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== Notation == | == Notation == | ||
| − | As a | + | As a maths parser is currently not used in this Wiki, we write math equations as formatted text: |
* Scalars are written as small italic letters, e.g. ''f'' | * Scalars are written as small italic letters, e.g. ''f'' | ||
* Vectors are written as small bold letters, e.g. '''x'''. | * Vectors are written as small bold letters, e.g. '''x'''. | ||
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:{| cellpadding="5" width="95%" | :{| cellpadding="5" width="95%" | ||
| + | |- | ||
| + | | ''a'', '''a''' | ||
| + | | additionals, i.e. additional value(s) | ||
|- | |- | ||
| ''f'', '''f''' | | ''f'', '''f''' | ||
| objective function(s) | | objective function(s) | ||
| + | |- | ||
| + | | ''g'', '''g''' | ||
| + | | constraint(s) | ||
|- | |- | ||
| ''x'', '''x''' | | ''x'', '''x''' | ||
| design variable(s) | | design variable(s) | ||
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| − | |||
| − | |||
|- | |- | ||
|} | |} | ||
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:{| cellpadding="5" cellspacing="0" width="95%" border="1" | :{| cellpadding="5" cellspacing="0" width="95%" border="1" | ||
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| − | | ''additionals'' | + | | ''additionals'' <div id="additionals"></div> |
| Additionals '''a''' are additional values in a '''solution''' that are for information only and do not influence the optimization. For example, if an optimization problem returns an objective function that is a sum of several values, these values could be added to the solution as additional values '''a'''. | | Additionals '''a''' are additional values in a '''solution''' that are for information only and do not influence the optimization. For example, if an optimization problem returns an objective function that is a sum of several values, these values could be added to the solution as additional values '''a'''. | ||
|- | |- | ||
| − | | ''constraints'' | + | | ''constraints'' <div id="constraints"></div> |
| − | | Constraints ''' | + | | Constraints '''g''' are criteria that have to be fulfilled. OpenDino defines a constraint ''g'' as |
| − | * fulfilled, if '' | + | * fulfilled, if ''g'' =< 0 |
| − | * violated, if '' | + | * violated, if ''g'' > 0. |
The most simple constraint handling in optimization is to add a penalty to the objective function if the constraint is vialoted, resulting in | The most simple constraint handling in optimization is to add a penalty to the objective function if the constraint is vialoted, resulting in | ||
| − | ''f'' + max(0, '' | + | ''f'' + max(0, ''g''). |
|- | |- | ||
| − | | ''design variables'' | + | | ''design variables'' <div id="design_variables"></div> |
| The vector of design variables '''x''' may consist of real numbers (continuous variables), integers (discrete variables) or both (mixed variables). | | The vector of design variables '''x''' may consist of real numbers (continuous variables), integers (discrete variables) or both (mixed variables). | ||
|- | |- | ||
| − | | ''objectives'' | + | | ''objectives'' <div id="objectives"></div> |
| An optimization problem must have at least one objective or one constraint. OpenDino requires that the objective(s) '''f''' is/are to be minimized. A maximization of a function ''k'' can be converted into minimization by using ''f'' = -''k''. | | An optimization problem must have at least one objective or one constraint. OpenDino requires that the objective(s) '''f''' is/are to be minimized. A maximization of a function ''k'' can be converted into minimization by using ''f'' = -''k''. | ||
|- | |- | ||
| − | | ''solution'' | + | | ''solution'' <div id="solution"></div> |
| The term solution is used in optimization. One solution ('''x''','''f''','''g''','''a''') consists of a vector of design variables '''x''', the evaluated objective(s) '''f'''('''x'''), constraint(s) '''g'''('''x'''), and optionally of additional values '''a'''. | | The term solution is used in optimization. One solution ('''x''','''f''','''g''','''a''') consists of a vector of design variables '''x''', the evaluated objective(s) '''f'''('''x'''), constraint(s) '''g'''('''x'''), and optionally of additional values '''a'''. | ||
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Aktuelle Version vom 7. April 2019, 13:31 Uhr
Notation
As a maths parser is currently not used in this Wiki, we write math equations as formatted text:
- Scalars are written as small italic letters, e.g. f
- Vectors are written as small bold letters, e.g. x.
- Matrices are written in capital bold letters, e.g. C.
Symbols
a, a additionals, i.e. additional value(s) f, f objective function(s) g, g constraint(s) x, x design variable(s)
Terminology
additionals Additionals a are additional values in a solution that are for information only and do not influence the optimization. For example, if an optimization problem returns an objective function that is a sum of several values, these values could be added to the solution as additional values a. constraints Constraints g are criteria that have to be fulfilled. OpenDino defines a constraint g as - fulfilled, if g =< 0
- violated, if g > 0.
The most simple constraint handling in optimization is to add a penalty to the objective function if the constraint is vialoted, resulting in
f + max(0, g).
design variables The vector of design variables x may consist of real numbers (continuous variables), integers (discrete variables) or both (mixed variables). objectives An optimization problem must have at least one objective or one constraint. OpenDino requires that the objective(s) f is/are to be minimized. A maximization of a function k can be converted into minimization by using f = -k. solution The term solution is used in optimization. One solution (x,f,g,a) consists of a vector of design variables x, the evaluated objective(s) f(x), constraint(s) g(x), and optionally of additional values a.
