power_2 module¶
A quadratic analytical problem.
- class gemseo.problems.optimization.power_2.Power2(exception_error=False, initial_value=1.0)[source]¶
Bases:
OptimizationProblem
Power2 is a very basic quadratic analytical
OptimizationProblem
.Objective to minimize: \(x_0^2 + x_1^2 + x_2^2\)
Inequality constraint 1: \(x_0^3 - 0.5 > 0\)
Inequality constraint 2: \(x_1^3 - 0.5 > 0\)
Equality constraint: \(x_2^3 - 0.9 = 0\)
Analytical optimum: \(x^*=(0.5^{1/3}, 0.5^{1/3}, 0.9^{1/3})\)
- Parameters:
- AggregationFunction¶
alias of
EvaluationFunction
- class ApproximationMode(value)¶
Bases:
StrEnum
The approximation derivation modes.
- capitalize()¶
Return a capitalized version of the string.
More specifically, make the first character have upper case and the rest lower case.
- casefold()¶
Return a version of the string suitable for caseless comparisons.
- center(width, fillchar=' ', /)¶
Return a centered string of length width.
Padding is done using the specified fill character (default is a space).
- count(sub[, start[, end]]) int ¶
Return the number of non-overlapping occurrences of substring sub in string S[start:end]. Optional arguments start and end are interpreted as in slice notation.
- encode(encoding='utf-8', errors='strict')¶
Encode the string using the codec registered for encoding.
- encoding
The encoding in which to encode the string.
- errors
The error handling scheme to use for encoding errors. The default is ‘strict’ meaning that encoding errors raise a UnicodeEncodeError. Other possible values are ‘ignore’, ‘replace’ and ‘xmlcharrefreplace’ as well as any other name registered with codecs.register_error that can handle UnicodeEncodeErrors.
- endswith(suffix[, start[, end]]) bool ¶
Return True if S ends with the specified suffix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. suffix can also be a tuple of strings to try.
- expandtabs(tabsize=8)¶
Return a copy where all tab characters are expanded using spaces.
If tabsize is not given, a tab size of 8 characters is assumed.
- find(sub[, start[, end]]) int ¶
Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Return -1 on failure.
- format(*args, **kwargs) str ¶
Return a formatted version of S, using substitutions from args and kwargs. The substitutions are identified by braces (‘{’ and ‘}’).
- format_map(mapping) str ¶
Return a formatted version of S, using substitutions from mapping. The substitutions are identified by braces (‘{’ and ‘}’).
- index(sub[, start[, end]]) int ¶
Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Raises ValueError when the substring is not found.
- isalnum()¶
Return True if the string is an alpha-numeric string, False otherwise.
A string is alpha-numeric if all characters in the string are alpha-numeric and there is at least one character in the string.
- isalpha()¶
Return True if the string is an alphabetic string, False otherwise.
A string is alphabetic if all characters in the string are alphabetic and there is at least one character in the string.
- isascii()¶
Return True if all characters in the string are ASCII, False otherwise.
ASCII characters have code points in the range U+0000-U+007F. Empty string is ASCII too.
- isdecimal()¶
Return True if the string is a decimal string, False otherwise.
A string is a decimal string if all characters in the string are decimal and there is at least one character in the string.
- isdigit()¶
Return True if the string is a digit string, False otherwise.
A string is a digit string if all characters in the string are digits and there is at least one character in the string.
- isidentifier()¶
Return True if the string is a valid Python identifier, False otherwise.
Call keyword.iskeyword(s) to test whether string s is a reserved identifier, such as “def” or “class”.
- islower()¶
Return True if the string is a lowercase string, False otherwise.
A string is lowercase if all cased characters in the string are lowercase and there is at least one cased character in the string.
- isnumeric()¶
Return True if the string is a numeric string, False otherwise.
A string is numeric if all characters in the string are numeric and there is at least one character in the string.
- isprintable()¶
Return True if the string is printable, False otherwise.
A string is printable if all of its characters are considered printable in repr() or if it is empty.
- isspace()¶
Return True if the string is a whitespace string, False otherwise.
A string is whitespace if all characters in the string are whitespace and there is at least one character in the string.
- istitle()¶
Return True if the string is a title-cased string, False otherwise.
In a title-cased string, upper- and title-case characters may only follow uncased characters and lowercase characters only cased ones.
- isupper()¶
Return True if the string is an uppercase string, False otherwise.
A string is uppercase if all cased characters in the string are uppercase and there is at least one cased character in the string.
- join(iterable, /)¶
Concatenate any number of strings.
The string whose method is called is inserted in between each given string. The result is returned as a new string.
Example: ‘.’.join([‘ab’, ‘pq’, ‘rs’]) -> ‘ab.pq.rs’
- ljust(width, fillchar=' ', /)¶
Return a left-justified string of length width.
Padding is done using the specified fill character (default is a space).
- lower()¶
Return a copy of the string converted to lowercase.
- lstrip(chars=None, /)¶
Return a copy of the string with leading whitespace removed.
If chars is given and not None, remove characters in chars instead.
- static maketrans()¶
Return a translation table usable for str.translate().
If there is only one argument, it must be a dictionary mapping Unicode ordinals (integers) or characters to Unicode ordinals, strings or None. Character keys will be then converted to ordinals. If there are two arguments, they must be strings of equal length, and in the resulting dictionary, each character in x will be mapped to the character at the same position in y. If there is a third argument, it must be a string, whose characters will be mapped to None in the result.
- partition(sep, /)¶
Partition the string into three parts using the given separator.
This will search for the separator in the string. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.
If the separator is not found, returns a 3-tuple containing the original string and two empty strings.
- removeprefix(prefix, /)¶
Return a str with the given prefix string removed if present.
If the string starts with the prefix string, return string[len(prefix):]. Otherwise, return a copy of the original string.
- removesuffix(suffix, /)¶
Return a str with the given suffix string removed if present.
