Optimization in Scilab

The free and open source software for numerical computation
Optimization with Scilab
June 29th
2011
Michaël BAUDIN & Vincent COUVERT
Scilab Consortium

The free and open source software for numerical computationThe free software for numerical computation
Part 1 - What's new in Scilab 5 ?
Focus on the Nelder-Mead component
Part 2 - Optimization in Scilab: Matlab® compatibility
Part 3 - OMD2 project: Scilab Platform Development
Part 4 - Conclusion
What is missing in Scilab ?
Outline

1. Introduction
1.1 What's in Scilab ?
1.2 What's new in Scilab v5 ?
1.3 What's new on Atoms ?
Part 1 – What's new in Scilab 5 ?
2. The Nelder-Mead Component
2.1 Introduction
2.2 The algorithm
2.3 Test cases
2.4 Conclusions

Objective Bounds Equality Inequalities Size Gradient
Needed
Solver
Linear yes linear linear medium - linpro
Quadradic yes linear linear medium - quapro
Quadratic yes linear linear large - qpsolve
Quadratic yes linear linear medium - qld
Non-Linear yes large yes optim
Non-Linear small no fminsearch
Non-Linear yes small no neldermead
Non-Linear yes small no optim_ga
Non-Linear small no optim_sa
N.Li.Lea.Sq. large optional lsqrsolve
N.Li.Lea.Sq. large optional leastsq
Min-Max yes medium yes optim/nd
Multi-Obj yes small no optim_moga
Semi-Def. lin. (spectral) large no semidef
L.M.I. lin. (spectral) lin. (spectral) large no lmisolve

● Genetic Algorithms: nonlinear objective, bounds, global optimization
● Simulated Annealing: nonlinear objective, global optimization
● The Nelder-Mead component: nonlinear objective, unconstrained,
derivative-free, local optimization
● fminsearch: Matlab® compatible
1.2 What's new in Scilab 5 ?

● Optimization Solvers:
– Quapro: linear or quadratic objective, linear constraints (full
matrices),
– SciIpopt: an interface to Ipopt. Nonlinear objective, nonlinear
constraints (beta version),
– Fmincon: nonlinear objective, nonlinear constraints (alpha
version) – Matlab® compatible,
– Other modules: Cobyla, Particle Swarm Optimization,
Optkelley, …
● Test Problems:
– Uncprb: 35 unconstrained optimization problems,
– AMPL: load AMPL problems into Scilab,
– And also: CUTEr.
1.3 What's new on ATOMS ?

Outline
1. Introduction
1.2 What's new in Scilab v5 ?
1.3 What's new on Atoms ?
2.1 Introduction
2.2 The algorithm
2.3 Test cases
2.4 Conclusions

John Ashworth Nelder (8 October 1924 – 7 August 2010)
Source: http://www.guardian.co.uk/technology/2010/sep/23/john-nelder-obituary
2.1 Introduction

● We are interested in solving the unconstrained continuous
optimization problem:
Minimize f(x)
with unbounded, real, multidimensional, variable x.
● A direct search algorithm:
– Uses only function values (no gradient needed),
– Does not approximate the gradient.
● « A simplex method for function minimization », John Nelder, Roger
Mead, Computer Journal, vol. 7, no 4, 1965, p. 308-313
2. The Nelder-Mead Algorithm
2.1 Introduction

2.1 Introduction
Virginia Torczon (1989) writes:
"Margaret Wright has stated that over fifty percent of the calls
received by the support group for the NAG software library
concerned the version of the Nelder-Mead simplex algorithm to be
found in that library."

A simplex: a set of n+1 vertices, in n dimensions.
In 2 dimensions.
In 3 dimensions.
2.2 The algorithm

● Steps in the Nelder-Mead algorithm
● Inputs:
– the n+1 vertices v(1), v(2), ..., v(n+1) of a nondegenerate
simplex in n dimensions,
– the associated function values f(1),...,f(n+1),
– the coefficients ρ (reflection), χ (expansion), γ (contraction),
and σ (shrinkage).
● Standard Nelder-Mead: ρ=1, χ=2, γ=1/2, and σ=1/2.
2.2 The algorithm

2.2 The algorithm

f x1, x2=x1
2
x2
2
−x1 x2
function [ y , index ] = quadratic ( x , index )
y = x(1)^2 + x(2)^2 - x(1) * x(2);
endfunction
nm = neldermead_new ();
nm = neldermead_configure(nm,"-numberofvariables",2);
nm = neldermead_configure(nm,"-function",quadratic);
nm = neldermead_configure(nm,"-x0",[2 2]');
nm = neldermead_search(nm);
xopt = neldermead_get(nm,"-xopt");
nm = neldermead_destroy(nm);
2.3 Test cases

