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Max Gams

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Arguments in square brackets may be omitted: the default values used in such cases are specified in the function description provided.

Mathematical functions may be used as expressions in assignment statements and in equation definitions.

We start with some simple examples to illustrate. The GAMS function log x returns the natural logarithm of its argument. The assignment above evaluates the logarithm once for each element of the controlling domain j.

The function max returns the maximum of a set of expressions or terms. If the argument DECPL is not specified, it defaults to zero, and the function rounds its argument to the nearest integer value.

For example,. In the first assignment q is rounded to the nearest integer value, while in the second q is rounded to the number of decimals specified by d.

The result of the third assignment is Table 3: Mathematical Functions. GAMS provides some string manipulation capability by extending the card and ord functions to work on strings as well as sets.

In Table 4 the extended behavior is described. In this context, the functions take strings and places as arguments, and the numeric places argument must be a constant.

This behavior only applies to execution-time usage of these functions. Table 4: String Manipulation Functions. There are four types or sources of strings in this context.

A string may be a string literal, i. It may be the symbol text aka the explanatory text associated with any symbol.

Or it may be the labels or text associated with the elements of a set. The following table gives an overview:. Table 5: String Types.

The string types are illustrated in the following example. Note that the directive in the first line activates the end-of-line comment option, see eolCom.

Note that the strings teas. This is a typical usage pattern. Logical functions may be used as expressions in assignment statements as in the following example.

Note that logical functions may also be used in conditions and logic equations, see sections Logical Conditions and Logic Equations respectively.

Most of the logical functions can also be indicated using the familiar operator notation, e. In such cases, the operator notation is allowed at compile time.

The last column in Table 6 indicates if a logical function can be used at compile time yes or if only its alternative notation is available alt.

As expected, false becomes 0 and true becomes 1 when converting logical values to numeric, and 0 becomes false and nonzero becomes true when numeric values are converted to logicals.

Table 6: Logical Functions. GAMS offers several functions that relate to time and dates. This serial day number is a real number whose integer part contains a unique number for each day and whose fractional part contains information about hours, minutes, and seconds.

We can think of the serial day number as being a date. The day information extracted from serial day numbers is based on the Gregorian calendar.

All of the functions in Table 7 can be used at compile time. Table 7: Time and Calendar Functions. Apart from the import syntax, the imported functions may be used in the same way as intrinsic functions.

In particular, they may be used in equation definitions. Some function libraries are included with the standard GAMS software distribution, but GAMS users can also create their own libraries using an open programming interface.

For details on using and creating extrinsic function libraries, and on the extrinsic function libraries that are provided with GAMS, see the chapter on Extrinsic Functions.

Up to this point, this section has described and discussed functions without mentioning their derivatives. These derivatives are very important, though: most of the nonlinear solvers integrated with GAMS will require first derivatives to solve models, and many will also use or require second derivatives as well.

It is sometimes useful e. This can be done via function suffixes. Function suffixes can specify the evaluation of a gradient i. A full list of function suffixes is given in Table 9.

For functions whose arguments are constant, the derivatives are zero, so typically func is a mathematical function listed in Table 3 above.

Note that function suffixes are not defined for functions without arguments for example, pi , so specifying something like pi. Table 9: Function Suffixes.

For univariate functions like sine or cosine, there is no need to specify a variable index when evaluating derivatives.

For multivariate functions, the default is to take partial derivatives w. To specify other variables, the colon syntax in the example below is used.

To compute derivatives numerically, GAMS uses finite difference approximations. This computation can be controlled with two options: the FDOpt option controls which variant of the finite difference method is used, while the FDDelta option controls the step size.

GAMS uses an extended range arithmetic to handle missing data, the results of undefined operations, infinite values, and zeros that are stored explicitly.

The special values used in this arithmetic are listed and described in Table 10 below, along with the value of the mapVal function that corresponds to these values.

We can think of special values as any value for which mapVal does not return 0. GAMS has defined the results of all arithmetic operations and all function evaluations that use these special values.

The behavior is designed to both maximize utility and minimize surprise, and is illustrated in the library model [CRAZY].

The following table shows a selection of results for exponentiation and division for a variety of input parameters including NA and INF. Note that most extended range identifiers may be used in assignment statements, as illustrated below.

The special value EPS is numerically equal to zero, so when used in the context of numerical comparisons, it behaves as zero. However, EPS is a stored zero, so setting a parameter to zero e.

This is sometimes useful, especially if a is used later in loops see Programming Flow Control Features or dollar conditions see Conditional Expressions, Assignments and Equations.

For example, consider the statement. In the case where a was originally assigned to be zero, the dollar-condition is always false so no infinities are assigned.

If a was originally assigned the value EPS, the dollar-condition is always true and every tuple in a will be set to infinity. When an attempted arithmetic operation is illegal or has undefined results because of the value of arguments division by zero is the normal example , an error is reported and the result is set to undefined UNDF.

GAMS will also report the line number of the offending statement and give as much detail as possible about the cause.

From there on, the resulting UNDF is treated as a proper data value and does not trigger additional error messages.

It is therefore wise to anticipate and avoid operations like division by zero that will cause errors. This is most easily done with the dollar control, and is discussed in section The Dollar Condition.

With the exception of sameAs , diag , and sortedUels these predefined symbols are for very special and mostly internal purposes, but the way they work is identical.

These symbols can be used in a GAMS program without being declared but work in an idiosyncratic way.

For example, the complete program. However, if labels have been declared in any set that corresponds to a solver name, then set solvers will contain this label.

Consider the following example:. Note that Antigone and Baron are the names of solvers in GAMS and therefore they will be displayed as a member of the set solvers.

