quasi-convex function and invex function. So the study of prequasi-invex functions has some theoretical and practical singnificance. Professor Yang had obtained the necessary and sufficient conditions of prequasi-invex functions under condition of lower semicontinuity and Condition D. In this thesis, we obtained many equivalent conditions of prequasi-invex functions under much weaker conditions by applying nearly convexity of sets, that is to say we can obtain the prequasi-invexity in terms of the closeness of sets. Yang established many good properties of prequasi-invex functions and semistrictly prequasi-invex functions. In this thesis we obtained a sufficient condition and other properties of semistrictly prequasi-invex functions by weaking the conditions.The second generalized convex function we study in this thesis is B-preinvex functions. Bector and Singh introduced the B-vex functions by relaxing the defi-nition of convexity of a function. Sujea introduced the B-preinvex functions, thus united the B-vex functions and preinvex functions. In this thesis, we discuss other sufficient conditions and new properties. Futhermore, the sufficient optimality conditions and Mond-Weir type weak ang strong duality results are obtained for a nonlinear programming.(v, F, p, 0)-convex functions is the third functions we considered in this thesis. In recent years, optimality and duality for muli-objective fractional programs have been studed by many authors. Bector et al. derived Fritz John and Karush-Kuhn-Tucker necessary and sufficient optimality condition for a class of non-differentiable convex multi-objective fractional programming problems, also establised the duality results. Produ introduced the concept of (F, p)-convexity, an extension of F—convexity. Liu obtained necessary and sufficient conditions and derived duality theorems for a class of nonsmooth multiobjective fractional programming problems involving (F, p)-convex functions. KuK et al. denned the concept of (v, p)-invexity for vectorvalued functions, and they proved the generalized Karush-Kuhn-Tucker necessary and sufficient optimality theorem, weak and strong duality for nonsmooth multiobjective programs under the (v, p)-invexity assumptions. Bector et al. and Xu gave a mixed type duality for fractional programming, established some sufficient optimality conditions and obtained various duality results between the mixed duality problem and primal problem. In this thesis, we introduced a class of functions called (v, F, p, 0)convex functions which involvs the (F, p)convex functions and (v, p)-invex functions as special cases. And we establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality conditions and introduce the mixed duality problem (MFXD) of non-smooth multi-objective fractional programming problems (MFP), in which functions are locally Lipschitz, obtain various duality results between themixed duality problem (MFXD) and primal problems (MFP) under the assumptions of (v, F, p, #)-convexity.The monotone optimization is a global optimization problem in which objective function and constrained functons are all monotone. This thesis proposes a new convexification or concavification transformation method to convert a monotone functioin into a convex or concave function. Then the monotone optimization problem can be converted into an equivalent concave minimization problem or reverse convex programming problem or canoical D.C. programming problem.更多还原