Global Optimization Software. Doron Pearl Jonathan Li Olesya Peshko Xie Feng . What is global optimization?. Global optimization is aimed at finding the best solution of constrained optimization problem which (may) also have various local optima. .
Global optimization is aimed at finding the
best solution of constrained optimization
problem which (may) also have various
global min f(x), subject to the constraint x Є D
robust set : the closure of its nonempty interior.
No single optimization package can
solve all global optimization
-Branch and bound
Actually, We can further classify deterministic class into two different sub-
Such as ..Baron
Such as ..LGO( Lipschitz Global Optimization ).
then in second class.
2. Even though LGO is regard as using deterministic way to solve the problem,
the solution isn’t always guaranteed to be “deterministic” global optimal.
3. There are some more solvers in first class won’t be discussed in detail, but
in later slides, they will be included in comparison.
represents a ‘simple’ explicit constraint set:
frequently, it’s a finite n-dimensional interval or simplex, or Rn.
Furthermore, the objective function and constraint functions
are Lipschitz-continuous on D0.That is,they satisfy the relation
Three Key Components in the approach:
With Lipschitz Continuous Property :
We can conclude the following observation of the function:
domain, it’s guaranteed that the bound of the function
of sample points and corresponding function values, one
cannot provide a lower boundafterany finite number
of function evaluations of D.
In Lipschitz continuous function , the
more sample points we have , the more
accurate approximation of the lower
bound we can obtain.
-Interval set: a<x<b (x, a, b is vector)
The strategy is to partition the interval into sub-interval by bisection.
In high dimension, it could be regard as a box.
The strategy is to partition the simplex into sub-simplex by each time cutting
one vertex out.
-Convex Cone set:
The strategy is to partition the cone into sub-cone.
-Create linear bound constraint of each partition
-Fulfill “exhaustive search”
(single & multi-start)
The random option of approach is also usually used to
handle black-box function.
Integrated technical computing systems:
GLOBALLib is a collection of nonlinear models that provides GO solver developers with a large and varied set of theoretical and practical test models.
The entire test set used consists of 117 models.
The test models included have up to 142 variables, 109 constraints, 729 non-zero and 567 nonlinear-non-zero model terms.
(for brevity we shall use opmode)
.opmode 0: local search from a given nominal solution, without a preceding global search mode (LS)
·opmode 1: global branch-and-bound search and local search (BB+LS)
· opmode 2: global adaptive random search and local search (GARS+LS)
· opmode 3: global multi-start random search and local search (MS+LS)
Efficiency profiles: all LGO solver modes are applied to GLOBALLib models.
There are usually five stages while using LGO, they are
problem definition, problem compilation, model
parameters, model solution, and result analysis
-Problem definition: Define the function
-Problem Compilation : Link to obj and lib
-Model parameters: Set up lower bound, upper bound , and number of
-Model solution: There is automatic model and interactive model
Automatic model :
Program determine which of the four module to use
to compute with respect to the input file
User determine which module to use and in which order
,maximum search effort
Naturally, in such cases only a statistical guarantee can be given for the global lower bound estimate. The lower bound generated by LGO is statistical in all cases, since it is based partially on pseudo-random sampling.
Solvers being compared:
Outline of test set:
but currently most problems on this site are without computational
nonempty feasible domain
almost all being feasible.
Those excluded from libraries:
1.Certain difficult ones for testing, but the difficulties is unrelated to solver
2.Those contain function which aren’t support by ampl2dag converter.
3. Problem actually contain objective function in Library3.
4. Showing strange behavior, which might caused by bug in converter
5. No solver can get optimal solution
Different solvers have different stopping criteria, Which should also be considered.
Baron, Lingo : stop while time
LGO,OQNLP : stop based on
Janos D. Pinter (LGO ‘s creator) website
Website maintained by Arnold Neumaier
p.s. If you like to check above two books, go asking Prof.Tamas. He will be generous to who like to learn.