2.3 Test cases

● Mc Kinnon, « Convergence of the neldermead simplex method to a nonstationary
point ». SIAM J. on Optimization, 1998
● Failure by repeated inside contraction
2.3 Test cases

● C. T. Kelley. « Detection and remediation of stagnation in the neldermead algorithm
using a sufficient decrease condition » SIAM J. on Optimization, 1999
● Restart the algorithm...
2.3 Test cases

● Some general facts:
– Memory requirement is O(n²)
– Shrink steps are rare
– Generally 1 or 2 function evaluations by iteration
– Convergence is slow. Typical number of iterations is 100n,
where n is the number of dimensions
– Hundreds of iterations are not rare
– Convergence can be even slower when n > 10 (Han &
Neumann, 2006)
– Restart the algorithm when in doubt for convergence (Kelley,
1999)
– Convergence is guaranteed in 1 dimension (Lagarias et al.,
1999)
2.4 Conclusions

● We should not use this algorithm just because the gradient is not
required:
– For example, if f is smooth, Quasi-Newton methods (optim)
with numerical derivatives converge much faster.
● We may use this algorithm when:
– No other property of the problem can be used (e.g. non linear
least squares can be solved by lsqrsolve),
– The objective function is nonsmooth or "noisy" (Kelley, 1999),
– We do not need too much accuracy (Torzcon, 1989),
– The number of parameters is moderate (Han & Neumann,
2006).
2.4 Conclusions

1. Introduction
2. Scilab Coverage
3. Overview
Part 3 –
Optimization in Scilab: Matlab® compatibility

● Matlab® has many functions for optimization:
– Minimization,
– Equation solving,
– Datafitting and nonlinear least squares,
– Global optimization.
● Scilab has often similar functions: let's see which ones.
Matlab is a registered trademark of The Mathworks, Inc.
1. Introduction

● For each Matlab® function, we search:
– Scilab function, if available,
– Differences of features, differences of algorithms.
● (*) : Function will be reviewed at the end of the talk,
● For most functions, the match is not 100% identical,
– But some other functions can do it : which ones ?
● We consider only Scilab Industrial Grade solvers:
– Scilab internal modules,
– ATOMS modules,
– Portables on all OS,
– Well documented,
– Tested.
1. Introduction

1. Introduction
Main differences
● Design:
– Matlab®: problem oriented (may be with several solvers),
– Scilab: solver oriented (may be several solvers).
● Function arguments:
– Matlab® nearly always provides common options,
– Scilab is less homogeneous.
● Management of the callbacks/extra-arguments:
– Matlab®: M-file or @
– Scilab: list

● Minimization:
– fminbnd Not 100% identical,
But optim can do it.
– fmincon ATOMS/fmincon (alpha version)
– fminimax Not 100% identical,
but optim/''nd'' is designed for it.
– fminsearch fminsearch 90% identical in Scilab 5.3.2.
fminsearch 99% identical in 5.4.0
2. Scilab Coverage

– fminunc Not 100% identical,
but optim/''qn'' or optim/''gc'' are designed for it.
No sparsity pattern of Hessian in Scilab.
No PCG in optim: L-BFGS instead.
– linprog 100% for full matrices: karmarkar
ATOMS/quapro: linpro
No known solver for sparse matrices (*).
– quadprog 100% for full matrices: qpsolve, qp_solve
ATOMS/quapro: quapro
No known solver for sparse matrices.
2. Scilab Coverage

● Equation Solving:
– fsolve fsolve 100% for full matrices.
No known solver with sparse Jacobian (*).
– fzero No identical function.
But fsolve can do it.
● Least Squares (Curve Fitting):
– lsqcurvefit datafit
– lsqnonlin lsqrsolve (leastsq)
2. Scilab Coverage

● Global Optimization Toolbox:
● Genetic Algorithm Not 100% identical,
But optim_ga is built-in Scilab.
No linear equality and inequality in Scilab,
but bounds are managed.
● Simulated Annealing Not 100% identical,
But optim_sa is built-in Scilab
No bounds in Scilab SA, but user can
customize the neighbour function.
2. Scilab Coverage

