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Inexact SQP Methods for Equality Constrained OptimizationPowerPoint Presentation

Inexact SQP Methods for Equality Constrained Optimization

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### Inexact SQP Methods for Equality Constrained Optimization

Frank Edward Curtis

Department of IE/MS, Northwestern University

with Richard Byrd and Jorge Nocedal

November 6, 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

Equality constrained optimization

Goal: solve the problem

Define: the derivatives

Define: the Lagrangian

Goal: solve KKT conditions

Equality constrained optimization

- Two “equivalent” step computation techniques

Algorithm: Newton’s method

Algorithm: the SQP subproblem

Equality constrained optimization

- Two “equivalent” step computation techniques

Algorithm: Newton’s method

Algorithm: the SQP subproblem

- KKT matrix
- Cannot be formed
- Cannot be factored

Equality constrained optimization

- Two “equivalent” step computation techniques

Algorithm: Newton’s method

Algorithm: the SQP subproblem

- KKT matrix
- Cannot be formed
- Cannot be factored

- Linear system solve
- Iterative method
- Inexactness

Unconstrained optimization

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)

Nonlinear equations

Goal: solve a nonlinear system

Algorithm: Newton’s method

Note: choosing any step with

and

ensures global convergence

(Dembo, Eisenstat, and Steihaug, 1982)

(Eisenstat and Walker, 1994)

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

- Two “equivalent” step computation techniques

Algorithm: Newton’s method

Algorithm: the SQP subproblem

Question: can we ensure convergence to a local solution by choosing any step into the ball?

Globalization strategy

- Step computation: inexact SQP step

- Globalization strategy: exact merit function
… with Armijo line search condition

First attempt… not robust

- Proposition: sufficiently small residual

- … not enough for complete robustness
- We have multiple goals (feasibility and optimality)
- Lagrange multipliers may be completely off

Second attempt

- Step computation: inexact SQP step

- Recall the line search condition

- We can show

... but how negative should this be?

Exact case

Exact step minimizes the objective on the linearized constraints

Exact case

Exact step minimizes the objective on the linearized constraints

… which may lead to an increase in the objective (but that’s ok)

Option #1: current penalty parameter

Step is acceptable if for

Option #2: new penalty parameter

Step is acceptable if for

Option #2: new penalty parameter

Step is acceptable if for

Algorithm outline

- for k = 0, 1, 2, …
- Iteratively solve
- Until
- Update penalty parameter
- Perform backtracking line search
- Update iterate

or

Termination test

- Observe KKT conditions

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

Assumptions

- 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

Sufficient reduction to sufficient decrease

- Taylor expansion of merit function yields

- Accepted step satisfies

Step in dual space

- We converge to an optimal primal solution, and

(for sufficiently small and )

Therefore,

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

- 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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