If the string ends with the suffix string and that suffix is not empty, return string[:-len(suffix)]. Otherwise, return a copy of the original string.
- replace(old, new, count=-1, /)¶
Return a copy with all occurrences of substring old replaced by new.
- count
Maximum number of occurrences to replace. -1 (the default value) means replace all occurrences.
If the optional argument count is given, only the first count occurrences are replaced.
- rfind(sub[, start[, end]]) int ¶
Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Return -1 on failure.
- rindex(sub[, start[, end]]) int ¶
Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Raises ValueError when the substring is not found.
- rjust(width, fillchar=' ', /)¶
Return a right-justified string of length width.
Padding is done using the specified fill character (default is a space).
- rpartition(sep, /)¶
Partition the string into three parts using the given separator.
This will search for the separator in the string, starting at the end. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.
If the separator is not found, returns a 3-tuple containing two empty strings and the original string.
- rsplit(sep=None, maxsplit=-1)¶
Return a list of the words in the string, using sep as the delimiter string.
- sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
- maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.
Splits are done starting at the end of the string and working to the front.
- rstrip(chars=None, /)¶
Return a copy of the string with trailing whitespace removed.
If chars is given and not None, remove characters in chars instead.
- split(sep=None, maxsplit=-1)¶
Return a list of the words in the string, using sep as the delimiter string.
- sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
- maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.
- splitlines(keepends=False)¶
Return a list of the lines in the string, breaking at line boundaries.
Line breaks are not included in the resulting list unless keepends is given and true.
- startswith(prefix[, start[, end]]) bool ¶
Return True if S starts with the specified prefix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. prefix can also be a tuple of strings to try.
- strip(chars=None, /)¶
Return a copy of the string with leading and trailing whitespace removed.
If chars is given and not None, remove characters in chars instead.
- swapcase()¶
Convert uppercase characters to lowercase and lowercase characters to uppercase.
- title()¶
Return a version of the string where each word is titlecased.
More specifically, words start with uppercased characters and all remaining cased characters have lower case.
- translate(table, /)¶
Replace each character in the string using the given translation table.
- table
Translation table, which must be a mapping of Unicode ordinals to Unicode ordinals, strings, or None.
The table must implement lookup/indexing via __getitem__, for instance a dictionary or list. If this operation raises LookupError, the character is left untouched. Characters mapped to None are deleted.
- upper()¶
Return a copy of the string converted to uppercase.
- zfill(width, /)¶
Pad a numeric string with zeros on the left, to fill a field of the given width.
The string is never truncated.
- CENTERED_DIFFERENCES = 'centered_differences'¶
The centered differences method used to approximate the Jacobians by perturbing each variable with a small real number.
- COMPLEX_STEP = 'complex_step'¶
The complex step method used to approximate the Jacobians by perturbing each variable with a small complex number.
- FINITE_DIFFERENCES = 'finite_differences'¶
The finite differences method used to approximate the Jacobians by perturbing each variable with a small real number.
- class DifferentiationMethod(value)¶
Bases:
StrEnum
The differentiation methods.
- capitalize()¶
Return a capitalized version of the string.
More specifically, make the first character have upper case and the rest lower case.
- casefold()¶
Return a version of the string suitable for caseless comparisons.
- center(width, fillchar=' ', /)¶
Return a centered string of length width.
Padding is done using the specified fill character (default is a space).
- count(sub[, start[, end]]) int ¶
Return the number of non-overlapping occurrences of substring sub in string S[start:end]. Optional arguments start and end are interpreted as in slice notation.
- encode(encoding='utf-8', errors='strict')¶
Encode the string using the codec registered for encoding.
- encoding
The encoding in which to encode the string.
- errors
The error handling scheme to use for encoding errors. The default is ‘strict’ meaning that encoding errors raise a UnicodeEncodeError. Other possible values are ‘ignore’, ‘replace’ and ‘xmlcharrefreplace’ as well as any other name registered with codecs.register_error that can handle UnicodeEncodeErrors.
- endswith(suffix[, start[, end]]) bool ¶
Return True if S ends with the specified suffix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. suffix can also be a tuple of strings to try.
- expandtabs(tabsize=8)¶
Return a copy where all tab characters are expanded using spaces.
If tabsize is not given, a tab size of 8 characters is assumed.
- find(sub[, start[, end]]) int ¶
Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Return -1 on failure.
- format(*args, **kwargs) str ¶
Return a formatted version of S, using substitutions from args and kwargs. The substitutions are identified by braces (‘{’ and ‘}’).
- format_map(mapping) str ¶
Return a formatted version of S, using substitutions from mapping. The substitutions are identified by braces (‘{’ and ‘}’).
- index(sub[, start[, end]]) int ¶
Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Raises ValueError when the substring is not found.
- isalnum()¶
Return True if the string is an alpha-numeric string, False otherwise.
A string is alpha-numeric if all characters in the string are alpha-numeric and there is at least one character in the string.
- isalpha()¶
Return True if the string is an alphabetic string, False otherwise.
A string is alphabetic if all characters in the string are alphabetic and there is at least one character in the string.
- isascii()¶
Return True if all characters in the string are ASCII, False otherwise.
ASCII characters have code points in the range U+0000-U+007F. Empty string is ASCII too.
- isdecimal()¶
Return True if the string is a decimal string, False otherwise.
A string is a decimal string if all characters in the string are decimal and there is at least one character in the string.
- isdigit()¶
Return True if the string is a digit string, False otherwise.
A string is a digit string if all characters in the string are digits and there is at least one character in the string.
- isidentifier()¶
Return True if the string is a valid Python identifier, False otherwise.
Call keyword.iskeyword(s) to test whether string s is a reserved identifier, such as “def” or “class”.
- islower()¶
Return True if the string is a lowercase string, False otherwise.
A string is lowercase if all cased characters in the string are lowercase and there is at least one cased character in the string.
- isnumeric()¶
Return True if the string is a numeric string, False otherwise.