Matlab Problem Scilab
bintprog Binary Integer Programming -
fgoalattain Multiobjective goal attainment -
fminbd Single-variable, on interval optim
fmincon Constrained, nonlinear, multivariable ATOMS/fmincon
fminimax Minimax, constrained optim/''nd''
fminsearch Unconstrained, multivariable, derivative-free fminsearch (100%)
fminunc Unconstrained, multivariable optim/''qn'',''gc''
fseminf Semi-infinitely constrained, multivariable,
nonlinear
-
ktrlink Constrained or unconstrained, nonlinear,
multivariable using Knitro
-
linprog Linear programming karmarkar,
ATOMS/quapro
quadprog Quadratic programming qpsolve,
ATOMS/quapro
3. Overview

Matlab Problem Scilab
fsolve Solve systems of nonlinear equations fsolve
fzero Root of continuous function of one variable -
lsqcurvefi t Nonlinear least squares curve fitting datafit
lsqlin Constrained linear least squares -
lsqnonlin Nonlinear least-squares (nonlinear data-fitting) lsqrsolve, leastsq
lsqnonneg Nonnegative least squares -
optimtool GUI to select solvers, options and run problems -
Global Search Solve GlobalSearch problems -
Multi Start Solve MultiStart problems -
Genetic Algorithm Genetic Algorithms optim_ga
Direct Search Pattern Search -
Simulated Annealing Simulated Annealing optim_sa
3. Overview

Part 4 -
OMD2 project: Scilab Platform Development
1. Overview
2. Modules
2.1 Data Management
2.2 Modeling
2.3 Optimization

● OMD2 / CSDL projects collaboration
● Will be available on Scilab forge:
– http://forge.scilab.org/index.php/p/omd2/
– Private project up to first release.
● Scilab Optimization Platform:
– Batch mode (script edition, large scale execution),
– GUI mode (interactive edition, prototyping).
● Future Scilab external module available through ATOMS.
Overview

● Project management (Save & Load working data as HDF5 files)
● Wrappers:
– Scilab algorithms,
– External tools,
– Proactive.
● Mask complexity for users
● Modules:
– Data Management,
– Modeling,
– Optimization,
– Visualization.
Main functionalities

● Factors / Parameters:
– Load existing Design Of Experiments (Isight .db files, …)
– Generate Design Of Experiments:
● DoE generator wrappers (LHS, …),
● DoE generator settings.
● Responses simulation using:
– External tool (openFOAM, Catia, CCM+, …),
– Scilab function.
● 2-D visualization:
– Factor / Factor,
– Response / Factor.
Data Management Module (1/2)

Data Management Module (2/2)

● Point selection:
– Learning points used for modeling,
– Validation points used to validate model,
– Bad points (simulation issue, …).
● Modeler:
– Selected among modeler wrappers (DACE, Lolimot, …),
– Parameters configuration,
– Multiple model management with best model user selection.
● Visualization:
– 2-D models,
– Cross correlation,
– Sensibility analysis.
Modeling module (1/2)

Modeling module (2/2)

● Responses coefficients values setting
● Optimizer:
– Selection among generic wrappers (optim, fmincon, genetic
algorithms, …),
– Optimizer configuration,
– Enable two chained optimizers.
● Visualization:
– Optimal point,
– Paretos,
– Robustness.
Optimization Module (1/2)

Optimization Module (2/2)

1. What is missing ?
2. Bibliography
Part 4 - Conclusion

● High Performance Optimization:
– Use BLAS/LAPACK within optim ?
● Sparse Linear Programming:
– Update LIPSOL ?
● Non Linear Programming:
– Improve fmincon ?
● Non Linear Programming Test Cases:
– CUTEr requires a compiler on the test machine,
– Connect the Hock-Schittkowski collection ?
Conclusion
1. What is missing ?

● « Nelder-Mead User's Manual », Michaël Baudin, Consortium Scilab
– DIGITEO, 2010
● « Optimization in Scilab », Baudin, Couvert, Steer, Consortium Scilab
- DIGITEO – INRIA, 2010
● « Optimization with scilab, present and future », Michaël Baudin and
Serge Steer, 2009 IEEE International Workshop on Open Source
Software for Scientific Computation, pp.99-106, 18-20 Sept. 2009
● « Introduction to Optimization with Scilab », Michaël Baudin,
Consortium Scilab – DIGITEO, 2010
● « Unconstrained Optimality Conditions with Scilab », Michaël Baudin,
Consortium Scilab – DIGITEO, 2010
Conclusion
2. Bibliography

