Newton Type Constrained Optimization in a Nutshell Moritz Diehl - PowerPoint PPT Presentation

Newton Type Constrained Optimization in a Nutshell Moritz Diehl Optimization in Engineering Center (OPTEC) K.U. Leuven

Overview • Equality Constrained Optimization • Optimality Conditions and Multipliers • Newton’s Method = SQP • Inequality Constraints • Constrained Gauss Newton Method • Relation to Optimal Control

General Nonlinear Program (NLP) In direct methods, we have to solve the discretized optimal control problem, which is a Nonlinear Program (NLP) � G ( w ) = 0 , min w F ( w ) s.t. H ( w ) ≥ 0 . We first treat the case without inequalities. w F ( w ) s.t. min G ( w ) = 0 ,

Lagrange Function and Optimality Conditions Introduce Lagrangian function L ( w, λ ) = F ( w ) − λ T G ( w ) Then for an optimal solution w ∗ exist multipliers λ ∗ such that ∇ w L ( w ∗ , λ ∗ ) = 0 , G ( w ∗ ) = 0 ,

Newton’s Method on Optimality Conditions How to solve nonlinear equations ∇ w L ( w ∗ , λ ∗ ) = 0 , G ( w ∗ ) = 0 , ? Linearize! + ∇ 2 ∇ w L ( w k , λ k ) w L ( w k , λ k )∆ w −∇ w G ( w k )∆ λ = 0 , + ∇ w G ( w k ) T ∆ w G ( w k ) = 0 , This is equivalent, due to ∇L ( w k , λ k ) = ∇ F ( w k ) −∇ G ( w k ) λ k , with the shorthand λ + = λ k + ∆ λ , to ∇ w F ( w k ) + ∇ 2 w L ( w k , λ k )∆ w −∇ w G ( w k ) λ + = 0 , + ∇ w G ( w k ) T ∆ w G ( w k ) = 0 ,

Newton Step = Quadratic Program Conditions ∇ w F ( w k ) + ∇ 2 w L ( w k , λ k )∆ w −∇ w G ( w k ) λ + = 0 , + ∇ w G ( w k ) T ∆ w G ( w k ) = 0 , are optimality conditions of a quadratic program (QP), namely: ∆ w ∇ F ( w k ) T ∆ w +1 2∆ w T A k ∆ w s.t. G ( w k ) + ∇ G ( w k ) T ∆ w min = 0 , with A k = ∇ 2 w L ( w k , λ k )

Newton’s Method The full step Newton’s Method iterates by solving in each iteration the Quadratic Progam ∆ w ∇ F ( w k ) T ∆ w +1 2∆ w T A k ∆ w s.t. G ( w k ) + ∇ G ( w k ) T ∆ w min = 0 , w L ( w k , λ k ) . This obtains as solution the step ∆ w k and with A k = ∇ 2 QP = λ k + ∆ λ k . the new multiplier λ + Then we iterate: w k + ∆ w k w k +1 = λ k + ∆ λ k = λ + λ k +1 = QP This Newton’s method is also called “Sequential Quadratic Program- ming (SQP) for equality constrained optimization” (with “exact Hes- sian” and “full steps”)

NLP with Inequalities Regard again NLP with both, equalities and inequalities: � G ( w ) = 0 , w F ( w ) s.t. min H ( w ) ≥ 0 . Introduce Lagrangian function L ( w, λ, µ ) = F ( w ) − λ T G ( w ) − µ T H ( w )

Optimality Conditions with Inequalities THEOREM (Karush-Kuhn-Tucker (KKT) conditions) For an optimal solution w ∗ exist multipliers λ ∗ and µ ∗ such that ∇ w L ( w ∗ , λ ∗ , µ ∗ ) = 0 , G ( w ∗ ) = 0 , H ( w ∗ ) ≥ 0 , ≥ 0 , µ ∗ H ( w ∗ ) T µ ∗ = 0 , These contain nonsmooth conditions (the last three) which are called “complementarity conditions”. This system cannot be solved by New- ton’s Method. But still with SQP ...

Sequential Quadratic Programming (SQP) By Linearizing all functions within the KKT Conditions, and setting λ + = λ k + ∆ λ and µ + = µ k + ∆ µ , we obtain the KKT conditions of a Quadratic Program (QP) (we omit these conditions). This QP is G ( w k ) + ∇ G ( w k ) T ∆ w � ∆ w ∇ F ( w k ) T ∆ w +1 = 0 , 2∆ w T A k ∆ w s.t. min H ( w k ) + ∇ H ( w k ) T ∆ w ≥ 0 , with A k = ∇ 2 w L ( w k , λ k , µ k ) and its solution delivers λ + µ + ∆ w k , QP , QP

Constrained Gauss-Newton Method In special case of least squares objectives F ( w ) = 1 2 � R ( w ) � 2 2 can approximate Hessian ∇ 2 w L ( w k , λ k , µ k ) by much cheaper A k = ∇ R ( w ) ∇ R ( w ) T . Need no multipliers to compute A k ! QP= linear least squares: G ( w k ) + ∇ G ( w k ) T ∆ w � 1 = 0 , 2 � R ( w k )+ ∇ R ( w k ) T ∆ w � 2 min 2 s.t. H ( w k ) + ∇ H ( w k ) T ∆ w ≥ 0 , ∆ w Convergence: linear (better if � R ( w ∗ ) � small)

Discrete Time Optimal Control Problem N − 1 � l i ( s i , q i ) + E ( s N ) minimize s, q i =0 subject to s 0 − x 0 = 0 , (initial value) s i +1 − f i ( s i , q i ) = 0 , i = 0 , . . . , N − 1 , (discrete system) (path constraints) h i ( s i , q i ) ≥ 0 , i = 0 , . . . , N, r ( s N ) ≥ 0 . (terminal constraints) Can arise also from direct multiple shooting parameterization of con- tinous optimal control problem. This NLP can be solved by SQP or Constrained Gauss-Newton method.

Summary • Nonlinear Programs (NLP) have nonlinear but differentiable problem functions. • Sequential Quadratic Programming (SQP) is a Newton type method to solve NLPs that • solves in each iteration a Quadratic Program (QP) • obtains this QP by linearizing all nonlinear problem functions • an important SQP variant is the Constrained Gauss-Newton Method • SQP can be generalized to Sequential Convex Programming (SCP) • Discrete time optimal control problems are a special case of NLPs.

Literature • J. Nocedal and S. Wright: Numerical Optimization, Springer, 2006 (2nd edition)