本文選題：均衡約束數(shù)學(xué)規(guī)劃 + 同倫方法　； 參考：《吉林大學(xué)》2016年博士論文

【摘要】：均衡約束數(shù)學(xué)規(guī)劃問(wèn)題(Mathematical Programs with Equilibrium Constraints,簡(jiǎn)稱(chēng)MPEC)是指約束集中含有參數(shù)變分不等式、互補(bǔ)問(wèn)題和廣義方程的約束規(guī)劃問(wèn)題.該問(wèn)題廣泛應(yīng)用于數(shù)理經(jīng)濟(jì),工程設(shè)計(jì),化學(xué)工程,交通科學(xué)等領(lǐng)域,并且與變分不等式問(wèn)題、Nash均衡、互補(bǔ)問(wèn)題等有著緊密的聯(lián)系.然而由于MPEC問(wèn)題的可行域不滿(mǎn)足大部分的約束規(guī)范,尤其是M-F約束規(guī)范(Magasarian-Fromvitz),在可行域的任意一點(diǎn)處都不滿(mǎn)足,所以這類(lèi)問(wèn)題在理論分析和算法求解中都是非常困難的.在過(guò)去的二十多年里,關(guān)于MPEC問(wèn)題在理論和算法方面的研究取得了豐碩的成果,但是仍有許多問(wèn)題有待于解決.在本文中,我們基于投影函數(shù)和光滑化理論,利用組合同倫算法深入研究了幾類(lèi)帶有均衡約束的數(shù)學(xué)規(guī)劃問(wèn)題和均衡約束多目標(biāo)優(yōu)化問(wèn)題,主要取得了以下成果.1、研究帶有有界箱式約束變分不等式的數(shù)學(xué)規(guī)劃問(wèn)題.首先將所求問(wèn)題中的變分不等式價(jià)轉(zhuǎn)化為帶有投影函數(shù)的非光滑等式,再基于箱式約束集合的特點(diǎn),利用Cabriel-More光滑函數(shù)逼近等式中的非光滑部分,構(gòu)造一個(gè)帶參數(shù)的等式約束,對(duì)轉(zhuǎn)化后的數(shù)學(xué)規(guī)劃問(wèn)題的KKT系統(tǒng)構(gòu)造同倫方程.這種做法既不需要假設(shè)函數(shù)F具有強(qiáng)單調(diào)性,也不需要引入額外的變量,而且在后續(xù)的計(jì)算中方便了初始點(diǎn)的選取.最后證明了同倫路徑的存在性和大范圍的收斂性,并通過(guò)數(shù)值實(shí)驗(yàn)驗(yàn)證了算法的可行性和有效性.2、建立了互補(bǔ)約束數(shù)學(xué)規(guī)劃問(wèn)題的同倫算法.首先將帶有互補(bǔ)約束的數(shù)學(xué)規(guī)劃問(wèn)題轉(zhuǎn)化為一般的非光滑函數(shù)約束的非線(xiàn)性規(guī)劃問(wèn)題.然后利用光滑化手段把其中的非光滑等式約束轉(zhuǎn)化為光滑函數(shù)的等式約束.從而將前述的數(shù)學(xué)規(guī)劃問(wèn)題轉(zhuǎn)化為光滑函數(shù)約束的數(shù)學(xué)規(guī)劃問(wèn)題,這樣避免了引入更多的乘子變量.對(duì)最后得到的光滑規(guī)劃問(wèn)題的KKT系統(tǒng)構(gòu)造同倫方程,證明了同倫路徑的存在性和收斂性,同時(shí)證明了所求得到的KKT點(diǎn)是原問(wèn)題的C-穩(wěn)定點(diǎn),并且利用數(shù)值算例驗(yàn)證了算法的可行性和有效性.3、構(gòu)建了求解帶有均衡約束的多目標(biāo)優(yōu)化問(wèn)題的新的同倫算法.首先利用SBCQ約束規(guī)范將原問(wèn)題等價(jià)轉(zhuǎn)化帶有KKT系統(tǒng)的一般的多目標(biāo)問(wèn)題,將上述的KKT系統(tǒng)轉(zhuǎn)化為一個(gè)光滑的等式約束,進(jìn)而得到一個(gè)帶有等式和不等式約束的多目標(biāo)規(guī)劃問(wèn)題,最后對(duì)轉(zhuǎn)化后的等價(jià)問(wèn)題的KKT系統(tǒng)構(gòu)造同倫方程,證明了同倫路徑地存在性和收斂性.最后用數(shù)值實(shí)驗(yàn)證明了所提出的算法的可行性和有效性.4、討論了約束條件中變分不等式定義在一般閉凸集上的均衡約束規(guī)劃問(wèn)題.通過(guò)引入無(wú)窮遠(yuǎn)解的概念,將所求的MPEC問(wèn)題轉(zhuǎn)化為帶有投影函數(shù)的單層優(yōu)化問(wèn)題。再利用光滑化手段將最后得到的單層優(yōu)化問(wèn)題轉(zhuǎn)化為光滑函數(shù)約束的優(yōu)化問(wèn)題,對(duì)其KKT系統(tǒng)構(gòu)造同倫方程,證明了同倫路徑的存在性和收斂性,并給出計(jì)算實(shí)例.