Thanks for your attention
www.scilab.org

Some slides you won't see, unless you ask...
Extra-Slides

Some Historical References:
● Spendley, Hext, Himsworth (1962): fixed shape simplex algorithm
● Nelder, Mead (1965): variable shape algorithm
● Box (1965): simplex algo., with constraints
● O'Neill (1971): Fortran 77 implementation.
● Torczon (1989): Multi Directional Search.
● Mc Kinnon (1998): Counter examples of N-M.
● Lagarias, Reeds, Wright, Wright (1998): Proof of convergence in
dimensions 1 and 2 for strictly convex functions.
● Han, Neumann (2006): More counter examples of N-M.
The Nelder-Mead Algorithm

In what softwares N-M can be found ?
● Matlab (fminsearch)
● NAG (E04CBF)
● Numerical Recipes (amoeba)
● IMSL (UMPOL)
● … and Scilab since v5.2.0 in 2009
● … and R after Sébastien Bihorel's port of Scilab's source code.

1. Sort by function value. Order the vertices: f(1) ≤ · · · ≤ f(n) ≤ f(n+1)
2. Calculate centroid. B = (v(1)+...+v(n))/n
3. Reflection. Compute R = (1+ρ)B − ρv(n+1) and evaluate f(R).
4. Expansion. If f(R)<f(1), compute E=(1+ρχ)B − ρχv(n+1) and evaluate f(E).
If f(E)<f(R), accept E, else accept R and goto 1.
5. Accept R. If f(1) ≤ f(R) < f(n), accept R and goto 1.
6. Outside Contraction. If f(n)≤f(R)<f(n+1), compute Co=(1+ργ)B − ργv(n+1)
and evaluate f(Co). If f(Co)<f(R), then accept Co and goto 1 else, goto 8.
7. Inside Contraction. If f(n+1)≤f(R), compute Ci=(1-γ)B +γv(n+1) and
evaluate f(Ci). If f(Ci)<f(n+1), then accept Ci and goto 1 else, goto 8.
8. Shrink. Compute the points v(i)=v(1)+σ(v(i)-v(1)) and evaluate
f(i)=f(x(i)), for i=2,3,...,n+1. Goto 1.

● Lagarias, Reeds, Wright, Wright (1998)
1. In dimension 1, the Nelder-Mead method converges to a
minimizer, and convergence is eventually M-step linear, when
the reflection parameter ρ = 1.
2. In dimension 2, the function values at all simplex vertices in the
standard Nelder-Mead algorithm converge to the same value.
3. In dimension 2, the simplices in the standard Nelder-Mead
algorithm have diameters converging to zero.
● Note that Result 3 does not implies that the simplices converge to a
single point x*.

● Minimization:
– bintprog Solve binary integer programming problems
– fgoalattain Solve multiobjective goal attainment problems
– fminbnd Find minimum of single-variable function on
fixed interval
– fmincon Find minimum of constrained nonlinear
multivariable function
– fminimax Solve minimax constraint problem
– fminsearch Find minimum of unconstrained multivariable
function using derivative-free method
What's in Matlab® ?

– fminunc Find minimum of unconstrained multivariable
function
– fseminf Find minimum of semi-infinitely constrained
multivariable nonlinear function
– ktrlink Find minimum of constrained or unconstrained
nonlinear multivariable function using KNITRO
third-party libraries
– linprog Solve linear programming problems
– quadprog Quadratic programming

● Equation Solving:
– fsolve Solve system of nonlinear equations
– fzero Find root of continuous function of one variable
● Least Squares (Curve Fitting):
– lsqcurvefit Solve nonlinear curve-fitting (data-fitting)
problems in least-squares sense
– lsqlin Solve constrained linear least-squares problems
– lsqnonlin Solve nonlinear least-squares problems
– lsqnonneg Solve nonnegative least-squares constraint
problem

● Utilities:
– optimtool GUI to select solver, optimization options, and
run problems
– optimget Optimization options values
– optimset Create or edit optimization options structure

● Global Optimization Toolbox:
– GlobalSearch Create and solve GlobalSearch
problems
– MultiStart Create and solve MultiStart problems
– Genetic Algorithm Use genetic algorithm and Optimization
Tool, and modify genetic algorithm
options
– Direct Search Use direct search and Optimization Tool,
and modify pattern search options
– Simulated Annealing Use simulated annealing and
Optimization Tool, and modify simulated
annealing options
What's in Matlab® toolboxes ?

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