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What is global optimization? |
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For nonlinear optimization problems, local minima become a problem. Local
methods that depend on searching on a gradient from a starting point are likely
to terminate in a local minimum, which may not be satisfactory to the user. In
addition, multiple true minima can exist for nonlinear problems, but local
methods find only one solution. Global methods attempt to find the overall best
solution(s). Special techniques are needed to deal with constraints in a
nonlinear system--or when constraints are nonlinear.
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How does Global Optimization work?
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Global Optimization has several algorithms, each of which is suitable
for a particular type of problem. These are Multistart Gradient Search,
Interchange Method, "Tabu" Search, Adaptive Grid Refinement, and Constrained
Allocation. In addition, nonlinear regression and maximum likelihood functions,
which are essentially optimization problems, are included.
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| Q: |
What is Multistart Gradient Search?
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Multistart Gradient Search works by a process of hill-climbing in a space
defined by equality and inequality constraints. It is implemented via the
functions GlobalSearch, GlobalPenaltyFn, and
MultiStartMin. These functions are designed to be robust to local
minima. Nonlinear, as well as linear, equality and inequality constraints
can be used. The functions have been tested with good results on very noisy
and difficult functions. Problems with several hundred variables have been
run in a few seconds. The multistart feature can locate multiple true minima
if they exist. Any mix of real, integer, and discrete variables may be used.
This means that knapsack problems can be solved. Functions need not be
differentiable or analytic.
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| Q: |
What is IntervalMin?
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IntervalMin uses interval methods to search for a minimum of a
function. For a certain class of functions, this method is guaranteed to
find a solution. Inequality constraints are allowed.
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What is the nonlinear regression function?
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The function NLRegression solves nonlinear regression problems
using the function GlobalSearch as the minimizing solver. This
means that, with the use of constraints to prevent unreasonable results,
highly nonlinear problems can be solved successfully. Both L1 and L2 norms
can be used, as can least-squares and chi-square fit criteria. Confidence
intervals, parametric sensitivity analyses, and fit statistics are printed.
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| Q: |
What is the maximum likelihood function?
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The function MaxLikelihood solves log-likelihood problems using the
function GlobalSearch as the minimizing solver. This means that,
with the use of constraints to prevent unreasonable results, highly nonlinear
problems can be solved successfully. A library of speed-optimized built-in
functions are provided. Descriptive statistics are printed.
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What is Interchange Method?
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Interchange Method is an algorithm for solving 0-1 integer programming
problems such as occur in scheduling and network contexts. It is a fast
approach to network problems and, combined with a multistart feature, gives
good assurance of finding a global solution to these difficult problems. The
method is implemented via the function InterchangeMethodMin, which
has been demonstrated to solve minimum spanning tree, traveling salesman,
and other problems. The function TabuSearchMin extends this
algorithm with a "tabu" feature that prevents recently tried moves from
being repeated.
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What is Adaptive Grid Refinement?
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Adaptive Grid Refinement (AGR) works by successively refining an n-dimensional
grid in the regions that have the lowest set of function values. This means
that multiple true solutions can be found if they exist. It also means that
the final solution can define not just the best point but also the optimal
region around the best point. It is often the case for applied problems that
the region of optimality is as important as the value of the "best"
solution. AGR does not use derivatives, and thus the user-defined function
need not be differentiable or even strictly continuous. AGR is quite robust
to local minima. The AGR algorithm is implemented in the function
GlobalMinima, which will solve problems of up to eight dimensions
on most machines. AGR can accept linear or nonlinear inequality constraints.
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What is Constrained Allocation?
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Constrained Allocation is designed for allocation problems such as arise in
investment. Certain problems, including many finance problems, have a nonlinear
objective function, with the single constraint that the sum of the parameters
must equal a fixed value. This is often the amount of funds that are being
invested, but could also be such things as time or materials in a manufacturing
setting. This particular constraint allows an efficient solution method to be
applied, which means that large problems can be solved. The solution method
involves making small incremental steps, always along the constraint line in
the direction of optimality. The function has been tested on problems with
more than 600 variables. The Constrained Allocation algorithm is implemented
via the function MaxAllocation.
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Can equality constraints be used?
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Yes, they can. Equality constraints are a new feature of the functions
GlobalSearch, GlobalPenaltyFn, and MultiStartMin.
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How can Global Optimization be used?
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Global Optimization is well suited for nonlinear least-squares model
estimation, maximum likelihood analysis, financial analysis (investment
allocation and derivative hedging with quadratic programming), engineering
design, optimal control, scheduling, vehicle routing, and other applications.
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Who are current customers of Global Optimization?
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Current customers include faculty at many major universities, the World Bank,
the National Institutes of Health, financial consulting firms, the U.S. Army,
and high-tech corporations such as the Rand Corporation, General Electric,
Ericsson Microwave Systems, and Hewlett-Packard. The package has been
sold in over 22 countries around the world.
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What have customers said about the package?
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Customers have been very happy with the package. Many customers use Global
Optimization on a regular basis. Some have bought multiple copies for
both home and work or for their students. See the review and comments for more details.
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What do I need to run Global Optimization, and how do I order it?
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Global Optimization requires Mathematica 4.2 or
higher. The software is distributed on CD-ROM and includes complete
documentation. It can be purchased from the Wolfram Research online store.
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| Q: |
Where can I get help if I have technical questions about
Global Optimization?
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Support is available from the developer, who can be contacted by email at
info@loehleenterprises.com.
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