非線性規(guī)劃01基本概念與凸規(guī)劃

非線性規(guī)劃

NonlinearProgrammingLudongUniversity2023/9/2LudongUniversity2第四章非線性規(guī)劃由前幾章知道，線性規(guī)劃的目標(biāo)函數(shù)和約束條件都是其自變量的線性函數(shù)，如果目標(biāo)函數(shù)或約束條件中包含有自變量的非線性函數(shù)，則這樣的規(guī)劃問(wèn)題就屬于非線性規(guī)劃。有些實(shí)際問(wèn)題可以表達(dá)成線性規(guī)劃問(wèn)題，但有些實(shí)際問(wèn)題則需要用非線性規(guī)劃的模型來(lái)表達(dá)，借助于非線性規(guī)劃解法來(lái)求解。2023/9/2LudongUniversity3第四章非線性規(guī)劃基本概念凸函數(shù)和凸規(guī)劃一維搜索方法無(wú)約束最優(yōu)化方法約束最優(yōu)化方法2023/9/2LudongUniversity4基本概念非線性規(guī)劃問(wèn)題非線性規(guī)劃方法概述2023/9/2LudongUniversity5Example1Threecustomerswithknownlocationsonaplanedescribedbycoordinates(ai,bi),i=1,2,3.Problem:Tofindalocationforadepotsothatthetotaldistancetothethreecustomersisminimized.Variable:(x,y)thecoordinatesofthedepotModel:Unconstrained2023/9/2LudongUniversity6Example2Usingtheminimummaterialtomakeabox.ThevolumeoftheboxhastobeV=1000.Decision:Boxlength:x,width:y,height:z.Objective:Tominimizesurfaceareaofthebox.Model:Constrained2023/9/2LudongUniversity7數(shù)學(xué)規(guī)劃

約束集或可行域MP的可行解或可行點(diǎn)2023/9/2LudongUniversity8向量化表示當(dāng)p=0，q=0時(shí)，稱為無(wú)約束非線性規(guī)劃或無(wú)約束最優(yōu)化問(wèn)題。否則稱為約束非線性規(guī)劃或約束最優(yōu)化問(wèn)題。2023/9/2LudongUniversity9最優(yōu)解和極小點(diǎn)2023/9/2LudongUniversity10最優(yōu)解的幾何位置x1x22023/9/2LudongUniversity11非線性規(guī)劃方法概述2023/9/2LudongUniversity12非線性規(guī)劃基本跌代格式2023/9/2LudongUniversity13凸函數(shù)與凸規(guī)劃凸函數(shù)及其性質(zhì)凸規(guī)劃及其性質(zhì)2023/9/2LudongUniversity14ConvexSet2023/9/2LudongUniversity15凸函數(shù)及其性質(zhì)2023/9/2LudongUniversity16凸函數(shù)及其性質(zhì)2023/9/2LudongUniversity17凸函數(shù)及其性質(zhì)2023/9/2LudongUniversity18凸函數(shù)及其性質(zhì)注：該逆命題不成立。2023/9/2LudongUniversity19凸規(guī)劃及其性質(zhì)約束集如果(MP)的約束集X是凸集，目標(biāo)函數(shù)f是X上的凸函數(shù)，則(MP)叫做非線性凸規(guī)劃，或簡(jiǎn)稱為凸規(guī)劃。2023/9/2LudongUniversity20凸規(guī)劃及其性質(zhì)定理4.2.6

凸規(guī)劃的任一局部最優(yōu)解都是它的整體最優(yōu)解。2023/9