A string is numeric if all characters in the string are numeric and there is at least one character in the string.
- isprintable()¶
Return True if the string is printable, False otherwise.
A string is printable if all of its characters are considered printable in repr() or if it is empty.
- isspace()¶
Return True if the string is a whitespace string, False otherwise.
A string is whitespace if all characters in the string are whitespace and there is at least one character in the string.
- istitle()¶
Return True if the string is a title-cased string, False otherwise.
In a title-cased string, upper- and title-case characters may only follow uncased characters and lowercase characters only cased ones.
- isupper()¶
Return True if the string is an uppercase string, False otherwise.
A string is uppercase if all cased characters in the string are uppercase and there is at least one cased character in the string.
- join(iterable, /)¶
Concatenate any number of strings.
The string whose method is called is inserted in between each given string. The result is returned as a new string.
Example: ‘.’.join([‘ab’, ‘pq’, ‘rs’]) -> ‘ab.pq.rs’
- ljust(width, fillchar=' ', /)¶
Return a left-justified string of length width.
Padding is done using the specified fill character (default is a space).
- lower()¶
Return a copy of the string converted to lowercase.
- lstrip(chars=None, /)¶
Return a copy of the string with leading whitespace removed.
If chars is given and not None, remove characters in chars instead.
- static maketrans()¶
Return a translation table usable for str.translate().
If there is only one argument, it must be a dictionary mapping Unicode ordinals (integers) or characters to Unicode ordinals, strings or None. Character keys will be then converted to ordinals. If there are two arguments, they must be strings of equal length, and in the resulting dictionary, each character in x will be mapped to the character at the same position in y. If there is a third argument, it must be a string, whose characters will be mapped to None in the result.
- partition(sep, /)¶
Partition the string into three parts using the given separator.
This will search for the separator in the string. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.
If the separator is not found, returns a 3-tuple containing the original string and two empty strings.
- removeprefix(prefix, /)¶
Return a str with the given prefix string removed if present.
If the string starts with the prefix string, return string[len(prefix):]. Otherwise, return a copy of the original string.
- removesuffix(suffix, /)¶
Return a str with the given suffix string removed if present.
If the string ends with the suffix string and that suffix is not empty, return string[:-len(suffix)]. Otherwise, return a copy of the original string.
- replace(old, new, count=-1, /)¶
Return a copy with all occurrences of substring old replaced by new.
- count
Maximum number of occurrences to replace. -1 (the default value) means replace all occurrences.
If the optional argument count is given, only the first count occurrences are replaced.
- rfind(sub[, start[, end]]) int ¶
Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Return -1 on failure.
- rindex(sub[, start[, end]]) int ¶
Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Raises ValueError when the substring is not found.
- rjust(width, fillchar=' ', /)¶
Return a right-justified string of length width.
Padding is done using the specified fill character (default is a space).
- rpartition(sep, /)¶
Partition the string into three parts using the given separator.
This will search for the separator in the string, starting at the end. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.
If the separator is not found, returns a 3-tuple containing two empty strings and the original string.
- rsplit(sep=None, maxsplit=-1)¶
Return a list of the words in the string, using sep as the delimiter string.
- sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
- maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.
Splits are done starting at the end of the string and working to the front.
- rstrip(chars=None, /)¶
Return a copy of the string with trailing whitespace removed.
If chars is given and not None, remove characters in chars instead.
- split(sep=None, maxsplit=-1)¶
Return a list of the words in the string, using sep as the delimiter string.
- sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
- maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.
- splitlines(keepends=False)¶
Return a list of the lines in the string, breaking at line boundaries.
Line breaks are not included in the resulting list unless keepends is given and true.
- startswith(prefix[, start[, end]]) bool ¶
Return True if S starts with the specified prefix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. prefix can also be a tuple of strings to try.
- strip(chars=None, /)¶
Return a copy of the string with leading and trailing whitespace removed.
If chars is given and not None, remove characters in chars instead.
- swapcase()¶
Convert uppercase characters to lowercase and lowercase characters to uppercase.
- title()¶
Return a version of the string where each word is titlecased.
More specifically, words start with uppercased characters and all remaining cased characters have lower case.
- translate(table, /)¶
Replace each character in the string using the given translation table.
- table
Translation table, which must be a mapping of Unicode ordinals to Unicode ordinals, strings, or None.
The table must implement lookup/indexing via __getitem__, for instance a dictionary or list. If this operation raises LookupError, the character is left untouched. Characters mapped to None are deleted.
- upper()¶
Return a copy of the string converted to uppercase.
- zfill(width, /)¶
Pad a numeric string with zeros on the left, to fill a field of the given width.
The string is never truncated.
- CENTERED_DIFFERENCES = 'centered_differences'¶
- COMPLEX_STEP = 'complex_step'¶
- FINITE_DIFFERENCES = 'finite_differences'¶
- NO_DERIVATIVE = 'no_derivative'¶
- USER_GRAD = 'user'¶
- class ProblemType(value)¶
Bases:
StrEnum
The type of problem.
- capitalize()¶
Return a capitalized version of the string.
More specifically, make the first character have upper case and the rest lower case.
- casefold()¶
Return a version of the string suitable for caseless comparisons.
- center(width, fillchar=' ', /)¶
Return a centered string of length width.
Padding is done using the specified fill character (default is a space).
- count(sub[, start[, end]]) int ¶
Return the number of non-overlapping occurrences of substring sub in string S[start:end]. Optional arguments start and end are interpreted as in slice notation.
- encode(encoding='utf-8', errors='strict')¶
Encode the string using the codec registered for encoding.
- encoding
The encoding in which to encode the string.
- errors
The error handling scheme to use for encoding errors. The default is ‘strict’ meaning that encoding errors raise a UnicodeEncodeError. Other possible values are ‘ignore’, ‘replace’ and ‘xmlcharrefreplace’ as well as any other name registered with codecs.register_error that can handle UnicodeEncodeErrors.
