( x) min. Nonlinear optimization problem without constraints NPP: then. Global minimum of the function f(x)

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1 Objectve fucto f() : he optzato proble cossts of fg a vector of ecso varables belogg to the feasble set of solutos R such that It s eote as: Nolear optzato proble wthout costrats NPP: R f ( ) : R R f f R f ( ) ( ) = f Local u a global u of the fucto f() he vector s local u of the fucto f() the space R f there ests such ope space E R of a pot that E f ( ) f ( ) Atoaly f f ( ) < f ( ) for the there ests strct local u. he vector s global u of the fucto f() the space R f there ests such ope space R of a pot that R f ( ) f ( ) Atoaly f f ( ) < f ( ) for the there ests the strct global u ths pot. Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Nolear optzato proble wthout costrats: Defto. A recto the space R we eote ay -etoal colu vector. Let us assue that there ests a pot R a scalar τ [ ; + ). Ay pot y R Optzato theory a coputg belogg to half strght le whch starts pot recto wll be etere as follows: Lea. Let f : X= R R y= + τ be a fferetable fucto the pot Let us assue that there est a recto for whch: ( ) < X he there ests such σ > that for all τ ( σ ] the followg coto s fulflle f ( +τ) < f ( ). Proof: let us use the property of Gateau fferetal wth the rectoal ervatve efto. Faculty of Electrocs Nolear optzato proble wthout costrats: heore. Let f : X= R R wll be fferetable fucto. If ze fucto f() the for: f ( ) f ( ) X Proof: ot rectly. he pot f ( ˆ) f ( ) For each X f a oly f whe fulflls the followg coto: Optzato theory a coputg the (ˆ) = s ae as a statoary pot. heore. Let f : X = R R wll be cove a fferetable fucto. For the pot ˆ X fucto f() costtutes al value of the fucto: ( ˆ) = It s the ecessary coto for local etreu f() the pot. X Faculty of Electrocs Global u of the fucto f() Suffcat cotos for the olear optzato proble wthout costrats: Fucto f() s cotuous a twce fferetable. he fucto f() has the atr wth seco ervatves ae as hesja A heore: Matr A has soe a subeterats A If the fucto f : X = R R wll be strctly cove a fferetable the the vector ˆ X whch fulflls the ecessary coto ( ˆ) = s the oly oe global u of the fucto f(). A = ( ) ( ) A =.. ( ).... ( ) ( ) ( ) A = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs

2 he statoary cotos for olear prograg proble wthout costrats Let us assue that s a statoary pot for a fuctof(). he the followg relatos are fulflle:. If the hesja atr A s postve efe : A( ) > for = the a fucto f() has local u ths pot.. If the hesja atr A s egatve efe : ( ) A( ) > for = the a fucto f() has local au ths pot. 3. If the hesja s half-postve efe: or the hesja s half-egatve efe: A( ) for = a A( ) = ( ) A( ) for = a A( ) = heore. If the fucto f() s twce fferetable the each local u for the olear prograg proble wthout costrats the followg optalty cotos are fulflle: = A ( ) > A= f ( ) for I orer coto II orer coto Matr A has to be strctly postve I ths oet t s possble to ece what type of a etreu of a fucto f( s ths pot. 4. If two cotos a are ot fulflle (the the hesja A s ot etere) the the fucto f() has ot etreu the pot. Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Nolear prograg proble wth costrats NPP : Eaple of NP proble Let us f where: such that: f ( ˆ) = f ( ) X = { : g ( ) = } f : X = R g : X = R X R R. f ( ) = ( X 3 ) + 4 ( ) + ( ) X = {( ) ( ) (.4+ ) 3 3/ + ( 3 3 = )}. Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Nolear prograg proble wth costrats NPP : Defto. he recto startg the pot s feasble f there ests such σ > that for ay τ [ ; σ ] + τ X he set of all feasble rectos: D( ) = { : σ > such that τ [; σ ] + τ X }. Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs

