Inexact SQP Methods for Equality Constrained OptimizationFrank Edward CurtisDepartment of IE/MS, Northwestern University
with Richard Byrd and Jorge Nocedal
November 6, 2006
INFORMS Annual Meeting 2006
Outline Introduction
Problem formulation Motivation for inexactness Unconstrained optimization and nonlinear
equations Algorithm Development
Step computation Step acceptance
Global Analysis Merit function and sufficient decrease Satisfying first-order conditions
Conclusions/Final remarks
Outline Introduction
Problem formulation Motivation for inexactness Unconstrained optimization and nonlinear
equations Algorithm Development
Step computation Step acceptance
Global Analysis Merit function and sufficient decrease Satisfying first-order conditions
Conclusions/Final remarks
Goal: solve the problem
Equality constrained optimization
)()(),( xcxfxL T
0)(s.t.
)(min
xcxf
x ),(:),(
)(:)(
)(:)(
2 xLxW
xcxA
xfxg
xx
i
0)(
)()(
xc
xAxg T
Define: the Lagrangian
Define: the derivatives
Goal: solve KKT conditions
Equality constrained optimization
0s.t.
min 21
Adc
Wdddg TT
d
c
Agd
A
AW TT 0
Algorithm: Newton’s method
Algorithm: the SQP subproblem
Two “equivalent” step computation techniques
Equality constrained optimization
c
Agd
A
AW TT 0 0s.t.
min 21
Adc
Wdddg TT
d
Algorithm: Newton’s method
Algorithm: the SQP subproblem
Two “equivalent” step computation techniques
0A
AW T KKT matrix• Cannot be formed• Cannot be factored
Equality constrained optimization
c
Agd
A
AW TT 0 0s.t.
min 21
Adc
Wdddg TT
d
Algorithm: Newton’s method
Algorithm: the SQP subproblem
Two “equivalent” step computation techniques
0A
AW T KKT matrix• Cannot be formed• Cannot be factored
Linear system solve• Iterative method• Inexactness
Unconstrained optimization
)(min xfx
kx
)()(2kkk xfdxf
Goal: minimize a nonlinear objective
Algorithm: Newton’s method (CG)
Note: choosing any intermediate step ensures global convergence to a local solution of NLP
(Steihaug, 1983)
Note: choosing any step with
and
ensures global convergence
Nonlinear equations
kkkk rxFdxF )()(
0)( xF
kx
)()( kkk xFdxF
Goal: solve a nonlinear system
Algorithm: Newton’s method
10 ,)( kk xFr
(Eisenstat and Walker, 1994)
(Dembo, Eisenstat, and Steihaug, 1982)
Outline Introduction/Motivation
Unconstrained optimization Nonlinear equations Constrained optimization
Algorithm Development Step computation Step acceptance
Global Analysis Merit function and sufficient decrease Satisfying first-order conditions
Conclusions/Final remarks
Equality constrained optimization
0s.t.
min 21
Adc
Wdddg TT
d
c
Agd
A
AW TT 0
Algorithm: Newton’s method
Algorithm: the SQP subproblem
Two “equivalent” step computation techniques
),( kkx Question: can we ensure convergence to a local solution by choosing any step into the ball?
Globalization strategy: exact merit function
… with Armijo line search condition
Globalization strategy
)()()( xcxfx
Step computation: inexact SQP step
)()()( dDxdx
rc
Agd
A
AW TT 0
First attempt
10 ,
c
Ag
r
T
Proposition: sufficiently small residual
1e-8 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1
Success 100% 100% 97% 97% 90% 85% 72% 38%
Failure 0% 0% 3% 3% 10% 15% 28% 62%
Test: 61 problems from CUTEr test set
rc
Agd
A
AW TT 0
First attempt… not robust
10 ,
c
Ag
r
T
Proposition: sufficiently small residual
… not enough for complete robustness We have multiple goals (feasibility and optimality) Lagrange multipliers may be completely off
rc
Agd
A
AW TT 0
Recall the line search condition
Second attempt
rcdgdD T )(
Step computation: inexact SQP step
)()()( dDxdx
rc
Agd
A
AW TT 0
We can show
Recall the line search condition
Second attempt
rcdgdD T )(
Step computation: inexact SQP step
)()()( dDxdx
rc
Agd
A
AW TT 0
We can show
... but how negative should this be?