- endswith(suffix[, start[, end]]) bool ¶
Return True if S ends with the specified suffix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. suffix can also be a tuple of strings to try.
- expandtabs(tabsize=8)¶
Return a copy where all tab characters are expanded using spaces.
If tabsize is not given, a tab size of 8 characters is assumed.
- find(sub[, start[, end]]) int ¶
Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Return -1 on failure.
- format(*args, **kwargs) str ¶
Return a formatted version of S, using substitutions from args and kwargs. The substitutions are identified by braces (‘{’ and ‘}’).
- format_map(mapping) str ¶
Return a formatted version of S, using substitutions from mapping. The substitutions are identified by braces (‘{’ and ‘}’).
- index(sub[, start[, end]]) int ¶
Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Raises ValueError when the substring is not found.
- isalnum()¶
Return True if the string is an alpha-numeric string, False otherwise.
A string is alpha-numeric if all characters in the string are alpha-numeric and there is at least one character in the string.
- isalpha()¶
Return True if the string is an alphabetic string, False otherwise.
A string is alphabetic if all characters in the string are alphabetic and there is at least one character in the string.
- isascii()¶
Return True if all characters in the string are ASCII, False otherwise.
ASCII characters have code points in the range U+0000-U+007F. Empty string is ASCII too.
- isdecimal()¶
Return True if the string is a decimal string, False otherwise.
A string is a decimal string if all characters in the string are decimal and there is at least one character in the string.
- isdigit()¶
Return True if the string is a digit string, False otherwise.
A string is a digit string if all characters in the string are digits and there is at least one character in the string.
- isidentifier()¶
Return True if the string is a valid Python identifier, False otherwise.
Call keyword.iskeyword(s) to test whether string s is a reserved identifier, such as “def” or “class”.
- islower()¶
Return True if the string is a lowercase string, False otherwise.
A string is lowercase if all cased characters in the string are lowercase and there is at least one cased character in the string.
- isnumeric()¶
Return True if the string is a numeric string, False otherwise.
A string is numeric if all characters in the string are numeric and there is at least one character in the string.
- isprintable()¶
Return True if the string is printable, False otherwise.
A string is printable if all of its characters are considered printable in repr() or if it is empty.
- isspace()¶
Return True if the string is a whitespace string, False otherwise.
A string is whitespace if all characters in the string are whitespace and there is at least one character in the string.
- istitle()¶
Return True if the string is a title-cased string, False otherwise.
In a title-cased string, upper- and title-case characters may only follow uncased characters and lowercase characters only cased ones.
- isupper()¶
Return True if the string is an uppercase string, False otherwise.
A string is uppercase if all cased characters in the string are uppercase and there is at least one cased character in the string.
- join(iterable, /)¶
Concatenate any number of strings.
The string whose method is called is inserted in between each given string. The result is returned as a new string.
Example: ‘.’.join([‘ab’, ‘pq’, ‘rs’]) -> ‘ab.pq.rs’
- ljust(width, fillchar=' ', /)¶
Return a left-justified string of length width.
Padding is done using the specified fill character (default is a space).
- lower()¶
Return a copy of the string converted to lowercase.
- lstrip(chars=None, /)¶
Return a copy of the string with leading whitespace removed.
If chars is given and not None, remove characters in chars instead.
- static maketrans()¶
Return a translation table usable for str.translate().
If there is only one argument, it must be a dictionary mapping Unicode ordinals (integers) or characters to Unicode ordinals, strings or None. Character keys will be then converted to ordinals. If there are two arguments, they must be strings of equal length, and in the resulting dictionary, each character in x will be mapped to the character at the same position in y. If there is a third argument, it must be a string, whose characters will be mapped to None in the result.
- partition(sep, /)¶
Partition the string into three parts using the given separator.
This will search for the separator in the string. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.
If the separator is not found, returns a 3-tuple containing the original string and two empty strings.
- removeprefix(prefix, /)¶
Return a str with the given prefix string removed if present.
If the string starts with the prefix string, return string[len(prefix):]. Otherwise, return a copy of the original string.
- removesuffix(suffix, /)¶
Return a str with the given suffix string removed if present.
If the string ends with the suffix string and that suffix is not empty, return string[:-len(suffix)]. Otherwise, return a copy of the original string.
- replace(old, new, count=-1, /)¶
Return a copy with all occurrences of substring old replaced by new.
- count
Maximum number of occurrences to replace. -1 (the default value) means replace all occurrences.
If the optional argument count is given, only the first count occurrences are replaced.
- rfind(sub[, start[, end]]) int ¶
Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Return -1 on failure.
- rindex(sub[, start[, end]]) int ¶
Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.
Raises ValueError when the substring is not found.
- rjust(width, fillchar=' ', /)¶
Return a right-justified string of length width.
Padding is done using the specified fill character (default is a space).
- rpartition(sep, /)¶
Partition the string into three parts using the given separator.
This will search for the separator in the string, starting at the end. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.
If the separator is not found, returns a 3-tuple containing two empty strings and the original string.
- rsplit(sep=None, maxsplit=-1)¶
Return a list of the words in the string, using sep as the delimiter string.
- sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
- maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.
Splits are done starting at the end of the string and working to the front.
- rstrip(chars=None, /)¶
Return a copy of the string with trailing whitespace removed.
If chars is given and not None, remove characters in chars instead.
- split(sep=None, maxsplit=-1)¶
Return a list of the words in the string, using sep as the delimiter string.
- sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
- maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.
- splitlines(keepends=False)¶
Return a list of the lines in the string, breaking at line boundaries.
Line breaks are not included in the resulting list unless keepends is given and true.
- startswith(prefix[, start[, end]]) bool ¶
Return True if S starts with the specified prefix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. prefix can also be a tuple of strings to try.
- strip(chars=None, /)¶
Return a copy of the string with leading and trailing whitespace removed.
If chars is given and not None, remove characters in chars instead.
