RelativeResourceManager,Lec11

# RelativeResourceManager,Lec11 - A PART IIOPTIMIZATION...

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P ART II O PTIMIZ A L ECTURE 11 TION C ONSTRAINED O PTIMIZATION I CE 491/591 Transportation Systems Analysis The University of Alabama

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2/10/ 2 R ECAP : U NCONSTRAINED O PTIMIZATION 010 Trans p Mathematical Expression ortation Syst e   n R x x f , ) ( min Optimality Condition Necessary (first order) ms Analysis * y ( ) Sufficient (second order) cal and Global Optimum The Universi t 0 ) ( x f 0 ) ( * 2 x f Local and Global Optimum First and second order optimality constraints only arantee local optimality y of Alabama guarantee local optimality Global optimum is hard to find Yingyan Lou 2
2/10/ 2 R ECAP : U NCONSTRAINED O PTIMIZATION 010 Trans p General Solution Procedure Step 1: find the stationary points ortation Syst e 0 ) ( * x f Step 2: examine whether the sufficient condition is met ms Analysis 0 ) ( * 2 x f Newton’s Method The Universi t ) ( ' 1 n n n x f x x ) ( ' x f Easy to implement, converges very quickly y of Alabama ) ( ' ' n x f * x Only finds stationary points Solution largely depends on initial guess Yingyan Lou 0 x 1 x 2 x x gy p g 3

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2/10/ 2 R ECAP : C ONVEXITY 010 Trans p Convex Function ortation Syst e ms Analysis Nice Property The Universi t First order (necessary) condition is also sufficient A local optimal solution must be a global optimum y of Alabama Newton’s method finds true optimum Excel solver finds true optimum Yingyan Lou p Signal timing example in Lecture 07 4
2/10/ 2 C ONSTRAINED O PTIMIZATION 010 Trans p General Mathematical Format (Standard Format in This Course) ortation Syst e ms Analysis x f n ) ( min Objective function The Universi t m i x h i R x , , 1 , 0 ) ( s.t. m EC y

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RelativeResourceManager,Lec11 - A PART IIOPTIMIZATION...

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