[Abstract]:Equilibrium constrained Programs with Equilibrium Constraints, (MPECs) is a constrained programming problem with parametric variational inequalities, complementary problems and generalized equations in the constraint set. This problem is widely used in the fields of mathematical economics, engineering design, chemical engineering, traffic science and so on, and is closely related to the variational inequality problems such as Nash equilibrium and complementarity problems. However, due to the fact that the feasible domain of the MPEC problem does not satisfy most of the constraint specifications, especially the M-F constraint specification, it is not satisfied at any point in the feasible domain, so it is very difficult for this kind of problem to be solved in theory and algorithm. In the past twenty years, great achievements have been made in the research of MPEC problem in theory and algorithm, but there are still many problems to be solved. In this paper, based on projection function and smoothing theory, we study several kinds of mathematical programming problems with equilibrium constraints and multi-objective optimization problems with equilibrium constraints by using combined homotopy algorithm. In this paper, the following results are obtained. 1. The mathematical programming problem with bounded box constrained variational inequalities is studied. First, the variational inequality valence in the problem is transformed into a nonsmooth equation with projection function. Then, based on the characteristics of box constraint set, a parameter equality constraint is constructed by using Cabriel-More smooth function to approximate the nonsmooth part of the equation. The homotopy equation is constructed for the KKT system of the transformed mathematical programming problem. This method does not need to assume that the function F has strong monotonicity, nor does it need to introduce additional variables, and it also facilitates the selection of initial points in subsequent calculations. Finally, the existence of homotopy path and the convergence of a wide range are proved. The feasibility and validity of the algorithm are verified by numerical experiments, and a homotopy algorithm for complementary constrained mathematical programming is established. Firstly, the mathematical programming problem with complementary constraints is transformed into a general nonlinear programming problem with nonsmooth function constraints. Then the nonsmooth equality constraints are transformed into the equality constraints of smooth functions by smoothing method. Thus, the above mathematical programming problem is transformed into a smooth function constrained mathematical programming problem, thus avoiding the introduction of more multiplier variables. The homotopy equation is constructed for the KKT system of the final smooth programming problem. The existence and convergence of the homotopy path are proved. It is also proved that the obtained KKT point is the C-stable point of the original problem. A numerical example is used to verify the feasibility and validity of the algorithm, and a new homotopy algorithm is constructed to solve the multi-objective optimization problem with equilibrium constraints. Firstly, the original problem is equivalent to a general multiobjective problem with KKT system by using SBCQ constraint specification, and the KKT system mentioned above is transformed into a smooth equality constraint, and then a multiobjective programming problem with equality and inequality constraints is obtained. Finally, the homotopy equation is constructed for the KKT system of the transformed equivalent problem, and the existence and convergence of the homotopy path are proved. Finally, the feasibility and validity of the proposed algorithm are proved by numerical experiments. Finally, the equilibrium constrained programming problem defined by variational inequalities on a general closed convex set is discussed. By introducing the concept of infinite solution, the solved MPEC problem is transformed into a single-layer optimization problem with projection function. Finally, by using smoothing method, the final single-layer optimization problem is transformed into an optimization problem with smooth function constraints. The homotopy equation is constructed for its KKT system, and the existence and convergence of homotopy path are proved, and an example is given.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O221

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