- swapcase()¶
Convert uppercase characters to lowercase and lowercase characters to uppercase.
- title()¶
Return a version of the string where each word is titlecased.
More specifically, words start with uppercased characters and all remaining cased characters have lower case.
- translate(table, /)¶
Replace each character in the string using the given translation table.
- table
Translation table, which must be a mapping of Unicode ordinals to Unicode ordinals, strings, or None.
The table must implement lookup/indexing via __getitem__, for instance a dictionary or list. If this operation raises LookupError, the character is left untouched. Characters mapped to None are deleted.
- upper()¶
Return a copy of the string converted to uppercase.
- zfill(width, /)¶
Pad a numeric string with zeros on the left, to fill a field of the given width.
The string is never truncated.
- LINEAR = 'linear'¶
- NON_LINEAR = 'non-linear'¶
- add_callback(callback, each_new_iter=True, each_store=False)¶
Add a callback for some events.
The callback functions are attached to the database, which means they are triggered when new values are stored within the database of the optimization problem.
- Parameters:
callback (Callable[[ndarray], Any]) – A function to be called after some events, whose argument is a design vector.
each_new_iter (bool) –
Whether to evaluate the callback functions after evaluating all functions of the optimization problem for a given point and storing their values in the
database
.By default it is set to True.
each_store (bool) –
Whether to evaluate the callback functions after storing any new value in the
database
.By default it is set to False.
- Return type:
None
- add_constraint(function, value=0.0, constraint_type=None, positive=False)¶
Add an equality or inequality constraint to the optimization problem.
An equality constraint is written as \(c(x)=a\), a positive inequality constraint is written as \(c(x)\geq a\) and a negative inequality constraint is written as \(c(x)\leq a\).
- Parameters:
function (MDOFunction) – The function \(c\).
value (float) –
The value \(a\).
By default it is set to 0.0.
constraint_type (MDOFunction.ConstraintType | None) – The type of the constraint.
positive (bool) –
Whether the inequality constraint is positive.
By default it is set to False.
- Raises:
TypeError – When the constraint of a linear optimization problem is not an
MDOLinearFunction
.ValueError – When the type of the constraint is missing.
- Return type:
None
- add_eq_constraint(function, value=0)¶
Add an equality constraint \(c(x)=a\) to the optimization problem.
- Parameters:
function (MDOFunction) – The function \(c\).
value (float) –
The value \(a\).
By default it is set to 0.
- Return type:
None
- add_ineq_constraint(function, value=0, positive=False)¶
Add an inequality constraint to the optimization problem.
A positive inequality constraint is written as \(c(x)\geq a\) and a negative inequality constraint is written as \(c(x)\leq a\)
- Parameters:
function (MDOFunction) – The function \(c\).
value (float) –
The value \(a\) for which the constraint is active.
By default it is set to 0.
positive (bool) –
Whether the inequality constraint is positive.
By default it is set to False.
- Return type:
None
- add_observable(observable, new_iter=True)¶
Add a function to be observed.
When the
OptimizationProblem
is executed, the observables are called following this sequence:The optimization algorithm calls the objective function with a normalized
x_vect
.The
OptimizationProblem.preprocess_functions()
wraps the function as aNormDBFunction
, which unnormalizes thex_vect
before evaluation.The unnormalized
x_vect
and the result of the evaluation are stored in theOptimizationProblem.database
.The previous step triggers the
OptimizationProblem.new_iter_listeners
, which calls the observables with the unnormalizedx_vect
.The observables themselves are wrapped as a
NormDBFunction
byOptimizationProblem.preprocess_functions()
, but in this case the input is always expected as unnormalized to avoid an additional normalizing-unnormalizing step.Finally, the output is stored in the
OptimizationProblem.database
.
- Parameters:
observable (MDOFunction) – A function to be observed.
new_iter (bool) –
If
True
, then the observable will be called at each new iterate.By default it is set to True.
- Return type:
None
- aggregate_constraint(constraint_index, method=EvaluationFunction.MAX, groups=None, **options)¶
Aggregate a constraint to generate a reduced dimension constraint.
- Parameters:
constraint_index (int) – The index of the constraint in
constraints
.method (Callable[[NDArray[float]], float] | AggregationFunction) –
The aggregation method, e.g.
"max"
,"lower_bound_KS"
,"upper_bound_KS"``or ``"IKS"
.By default it is set to “MAX”.
groups (Iterable[Sequence[int]] | None) – The groups of components of the constraint to aggregate to produce one aggregation constraint per group of components; if
None
, a single aggregation constraint is produced.**options (Any) – The options of the aggregation method.
- Raises:
ValueError – When the given index is greater or equal than the number of constraints or when the constraint aggregation method is unknown.
- Return type:
None
- apply_exterior_penalty(objective_scale=1.0, scale_inequality=1.0, scale_equality=1.0)¶
Reformulate the optimization problem using exterior penalty.
Given the optimization problem with equality and inequality constraints:
\[ \begin{align}\begin{aligned}min_x f(x)\\s.t.\\g(x)\leq 0\\h(x)=0\\l_b\leq x\leq u_b\end{aligned}\end{align} \]The exterior penalty approach consists in building a penalized objective function that takes into account constraints violations:
\[ \begin{align}\begin{aligned}min_x \tilde{f}(x) = \frac{f(x)}{o_s} + s[\sum{H(g(x))g(x)^2}+\sum{h(x)^2}]\\s.t.\\l_b\leq x\leq u_b\end{aligned}\end{align} \]Where \(H(x)\) is the Heaviside function, \(o_s\) is the
objective_scale
parameter and \(s\) is the scale parameter. The solution of the new problem approximate the one of the original problem. Increasing the values ofobjective_scale
and scale, the solutions are closer but the optimization problem requires more and more iterations to be solved.- Parameters:
scale_equality (float | ndarray) –
The equality constraint scaling constant.
By default it is set to 1.0.
objective_scale (float) –
The objective scaling constant.