3 Lagrage fucto Actve costrats: Defto. he Lagrage fucto for olear prograg proble s a scalar fucto as follows:. g (ˆ + τ) A(ˆ) where τ [ σ ] L ( λ) = f ( ) + λ g( ). g (ˆ + τ) = g (ˆ) + τ g (ˆ) + O( τ ) where λ R s a vector of Lagrage ultplers. 3. he ecessary coto for the feasblty of recto vector s: g ( ˆ) A(ˆ) Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Farkas lea he set of rectos: Let us efe the set of -esoal vectors R space { b a = }. he relato set up for each R b whch fulflls a b+ = λ a =. λ = [ λ ] λ D ( ) = : <g( ) > A( ) < ( ) D ( ) = : <g( ) > A( ) < ( ) > >< D3( ) = : <g ( ) >> forsoe A( ) ( 3 D ) = D ( ) D ( ) D ( ) Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Kuh-ucker-Karusch ecessary cotos: Proof: ecessary cotos Kuh-ucker-Karusch heore: If a)fuctos ˆ Optzato theory a coputg f g are fferetable; b) s a local u of NPP there ests λ ˆ ˆ λ =. such that f a oly f (ˆ) + g (ˆ) = (ˆ) = = = D ( ˆ) =φ Faculty of Electrocs Proof. Let Optzato theory a coputg a a =g ( ˆ) A(ˆ) he accorg to the Farkas lea there ests Such that whe A the coto for the actve costrats has to be fulflle : g ˆ) f a oly f b= (ˆ) (ˆ) = ˆ λ ( g (ˆ)) A( ˆ ) (ˆ ) R ( A(ˆ) D ( ˆ) =φ ˆ λ A(ˆ) Faculty of Electrocs

4 Proof: ecessary cotos Kuh-ucker-Karusch Kuh-ucker_Karush ecessary cotos wth the help of Lagraga fucto For A(ˆ) let us assue ˆ = So two followg equatos are fulflle: (ˆ) + = λ g (ˆ) = (ˆ) = = Ck L ( λ ) = f ( ) + λ Necessary cotos : L = g ( ˆ ) = λ ˆ L(ˆ ) = λ λ L ( ˆ ) ˆ λ ( ). Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Suffcet cotos efe as the regularty coto:. All fuctos for the cotrats g () are lear the Karl regularty coto... All fuctos for the cotrats g () are cove a the feasble set of solutos s ot epty - the Slater regularty coto. 3. he graets of all actve costrats g ( ) for A) are leary epeet Facco a McCorck regularty coto. ( Kuh-Karusch-ucker theore ecessary a suffcet cotos for olear prograg proble wth costrats (NPP) heore. If a)fuctos f ag are fferetable; b) s local u of NPP ˆ c) Regularty coto for costrats s fulflle the pot ˆ he there ests =. Such that the pot the Kuh-Karusch-ucker ecessary cotos are fulflle: (ˆ) + ˆ λg (ˆ) = = (ˆ) = = Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Eaple I Nolear prograg proble wth costrats NPP wth atoal costrats for varables: akg uer coserato the K-K- cotos solve the followg olear optzato proble: f ( ) = X he solutos of NPP proble: + * +.5* + X = a f (ˆ) f ( ) X = { : g ( ) = } f ( ) : R R g ( ) : R R = = X where : = [ ] f ( ) = L ( λ ) = f ( ) + λ = g ( ). Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs

5 L λ L λ = Lagrage cotos for NNP wth λl λ L λ = λ λ λ Eaples for Kuh-ucker-Karusch cotos Eaple II akg uer coserato the K--K cotos solve the followg olear optzato proble: f ( ) = * X + * +.5* + X = he solutos of NPP proble: = [ ] f ( ) = Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Nolear prograg proble wth costrats NPP wth atoal equatos as costrats : where : X = { : g ( ) = p h ( ) = = p+ } Where: f ( ) : X = R R g ( ) : X = R R = p h ( ) : X = R R = p+ = X f ( ) f ( ) L ( λ ) = f ( ) + λ = g ( ). heore: If a) Fuctos f a g for = are fferetable; b) s local u of NPP ˆ he there ests: λ = p a λ = p+ such that the pot are fulflle: (ˆ) + = (ˆ) = ˆ wthout ay costrats o varables the Kuh-Karusch-ucker ecessary cotos g (ˆ) = = p Optzato theory a coputg Faculty of Electrocs Optzato theory a coputg Faculty of Electrocs Eaple III akg uer coserato the K-K- cotos solve the followg olear optzato proble: f ( ) = + + X he solutos of NPP proble: 4 X = = = [ ] ; λ= f ( ) = 4 Optzato theory a coputg Faculty of Electrocs