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Unformatted text preview: Convex Optimization Boyd & Vandenberghe 5. Duality Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized inequalities 51 Lagrangian standard form problem (not necessarily convex) minimize f ( x ) subject to f i ( x ) , i = 1 , . . . , m h i ( x ) = 0 , i = 1 , . . . , p variable x R n , domain D , optimal value p Lagrangian: L : R n R m R p R , with dom L = D R m R p , m p L ( x, , ) = f ( x ) + i f i ( x ) + i h i ( x ) i =1 i =1 weighted sum of objective and constraint functions i is Lagrange multiplier associated with f i ( x ) i is Lagrange multiplier associated with h i ( x ) = 0 Duality 52 Lagrange dual function Lagrange dual function: g : R m R p R , g ( , ) = inf L ( x, , ) x D m p = inf f ( x ) + i f i ( x ) + i h i ( x ) x D i =1 i =1 g is concave, can be for some , lower bound property: if , then g ( , ) p proof: if x is feasible and , then f ( x ) L ( x, , ) inf L ( x, , ) = g ( , ) x D minimizing over all feasible x gives p g ( , ) Duality 53 Leastnorm solution of linear equations minimize x T x subject to Ax = b dual function Lagrangian is L ( x, ) = x T x + T ( Ax b ) to minimize L over x , set gradient equal to zero: x L ( x, ) = 2 x + A T = 0 = x = (1 / 2) A T plug in in L to obtain g : g ( ) = L (( 1 / 2) A T , ) = 1 T AA T b T 4 a concave function of lower bound property : p (1 / 4) T AA T b T for all Duality 54 Standard form LP minimize c T x subject to Ax = b, x dual function Lagrangian is L ( x, , ) = c T x + T ( Ax b ) T x = b T + ( c + A T ) T x L is ane in x , hence b T A T + c = 0 g ( , ) = inf L ( x, , ) = x otherwise g is linear on ane domain { ( , )  A T + c = 0 } , hence concave lower bound property : p b T if A T + c Duality 55 Equality constrained norm minimization minimize x subject to Ax = b dual function g ( ) = inf ( x T Ax + b T ) = b T A T 1 x otherwise where v = sup bardbl u bardbl 1 u T v is dual norm of proof: follows from inf x ( x y T x ) = 0 if y 1 , otherwise if y 1 , then x y T x for all x , with equality if x = 0 if y > 1 , choose x = tu where u 1 , u T y = y > 1 : x y T x = t ( u y ) as t lower bound property: p b T if A T 1 Duality 56 Twoway partitioning minimize x T W x subject to x i 2 = 1 , i = 1 , . . . , n a nonconvex problem; feasible set contains 2 n discrete points interpretation: partition { 1 , . . . , n, ....
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This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.
 Fall '06
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