1. General Nonlinear Programming (NLP) Software
CAS 737 / CES 735
Kristin Davies
Hamid Ghaffari
Alberto Olvera-Salazar
Voicu Chis
January 12, 2006

2. Outline Intro to NLP Examination of:
IPOPT
PENNON
CONOPT
LOQO
KNITRO
Comparison of Computational Results
Conclusions

3. Intro to NLP The general problem:
Either the objective function or some of the constraints may be nonlinear

4. Intro to NLP (cont’d…) Recall: However:
The feasible region of any LP is a convex set
if the LP has an optimal solution, there is an extreme point of the feasible set that is optimal
However:
even if the feasible region of an NLP is a convex set, the optimal solution might not be an extreme point of the feasible region

5. Intro to NLP (cont’d…) Some Major approaches for NLP
Interior Point Methods
Use a log-barrier function
Penalty and Augmented Lagrange Methods
Use the idea of penalty to transform a constrained problem into a sequence of unconstrained problems.
Generalized reduced gradient (GRG)
Use a basic Descent algorithm.
Successive quadratic programming (SQP)
Solves a quadratic approximation at every iteration.

6. Summary of NLP Solvers

7. IPOPT SOLVER (Interior Point OPTimizer)
Creators
Andreas Wachter and L.T. Biegler at CMU (~2002)
Aims
Solver for Large-Scale Nonlinear Optimization problems
Applications
General Nonlinear optimization
Process Engineering, DAE/PDE Systems, Process Design and Operations, Nonlinear Model Predictive control, Design Under Uncertainty

8. IPOPT SOLVER (Interior Point OPTimizer)
Input Format
Can be linked to Fortran and C code MATLAB and AMPL.
Language / OS
Fortran 77, C++ (Recent Version IPOPT 3.x)
Linux/UNIX platforms and Windows
Commercial/Free
Released as open source code under the Common Public License (CPL).  
It is available from the COIN-OR repository

9. IPOPT SOLVER (Interior Point OPTimizer)
Key Claims
Global Convergence by using a Line Search.
Find a KKT point
Point that Minimizes Infeasibility (locally)
Exploits Exact Second Derivatives
AMPL (automatic differentiation)
If not Available use QN approx (BFGS)
Sparsity of the KKT matrix.
IPOPT has a version to solve problems with MPEC Constraints. (IPOPT-C)

10. IPOPT SOLVER (Interior Point OPTimizer)
Algorithm
Interior Point method with a novel line search filter.
Optimization Problem
The bounds are replaced by a logarithmic Barrier term. The method solves a sequence of barrier problems for decreasing values of ml

11. IPOPT SOLVER (Interior Point OPTimizer)
Algorithm
(For a fixed value of )
Solve the Barrier Problem
Search Direction (Primal-Dual IP)
Use a Newton method to solve the primal dual equations.
Hessian Approximation (BFGS update)
Line Search (Filter Method)
Feasibility Restoration Phase

12. IPOPT SOLVER (Interior Point OPTimizer)
Optimization Problem
Outer
Loop
The bounds are replaced by a logarithmic Barrier term.
The method solves a sequence of barrier problems for decreasing values of

13. IPOPT SOLVER (Interior Point OPTimizer)
Algorithm (For a fixed value of )
Solve the Barrier Problem
Search Direction (Primal-Dual IP)
Use a Newton method to solve the primal dual equations
Hessian Approximation (BFGS update)

14. IPOPT SOLVER (Interior Point OPTimizer)
Barrier
NLP
Inner
Loop
Optimality conditions
At a Newton's iteration (xk,lk,vk)
At a Newton's iteration (xk,lk,vk)
Algorithm Core: Solution of this Linear system

15. IPOPT SOLVER (Interior Point OPTimizer)
Algorithm (For a fixed value of )
Line Search (Filter Method)
A trial point is accepted if improves feasibility or if improves the barrier function
Assumes Newton directions are “Good” especially when using Exact 2nd Derivatives If

16. IPOPT SOLVER (Interior Point OPTimizer)
Line Search - Feasibility Restoration Phase
When a new trial point does not provides sufficient improvement.
Restore Feasibility
Minimize constraint
violation
Force Unique Solution
Find closest feasible point.
Add Penalty function

17. IPOPT SOLVER (Interior Point OPTimizer)
The complexity of the problem increases when complementarity conditions are introduced from:
The interior point will drive the barrier parameter mu to zero as part of the solution hence, the complementarity constraints are recovered in the limit.
The interior Point method for NLPs has been extended to handle complementarity problems. (Raghunathan et al. 2003).
is relaxed as