AdccWdddg
dmmdmredTT
21
)()0()(
AdcWdddgfdm TT 21)(
Quadratic/linear model of merit function
Create model
Quantify reduction obtained from step
0s.t.
min 21
Adc
Wdddg TT
d
rc
Agd
A
AW TT 0
AdcWdddgfdm TT 21)(
Quadratic/linear model of merit function
Create model
Quantify reduction obtained from step
0s.t.
min 21
Adc
Wdddg TT
d
AdccWdddg
dmmdmredTT
21
)()0()(
rc
Agd
A
AW TT 0
Exact case
AdccWdddgdmred TT 21)(
rc
Agd
A
AW TT 0
kx
Exact case
AdccWdddgdmred TT 21)(
rc
Agd
A
AW TT 0
kxExact step minimizes the objective on the linearized constraints
Exact case
AdccWdddgdmred TT 21)(
rc
Agd
A
AW TT 0
kxExact step minimizes the objective on the linearized constraints
… which may lead to an increase in the objective (but that’s ok)
Inexact case
rc
Agd
A
AW TT 0
kx
Option #1: current penalty parameter
10 ,)( cdmred
rc
Agd
A
AW TT 0
kx
Option #1: current penalty parameter
10 ,)( cdmred
rc
Agd
A
AW TT 0
kx
TAg
r
,
:0 ,10Step is acceptable if for
Option #2: new penalty parameter
10 ,)( cdmred
rc
Agd
A
AW TT 0
kx
Option #2: new penalty parameter
10 ,)( cdmred
rc
Agd
A
AW TT 0
kx
c
cr
,
:0 ,10
Step is acceptable if for
Option #2: new penalty parameter
10 ,)( cdmred
rc
Agd
A
AW TT 0
kx
c
cr
,
:0 ,10
Step is acceptable if for
10 ,)1(
21
rc
Wdddg TT
for k = 0, 1, 2, … Iteratively solve
Until
Update penalty parameter Perform backtracking line search Update iterate
Algorithm outline
cdmred
Ag
rT
)(
10 ,
0 ,
0 ,
10 ,
c
cr
rc
Agd
A
AW TT 0
or
Observe KKT conditions
Termination test
10 , )(,1 max
10 , ,1 max
0
feasfeas
optoptT
xcc
gAg
Outline Introduction/Motivation
Unconstrained optimization Nonlinear equations Constrained optimization
Algorithm Development Step computation Step acceptance
Global Analysis Merit function and sufficient decrease Satisfying first-order conditions
Conclusions/Final remarks
The sequence of iterates is contained in a convex set over which the following hold:
the objective function is bounded below the objective and constraint functions and their first and second derivatives are uniformly bounded in norm the constraint Jacobian has full row rank and its smallest singular value is bounded below by a positive constant the Hessian of the Lagrangian is positive definite with smallest eigenvalue bounded below by a positive constant
Assumptions
Sufficient reduction to sufficient decrease
rcdgdD T )(
cWddrcdg TT 21
cd
cWdddD T
2
21)(
10 ,)( cdmred
Taylor expansion of merit function yields
Accepted step satisfies
Intermediate results
0 ,
10 ,
c
cr
10 ,)1(
21
rc
Wdddg TT
cdmred
Ag
rT
)(
0 ,
0 ,
d
is bounded below by a positive constant
is bounded above
is bounded above
Sufficient decrease in merit function
cdxdD 2
),;(
0lim k
Tk
kgZ
0lim2
kkk
cd
cddxx 2);();(
Step in dual space
10 , TTAgAg
|||| c |||| d(for sufficiently small and )
0lim k
kc
0lim k
Tkk
kAg
Therefore,
We converge to an optimal primal solution, and
Outline Introduction/Motivation
Unconstrained optimization Nonlinear equations Constrained optimization
Algorithm Development Step computation Step acceptance
Global Analysis Merit function and sufficient decrease Satisfying first-order conditions
Conclusions/Final remarks
Conclusion/Final remarks Review
Defined a globally convergent inexact SQP algorithm
Require only inexact solutions of KKT system Require only matrix-vector products involving
objective and constraint function derivatives Results also apply when only reduced Hessian of
Lagrangian is assumed to be positive definite Future challenges
Implementation and appropriate parameter values
Nearly-singular constraint Jacobian Inexact derivative information Negative curvature etc., etc., etc….
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