By default it is set to 1.0.
scale_inequality (float | ndarray) –
The inequality constraint scaling constant.
By default it is set to 1.0.
- Return type:
None
- check()¶
Check if the optimization problem is ready for run.
- Raises:
ValueError – If the objective function is missing.
- Return type:
None
- check_design_point_is_feasible(x_vect)¶
Check if a design point is feasible and measure its constraint violation.
The constraint violation measure at a design point \(x\) is
\[\lVert\max(g(x)-\varepsilon_{\text{ineq}},0)\rVert_2^2 +\lVert|\max(|h(x)|-\varepsilon_{\text{eq}},0)\rVert_2^2\]where \(\|.\|_2\) is the Euclidean norm, \(g(x)\) is the inequality constraint vector, \(h(x)\) is the equality constraint vector, \(\varepsilon_{\text{ineq}}\) is the tolerance for the inequality constraints and \(\varepsilon_{\text{eq}}\) is the tolerance for the equality constraints.
If the design point is feasible, the constraint violation measure is 0.
- static check_format(function)¶
Check that a function is an instance of
MDOFunction
.- Parameters:
function (Any) – The function to be tested.
- Raises:
TypeError – If the function is not an
MDOFunction
.- Return type:
None
- clear_listeners()¶
Clear all the listeners.
- Return type:
None
- evaluate_functions(x_vect=None, eval_jac=False, eval_obj=True, eval_observables=True, normalize=True, no_db_no_norm=False, constraint_names=None, observable_names=None, jacobian_names=None)¶
Compute the functions of interest, and possibly their derivatives.
These functions of interest are the constraints, and possibly the objective.
Some optimization libraries require the number of constraints as an input parameter which is unknown by the formulation or the scenario. Evaluation of initial point allows to get this mandatory information. This is also used for design of experiments to evaluate samples.
- Parameters:
x_vect (ndarray) – The input vector at which the functions must be evaluated; if None, the initial point x_0 is used.
eval_jac (bool) –
Whether to compute the Jacobian matrices of the functions of interest. If
True
andjacobian_names
isNone
then compute the Jacobian matrices (or gradients) of the functions that are selected for evaluation (witheval_obj
,constraint_names
,eval_observables
and``observable_names``). IfFalse
andjacobian_names
isNone
then no Jacobian matrix is evaluated. Ifjacobian_names
is notNone
then the value ofeval_jac
is ignored.By default it is set to False.
eval_obj (bool) –
Whether to consider the objective function as a function of interest.
By default it is set to True.
eval_observables (bool) –
Whether to evaluate the observables. If
True
andobservable_names
isNone
then all the observables are evaluated. IfFalse
andobservable_names
isNone
then no observable is evaluated. Ifobservable_names
is notNone
then the value ofeval_observables
is ignored.By default it is set to True.
normalize (bool) –
Whether to consider the input vector
x_vect
normalized.By default it is set to True.
no_db_no_norm (bool) –
If
True
, then do not use the pre-processed functions, so we have no database, nor normalization.By default it is set to False.
constraint_names (Iterable[str] | None) – The names of the constraints to evaluate. If
None
then all the constraints are evaluated.observable_names (Iterable[str] | None) – The names of the observables to evaluate. If
None
andeval_observables
isTrue
then all the observables are evaluated. IfNone
andeval_observables
isFalse
then no observable is evaluated.jacobian_names (Iterable[str] | None) – The names of the functions whose Jacobian matrices (or gradients) to compute. If
None
andeval_jac
isTrue
then compute the Jacobian matrices (or gradients) of the functions that are selected for evaluation (witheval_obj
,constraint_names
,eval_observables
and``observable_names``). IfNone
andeval_jac
isFalse
then no Jacobian matrix is computed.
- Returns:
The output values of the functions of interest, as well as their Jacobian matrices if
eval_jac
isTrue
.- Raises:
ValueError – If a name in
jacobian_names
is not the name of a function of the problem.- Return type:
EvaluationType
- execute_observables_callback(last_x)¶
The callback function to be passed to the database.
Call all the observables with the last design variables values as argument.
- Parameters:
last_x (ndarray) – The design variables values from the last evaluation.
- Return type:
None
- classmethod from_hdf(file_path, x_tolerance=0.0, hdf_node_path='')¶
Import an optimization history from an HDF file.
- Parameters:
file_path (str | Path) – The file containing the optimization history.
x_tolerance (float) –
The tolerance on the design variables when reading the file.
By default it is set to 0.0.
hdf_node_path (str) –
The path of the HDF node from which the database should be imported. If empty, the root node is considered.
By default it is set to “”.
- Returns:
The read optimization problem.
- Return type:
- get_active_ineq_constraints(x_vect, tol=1e-06)¶
For each constraint, indicate if its different components are active.
- Parameters:
- Returns:
For each constraint, a boolean indicator of activation of its different components.
- Return type:
- get_all_function_name()¶
Retrieve the names of all the function of the optimization problem.
These functions are the constraints, the objective function and the observables.
- get_all_functions(original=False)¶
Retrieve all the functions of the optimization problem.
These functions are the constraints, the objective function and the observables.
- Parameters:
original (bool) –
Whether to return the original functions or the preprocessed ones.
By default it is set to False.
- Returns:
All the functions of the optimization problem.
- Return type:
- get_best_infeasible_point()¶
Retrieve the best infeasible point within a given tolerance.
- get_constraint_names()¶
Retrieve the names of the constraints.
- get_constraints_number()¶
Retrieve the number of constraints.
- Returns:
The number of constraints.
- Return type:
- get_data_by_names(names, as_dict=True, filter_non_feasible=False)¶
Return the data for specific names of variables.
- Parameters:
- Returns:
The data related to the variables.
- Return type:
- get_design_variable_names()¶
Retrieve the names of the design variables.
- get_dimension()¶
Retrieve the total number of design variables.
- Returns:
The dimension of the design space.
