Nonlinear Inequality Constraints Example Example max x 1 + 2 - PowerPoint PPT Presentation

Nonlinear Inequality Constraints Example

Example max x 1 + ≤ 2 2 s.t. x x 1 1 2 − − ≤ 3 ( 1 ) 0 x x 1 2 + x ≤ − − ≤ 2 2 3 ( x 1 ) x 0 x 1 1 2 1 2

Convert to standard form − min x 1 − − + ≥ 2 2 s.t. x x 1 0 1 2 − − + ≥ 3 ( x 1 ) x 0 1 2 ⎡ ⎤ [ ] 1 = λ = x * 1 , 0 * , 0 ⎢ ⎥ ⎣ ⎦ 2 Primal feasibility satisfied Both constraints active λ = − − λ − − − − λ − − + 2 2 3 Dual Feasibility: L ( x , ) x ( x x 1 ) ( ( x 1 ) x ) 1 1 1 2 2 1 2

Check FONC ∇ f ( x ) ⎡− − ⎡ ⎤ ⎡ ⎤ − − ⎤ 2 1 2 x 3 ( x 1 ) ∇ λ = − λ − λ 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ L ( x , ) − x 1 2 ⎣ ⎦ ⎣ ⎦ 0 2 x ⎣ ⎦ 1 2 Solution solves KKT cond ⎡ ⎤ [ ] 1 = λ = Assuming two constraints active x * 1 , 0 * , 0 ⎢ ⎥ ∂ L ⎣ ⎦ = − + λ + λ − = 2 2 1 2 x 3 ( x 1) 0 ∂ 1 1 2 1 x 1 ∂ λ ∂ L ( x *, *) L = − + + − = 2 = + λ − λ = 1 2 ( 1 )( 1 ) ( 3 )( 0 )( 1 1 ) 0 0 2 x 0 ∂ 2 x ∂ 1 2 2 x 1 2 ∂ λ L ( x *, *) = + − = − − + = 2 2 0 2 ( 1 )( 0 ) 0 0 x x 1 0 ∂ 2 1 2 x 2 − − + = 3 ( x 1) x 0 λ * ≥ 1 2 0 λ λ ≥ , 0 1 2

Complementarity λ = g i ( x ) 0 Complementarity: i = g ( x ) 0 1 thus complementarity holds = g ( x ) 0 2 λ = For inactive constraints, 0 i

More FONC, SONC Let’s check necessary conditions Is x* regular, i.e. are the gradients of the active constraints linearly independent? ⎡ ⎤ ⎡− ∇ ⎤ T 2 0 g ( x ) = 1 ⎢ ⎥ ⎢ ⎥ ∇ T ⎣ ⎦ ⎣ ⎦ 0 1 ( ) g x 2 So x* is regular, and x* is a KKT point, so FONC are satisfied. The null space is empty so ∇ λ 2 T p.s.d. is vacuously satisfied. SONC: Z L ( x , ) Z xx

A + ,Z + ,etc. [ ] = − − A 2 x 2 x + 1 2 Only first constraint is active And non-degenerate (positive multiplier) [ ] = − 2 0 ⎡− − ⎡ ⎤ ⎤ ⎡ ⎤ − − 2 1 2 x 3 ( x 1 ) ⎡ ⎤ ∇ λ = − λ − λ 1 1 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ L ( x , ) − x 1 2 = ⎣ ⎦ ⎣ ⎦ 0 2 x ⎣ ⎦ 1 ⎢ ⎥ Z 2 + ⎣ ⎦ 1 λ + λ − ⎡ ⎤ 2 6 ( x 1 ) 0 ∇ λ = 2 1 2 1 L ( x , ) ⎢ ⎥ xx ⎣ ⎦ 0 1 ⎡ ⎤ 1 0 = ⎢ ⎥ ⎣ ⎦ 0 1

Conclusion Just showed that x*, λ * is a KKT point. + ∇ λ T 2 Let’s check SOSC: Z L ( x , ) Z is positive definite + xx The general Jacobian of this problem for the active constraints: ⎡ ⎤ − − ∇ ⎡ ⎤ T 2 x 2 x g ( x ) = 1 2 1 ⎢ ⎥ ⎢ ⎥ − − ∇ 2 T ⎣ ⎦ ⎣ ⎦ 3 ( x 1 ) 1 g ( x ) 1 2 We only need non-degenerate active constraints ⎡ ⎤ ⎡ ⎤ 1 0 0 [ ] ∇ λ = = > T 2 ( , ) 0 1 ⎢ ⎥ ⎢ ⎥ 1 0 Z L x Z + + xx ⎣ ⎦ ⎣ ⎦ 0 1 1 so SOSC satisfied; x* is a strict local minimizer