18. IPOPT SOLVER (Interior Point OPTimizer)
Additional
IPOPT 3x. Is now programmed in C++.
Is the primary NLP Solver in an undergoing project for MINLP with IBM.
References
Ipopt homepage:
A. Wächter and L. T. Biegler, On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming, Research Report, IBM T. J. Watson Research Center, Yorktown, USA, (March accepted for publication in Mathematical Programming)

19. PENNON (PENalty method for NONlinear & semidefinite programming)
Creators
Michal Kocvara & Michael Stingl (~2001)
Aims
NLP, Semidefinite Programming (SDP), Linear & Bilinear Matrix Inequalities (LMI & BMI), Second Order Conic Programming (SOCP)
Applications
General purpose nonlinear optimization, systems of equations, control theory, economics & finance, structural optimization, engineering

20. SDP (SemiDefinite Programming)
Minimization of a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite
Linear Matrix Inequality (LMI)
defines a convex constraint on x

21. SDP (SemiDefinite Programming)
-always an optimal point on the boundary
-boundary consists of
piecewise algebraic
surfaces

22. SOCP (Second-Order Conic Programming)
Minimization of a linear function subject to a second-order cone constraint
Called a second-order cone constraint since the unit second-order cone of dimension k is defined as:
Which is called the
quadratic, ice-cream,
or Lorentz cone

23. PENNON (PENalty method for NONlinear & semidefinite programming)
Input Format
MATLAB function, routine called from C or Fortran, stand-alone program with AMPL
Language
Fortran 77
Commercial/Free
Variety of licenses ranging from
Academic – single user ($460 CDN) to Commercial – company ($40,500 CDN)

24. PENNON (PENalty method for NONlinear & semidefinite programming)
Key Claims
1st available code for combo NLP, LMI, & BMI constraints
Aimed at (very) large-scale problems
Efficient treatment of different sparsity patterns in problem data
Robust with respect to feasibility of initial guess
Particularly efficient for large convex problems

25. PENNON (PENalty method for NONlinear & semidefinite programming)
Algorithm
Generalized version of the Augmented Langrangian method (originally by Ben-Tal & Zibulevsky)
Augmented Problem
Augmented Lagrangian

26. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm
Consider only inequality constraints from (NLP)
Based on choice of a penalty function, φg, that penalizes the inequality constraints
Penalty function must satisfy multiple properties such that the original (NLP) has the same solution as the following “augmented” problem:
(NLPφ)
[3] Kocvara & Stingl

27. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm (Cont’d…)
The Lagrangian of (NLPφ) can be viewed as a (generalized) augmented Lagrangian of (NLP):
Inequality constraint
Penalty parameter
Lagrange multiplier
Penalty function
[3] Kocvara & Stingl

28. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm STEPS
[3] Kocvara & Stingl

29. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm STEPS
Initialization
Can start with an arbitrary primal variable , therefore, choose
Calculate initial multiplier values
Initial p= , typically between
[3] Kocvara & Stingl

30. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm STEPS
(Approximate) Unconstrained Minimization
Performed either by Newton with Line Search, or by Trust Region
Stopping criteria:
[3] Kocvara & Stingl

31. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm STEPS
Update of Multipliers
Restricted in order to satisfy:
with a positive
If left-side violated, let
If right side violate, let
[3] Kocvara & Stingl

32. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm STEPS
Update of Penalty Parameter
No update during first 3 iterations
Afterwards, updated by a constant factor dependent on initial penalty parameter
Penalty update is stopped if peps (10-6) is reached
[3] Kocvara & Stingl

33. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm
Choice of Penalty Function
Most efficient penalty function for convex NLP is the quadratic-logarithmic function:
[4] Ben-Tal & Zibulevsky

34. PENNON (PENalty method for NONlinear & semidefinite programming)
The Algorithm
Overall Stopping Criteria
[3] Kocvara & Stingl

35. PENNON (PENalty method for NONlinear & semidefinite programming)
Assumptions / Warnings
More tuning for nonconvex problems is still required
Slower at solving linear SDP problems since algorithm is generalized

36. PENNON (PENalty method for NONlinear & semidefinite programming)
References
Kocvara, Michal & Michael Stingl. PENNON: A Code for Convex and Semidefinite Programming. Optimization Methods and Software, 8(3): , 2003.
Kocvara, Michal & Michael Stingl. PENNON-AMPL User’s Guide August 2003.
Ben-Tal, Aharon & Michael Zibulevsky. Penalty/Barrier Multiplier Methods for Convex Programming Problems. Siam J. Optim., 7(2): , 1997.
Pennon Homepage. Available online January 2007.