- Return type:
- get_eq_constraints()¶
Retrieve all the equality constraints.
- Returns:
The equality constraints.
- Return type:
- get_eq_constraints_number()¶
Retrieve the number of equality constraints.
- Returns:
The number of equality constraints.
- Return type:
- get_eq_cstr_total_dim()¶
Retrieve the total dimension of the equality constraints.
This dimension is the sum of all the outputs dimensions of all the equality constraints.
- Returns:
The total dimension of the equality constraints.
- Return type:
- get_feasible_points()¶
Retrieve the feasible points within a given tolerance.
This tolerance is defined by
OptimizationProblem.eq_tolerance
for equality constraints andOptimizationProblem.ineq_tolerance
for inequality ones.
- get_function_dimension(name)¶
Return the dimension of a function of the problem (e.g. a constraint).
- Parameters:
name (str) – The name of the function.
- Returns:
The dimension of the function.
- Raises:
ValueError – If the function name is unknown to the problem.
RuntimeError – If the function dimension is not unavailable.
- Return type:
- get_function_names(names)¶
Return the names of the functions stored in the database.
- get_functions_dimensions(names=None)¶
Return the dimensions of the outputs of the problem functions.
- Parameters:
names (Iterable[str] | None) – The names of the functions. If
None
, then the objective and all the constraints are considered.- Returns:
The dimensions of the outputs of the problem functions. The dictionary keys are the functions names and the values are the functions dimensions.
- Return type:
- get_ineq_constraints()¶
Retrieve all the inequality constraints.
- Returns:
The inequality constraints.
- Return type:
- get_ineq_constraints_number()¶
Retrieve the number of inequality constraints.
- Returns:
The number of inequality constraints.
- Return type:
- get_ineq_cstr_total_dim()¶
Retrieve the total dimension of the inequality constraints.
This dimension is the sum of all the outputs dimensions of all the inequality constraints.
- Returns:
The total dimension of the inequality constraints.
- Return type:
- get_last_point()¶
Return the last design point.
- Returns:
The last point result, defined by:
the value of the objective function,
the value of the design variables,
the indicator of feasibility of the last point,
the value of the constraints,
the value of the gradients of the constraints.
- Raises:
ValueError – When the optimization database is empty.
- Return type:
tuple[ndarray, ndarray, bool, dict[str, ndarray], dict[str, ndarray]]
- get_nonproc_constraints()¶
Retrieve the non-processed constraints.
- Returns:
The non-processed constraints.
- Return type:
- get_nonproc_objective()¶
Retrieve the non-processed objective function.
- Return type:
- get_number_of_unsatisfied_constraints(design_variables, values=mappingproxy({}))¶
Return the number of scalar constraints not satisfied by design variables.
- Parameters:
- Returns:
The number of unsatisfied scalar constraints.
- Return type:
- get_objective_name(standardize=True)¶
Retrieve the name of the objective function.
- get_observable(name)¶
Return an observable of the problem.
- Parameters:
name (str) – The name of the observable.
- Returns:
The pre-processed observable if the functions of the problem have already been pre-processed, otherwise the original one.
- Return type:
- get_optimum()¶
Return the optimum solution within a given feasibility tolerances.
- Returns:
The optimum result, defined by:
the value of the objective function,
the value of the design variables,
the indicator of feasibility of the optimal solution,
the value of the constraints,
the value of the gradients of the constraints.
- Raises:
ValueError – When the optimization database is empty.
- Return type:
tuple[ndarray, ndarray, bool, dict[str, ndarray], dict[str, ndarray]]
- get_reformulated_problem_with_slack_variables()¶
Add slack variables and replace inequality constraints with equality ones.
Given the original optimization problem,
\[ \begin{align}\begin{aligned}min_x f(x)\\s.t.\\g(x)\leq 0\\h(x)=0\\l_b\leq x\leq u_b\end{aligned}\end{align} \]Slack variables are introduced for all inequality constraints that are non-positive. An equality constraint for each slack variable is then defined.
\[ \begin{align}\begin{aligned}min_{x,s} F(x,s) = f(x)\\s.t.\\H(x,s) = h(x)=0\\G(x,s) = g(x)-s=0\\l_b\leq x\leq u_b\\s\leq 0\end{aligned}\end{align} \]- Returns:
An optimization problem without inequality constraints.
- Return type:
- get_scalar_constraint_names()¶
Return the names of the scalar constraints.
- get_x0_normalized(cast_to_real=False, as_dict=False)¶
Return the initial values of the design variables after normalization.
- Parameters:
- Returns:
The current values of the design variables normalized between 0 and 1 from their lower and upper bounds.
- Return type:
- has_constraints()¶
Check if the problem has equality or inequality constraints.
- Returns:
True if the problem has equality or inequality constraints.
- Return type:
- has_eq_constraints()¶
Check if the problem has equality constraints.
- Returns:
True if the problem has equality constraints.
- Return type:
- has_ineq_constraints()¶
Check if the problem has inequality constraints.
- Returns:
True if the problem has inequality constraints.
- Return type:
- has_nonlinear_constraints()¶
Check if the problem has non-linear constraints.
- Returns:
True if the problem has equality or inequality constraints.
- Return type:
- static ineq_constraint2(x_dv)[source]¶
Compute the second inequality constraint \(x_1^3 - 0.5 > 0\).
- static ineq_constraint2_jac(x_dv)[source]¶
Compute the gradient of the second inequality constraint.
- is_max_iter_reached()¶
Check if the maximum amount of iterations has been reached.
- Returns:
Whether the maximum amount of iterations has been reached.
- Return type:
- is_point_feasible(out_val, constraints=None)¶
Check if a point is feasible.
Notes
If the value of a constraint is absent from this point, then this constraint will be considered satisfied.
- Parameters:
out_val (Mapping[str, RealOrComplexArray]) – The values of the objective function, and eventually constraints.
constraints (Iterable[MDOFunction] | None) – The constraints whose values are to be tested. If
None
, then take all constraints of the problem.
