Solving minimax problems with feasible sequential quadratic - PowerPoint PPT Presentation

Qianli Deng (Shally) E-mail: dqianli@umd.edu Advisor: Dr. Mark A. Austin Department of Civil and Environmental Engineering Institute for Systems Research University of Maryland, College Park, MD 20742 E-mail: austin@isr.umd.edu Solving minimax problems with feasible sequential quadratic programming 12/13/2013

Background � Feasible sequential quadratic programming (FSQP) refers to a class of sequential quadratic programming methods. � Engineering applications: number of variables is not large; evaluations of objective or constraint functions and of the gradients are time consuming � Advantages: � 1) Generating feasible iterates. � 2) Reducing the amount of computation. � 3) Enjoying the same global and fast local convergence properties. � ����������

Background � The constrained mini-max problem  ≤ ≤ bl x bu * Bounds  ≤ = g ( ) x 0, j 1,..., n  * Nonlinear inequality j i  ≡ − ≤ = + g ( ) x c , x d 0, j n 1,..., t  * Linear inequality − − j j n j n i i i i  = = ( ) 0, 1,..., h x j n  * Nonlinear equality j e  ≡ − = = + ( ) , 0, 1,..., h x a x b j n t  * Linear equality − − j j n j n e e e e � ����������

Algorithm - FSQP * Step 1. Initialization * Step 2. A search arc * Step 3. Arc search * Step 4. Updates � ����������

Initialization ( x, H, p, k ) Initial guess x 00 n − ∑ e { } Positive penalty parameters = f ( , x p ) max f x ( ) p h ( ) x m i j j f ∈ i I = j 1 � ����������

Computation of a search arc � ����������

Computation of a search arc  η  1 0 1 0 1 − − + γ min d d d , d  0 0 0 + min , '( , , ) d H d f x d p  k k 1 ℝ n ℝ 2 ∈ γ ∈ d , k k k  0 2 d  1  ≤ + ≤ s t . . bl x d bu  0 ≤ + ≤ k bl x d bu   . . k st 1 ≤ γ f '( x d , , p )   k k 0  + ∇ ≤ g ( x ) g ( x ), d 0, = j 1,..., t  j k j k 1  + ∇ ≤ γ = g ( x ) g ( x ), d j 1,..., n i j k j k i   0 = + ∇ ≤ j 1,..., n h x ( ) h x ( ), d 0, e 1 j k j k  + ≤ = −  c x , d d j 1,..., t n j k j i i = − j 1,..., t n   0 + = a x , d b e e  1  + ∇ ≤ γ = ( ) ( ), 1,..., j k j h x h x d j n j k j k e  1  + = = − a x , d b j 1,..., t n  j k j e e 0 κ 0 κ ρ = + || d || /(|| d || v ) k k k k 1 τ 1 = v max(0.5,|| d || ) k k � ����������

Quadratic programming Strictly convex quadratic programming: Unique global minimum • Matrix C need to be positive definite • m = number of constraints; n = number of variables × = X [ x x , ,..., x ]' n 1 1 2 n A ; B ; C ; D × × × × m n m 1 n n n 1 Extended Wolfe’s simplex method No derivative • � ���������� Reference: Wolfe, P. (1959). The simplex method for quadratic programming. Econometrica: Journal of the Econometric Society, 382-398.

Quadratic programming Lagrangian function: Karush-Kuhn-Tucker conditions:  ∂ L  = − ≤ + υ = AX B 0 AX B  ∂ λ   ∂ + λ − µ + = − ' ' CX A s D   L  = + + λ ≥ + + λ ≥ X C ' D ' A 0; CX D A ' ' 0  ∂  λυ = X 0   ∂  L µ = ' X 0 λ = λ − = ( AX B ) 0   ∂ λ    ≥ λ ≥ µ ≥ υ ≥ ≥ X 0, 0, 0, 0, s 0 ∂ L  = + + λ = X ' X CX '( D A ' ') 0  ∂ X � = slack variables; � = surplus variables s = artificial variables ( � , � ) and ( � ’ , X ) = complementary slack variables � ����������

Linear programming Quadratic programming: => Conditioned linear programming: Simplex tableau: � ��� � are basis variables �� ����������

Test example �� ����������

Arc search δ = '( , , ) f x d p k k k k �� ����������

Updates ( x, H, p, k ) ɶ 2 + = + + x x t d t d 1 k k k k k k  + µ ≥ ε p p =  k j , k j , j 1 p + k 1, j ε − µ δ max{ , } p otherwise  1 j k j , = + k k 1 �� ����������

Updates BFGS formula with Powell’s modification: f ( x , p ) H + is the Hessian of the Lagrangian function k 1 m k k η = − x x + + k 1 k 1 k γ = ∇ −∇ f ( x , p ) f ( x , p ) + + k 1 x m k 1 k x m k k  T T η γ ≥ η η 1, 0.2 H k + 1 k + 1 k + 1 k k + 1  =  ξ = θ ⋅ γ + − θ ⋅ δ (1 ) H θ T η η 0.8 H + + + + + k 1 k 1 k 1 k 1 k k 1 + k 1 + 1 + 1 k k k  otherwise T T  η η − η γ H + + + + k 1 k k 1 k 1 k 1 T T η η ξ ξ H H + + + + = − k k 1 k 1 k + k 1 k 1 H H + k 1 k T T η η η ξ H + + + + k 1 k k 1 k 1 k 1 �� ����������

Project Schedule • Literature review; October • Specify the implementation module details; • Structure the implementation; • Develop the quadratic programming module; • Unconstrained quadratic program; November • Strictly convex quadratic program; • Validate the quadratic programming module; • Develop the Gradient and Hessian matrix calculation module; December • Validate the Gradient and Hessian matrix calculation module; • Midterm project report and presentation; �� ����������

• Develop Armijo line search module; January • Validate Armijo line search module; • Develop the feasible initial point module; February • Validate the feasible initial point module; • Integrate the program; • Debug and document the program; March • Validate and test the program with case application; ɶ • Add arch search variable in; d April • Compare calculation efficiency of line search with arch search methods; • Develop the user interface if time available; May • Final project report and presentation; �� ����������

Bibliography �� ����������