- Returns:
The feasibility of the point.
- Return type:
- pow2(x_dv)[source]¶
Compute the objective \(x_0^2 + x_1^2 + x_2^2\).
- Parameters:
x_dv (ndarray) – The design variable vector.
- Returns:
The objective value.
- Raises:
ValueError – When
exception_error
isTrue
and the method has already been called three times.- Return type:
- preprocess_functions(is_function_input_normalized=True, use_database=True, round_ints=True, eval_obs_jac=False, support_sparse_jacobian=False)¶
Pre-process all the functions and eventually the gradient.
Required to wrap the objective function and constraints with the database and eventually the gradients by complex step or finite differences.
- Parameters:
is_function_input_normalized (bool) –
Whether to consider the function input as normalized and unnormalize it before the evaluation takes place.
By default it is set to True.
use_database (bool) –
Whether to wrap the functions in the database.
By default it is set to True.
round_ints (bool) –
Whether to round the integer variables.
By default it is set to True.
eval_obs_jac (bool) –
Whether to evaluate the Jacobian of the observables.
By default it is set to False.
support_sparse_jacobian (bool) –
Whether the driver support sparse Jacobian.
By default it is set to False.
- Return type:
None
- static repr_constraint(function, constraint_type, value=None, positive=False)¶
Express a constraint as a string expression.
- Parameters:
function (MDOFunction) – The constraint function.
constraint_type (MDOFunction.ConstraintType) – The type of the constraint.
value (float | None) – The value for which the constraint is active. If
None
, this value is 0.positive (bool) –
If
True
, then the inequality constraint is positive.By default it is set to False.
- Returns:
A string representation of the constraint.
- Return type:
- reset(database=True, current_iter=True, design_space=True, function_calls=True, preprocessing=True)¶
Partially or fully reset the optimization problem.
- Parameters:
database (bool) –
Whether to clear the database.
By default it is set to True.
current_iter (bool) –
Whether to reset the current iteration
OptimizationProblem.current_iter
.By default it is set to True.
design_space (bool) –
Whether to reset the current point of the
OptimizationProblem.design_space
to its initial value (possibly none).By default it is set to True.
function_calls (bool) –
Whether to reset the number of calls of the functions.
By default it is set to True.
preprocessing (bool) –
Whether to turn the pre-processing of functions to False.
By default it is set to True.
- Return type:
None
- to_dataset(name='', categorize=True, opt_naming=True, export_gradients=False, input_values=())¶
Export the database of the optimization problem to a
Dataset
.The variables can be classified into groups:
Dataset.DESIGN_GROUP
orDataset.INPUT_GROUP
for the design variables andDataset.FUNCTION_GROUP
orDataset.OUTPUT_GROUP
for the functions (objective, constraints and observables).- Parameters:
name (str) –
The name to be given to the dataset. If empty, use the name of the
OptimizationProblem.database
.By default it is set to “”.
categorize (bool) –
Whether to distinguish between the different groups of variables. Otherwise, group all the variables in
Dataset.PARAMETER_GROUP`
.By default it is set to True.
opt_naming (bool) –
Whether to use
Dataset.DESIGN_GROUP
andDataset.FUNCTION_GROUP
as groups. Otherwise, useDataset.INPUT_GROUP
andDataset.OUTPUT_GROUP
.By default it is set to True.
export_gradients (bool) –
Whether to export the gradients of the functions (objective function, constraints and observables) if the latter are available in the database of the optimization problem.
By default it is set to False.
input_values (Iterable[ndarray]) –
The input values to be considered. If empty, consider all the input values of the database.
By default it is set to ().
- Returns:
A dataset built from the database of the optimization problem.
- Return type:
- to_hdf(file_path, append=False, hdf_node_path='')¶
Export the optimization problem to an HDF file.
- Parameters:
file_path (str | Path) – The path of the file to store the data.
append (bool) –
If
True
, then the data are appended to the file if not empty.By default it is set to False.
hdf_node_path (str) –
The path of the HDF node in which the database should be exported. If empty, the root node is considered.
By default it is set to “”.
- Return type:
None
- OPTIM_DESCRIPTION: ClassVar[str] = ['minimize_objective', 'fd_step', 'differentiation_method', 'pb_type', 'ineq_tolerance', 'eq_tolerance']¶
- activate_bound_check: ClassVar[bool] = True¶
Whether to check if a point is in the design space before calling functions.
- property constraint_names: dict[str, list[str]]¶
The standardized constraint names bound to the original ones.
- constraints: list[MDOFunction]¶
The constraints.
- design_space: DesignSpace¶
The design space on which the optimization problem is solved.
- property is_mono_objective: bool¶
Whether the optimization problem is mono-objective.
- Raises:
ValueError – When the dimension of the objective cannot be determined.
- new_iter_observables: list[MDOFunction]¶
The observables to be called at each new iterate.
- nonproc_constraints: list[MDOFunction]¶
The non-processed constraints.
- nonproc_new_iter_observables: list[MDOFunction]¶
The non-processed observables to be called at each new iterate.
- nonproc_objective: MDOFunction¶
The non-processed objective function.
- nonproc_observables: list[MDOFunction]¶
The non-processed observables.
- property objective: MDOFunction¶
The objective function.
- observables: list[MDOFunction]¶
The observables.
- property parallel_differentiation_options: dict[str, int | bool]¶
The options to approximate the derivatives in parallel.
- pb_type: ProblemType¶
The type of optimization problem.
- solution: OptimizationResult | None¶
The solution of the optimization problem if solved; otherwise
None
.
- use_standardized_objective: bool¶
Whether to use standardized objective for logging and post-processing.
The standardized objective corresponds to the original one expressed as a cost function to minimize. A
DriverLibrary
works with this standardized objective and theDatabase
stores its values. However, for convenience, it may be more relevant to log the expression and the values of the original objective.