Also note that the set sameAs contains the diagonal elements for both sets:. User defined symbols can have the same name as a predefined symbol.

In this case the user symbol hides the predefined symbols as demonstrated in the following example:. The log will only contain the line diag is a set.

GAMS provides powerful facilities for data manipulation with parallel assignment statements, built-in functions and extended range arithmetic.

Data Manipulations with Parameters. Introduction In this chapter we explicitly cover only parameter manipulation, including all aspects of data handling.

Note An assignment cannot start with a reserved word. A semicolon is therefore required as a delimiter before all assignments.

Attention The index set s on the left hand side of an indexed assignment are referred to synonymously as the controlling indices, controlling sets, or controlling domain of the assignment.

Restricting the Domain: Explicit Labels The strongest restriction of the domain is assigning a value to just one element. Restricting the Domain: Subsets In general, wherever a set name may occur in an indexed assignment, a subset may be used instead.

Restricting the Domain: Conditionals Conditional assignments are discussed in detail in section Conditional Assignments in chapter Conditional Expressions, Assignments and Equations.

There should be no index on the right-hand side of the assignment that is not present on the left unless it is operated on by an indexed operator. For more on indexed operators, see section Indexed Operations.

Attention Each set is counted only once to determine the number of controlling indices. If the intent is for a set to appear independently more than once within the controlling domain, the second and subsequent occurrences of the set should be aliases of the original set, so that the number of controlling indices is equal to the number of indices.

Standard Arithmetic Operations The standard arithmetic symbols and operations and their order of precedence are given in Table 1.

When evaluating nonlinear functions and operators GAMS guards against evaluating at or very near singular points and in such cases signals an error or returns an appropriate function value.

Note As usual, operations within parentheses or brackets are evaluated before other numerical calculations, where the innermost parentheses are resolved first.

Note It is often better to use parentheses than to rely on the order of precedence of operators, since this prevents errors and clarifies intentions.

Operation Keyword Summation sum Product prod Minimum value smin Maximum value smax Table 2: Indexed Operations These four operations are performed over one or more controlling indices.

Note In the context of assignment statements, the attributes of variables and equations e. For more on variable and equations attributes, see sections Variable Attributes and Equation Attributes respectively.

In the context of equation definitions, scalars, parameters and variables may appear freely in indexed operations. For more on equation definitions, see section Defining Equations.

Intrinsic Functions GAMS provides many functions, ranging from commonly used standard functions like exponentiation, logarithms, and trigonometric functions to utility functions for controlling and querying the running GAMS job or process.

Mathematical Functions Mathematical functions may be used as expressions in assignment statements and in equation definitions. Max x1,x2,x3, NLP no eDist x1[,x2,x3, Note that the state of the RNG cannot typically be captured in one seed value; in such cases "reading" the seed involves harvesting a seed from the RNG, resetting the RNG with this seed, and returning the seed.

Maximum value of the arguments, where the number of arguments may vary. DNLP yes min x1,x2,x3, Minimum value of the arguments, where the number of arguments may vary.

The optional place entry defaults to 1 if omitted and identifies the character position within the string to be examined 1 for the first, 2 for the second, etc.

The following table gives an overview: Table 5: String Types Notation Description Comments characters A concatenation of characters and blanks: all legal GAMS characters are allowed, see section Characters for details.

The characters must be surrounded by double or single quotes. If the explanatory text is missing, the value of card is 0. Logical Functions Logical functions may be used as expressions in assignment statements as in the following example.

See example above. Note In all functions given in Table 7 , serial day 1 is January 1, Compile Time embeddedHandle Returns the handle for the last embedded code section executed, see section Syntax in chapter Embedded Code Facility for details.

For details, see Table 1 in section Grid Computing. The return value is 1 if this was successful, 0 otherwise. See also maxExecError. Function Suffixes Up to this point, this section has described and discussed functions without mentioning their derivatives.

Note The function suffixes value , grad , gradn , hess and hessn are also defined for extrinsic functions. When implementing an extrinsic function, be aware that missing derivatives will be computed numerically: see model [TRILIB01] for an example.

Extended Range Arithmetic and Error Handling GAMS uses an extended range arithmetic to handle missing data, the results of undefined operations, infinite values, and zeros that are stored explicitly.

Similar to IEEE plus infinity. Behaves in the expected way in computations, so that e. Similar to IEEE minus infinity. It is a sticky value: e.

For details, see the chapter on Dollar Control Options. For details on dollar control options, see chapter Dollar Control Options.

Note GAMS will not solve a model if an error has been detected, and it will terminate with an error condition. Minimum value. Maximum value.

Regularized beta function , see MathWorld. Reads or resets the seed for the random number generator RNG. Log Gamma function as discussed in MathWorld.

Common logarithm : logarithm base 10, see MathWorld. Binary logarithm : logarithm base 2, see MathWorld. Drive your way through hordes of zombies, collect hardware to upgrade and customize new vehicles.

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An expression is an arbitrarily complicated specification for a calculation, with parentheses nested as needed for clarity and intent.

We will cover standard arithmetic operations and indexed operations here, and functions and extended range arithmetic in the next two sections.

The standard arithmetic symbols and operations and their order of precedence are given in Table 1. Note that 1 denotes the highest order of precedence and 3 denotes the lowest order of precedence.

Parentheses can be used to override the default precedence order in the usual way. Operators including exponentiation on the same level are evaluated from left to right.

Note that the full GAMS operator precedence hierarchy also includes logical operators; it is given in section Mixed Logical Conditions. Note further that the symbols for addition, subtraction and multiplication have a different meaning if they are used in the context of sets.

Note that expressions may be freely continued over many lines: an end of line is permissible at any point where a blank may be used. Blanks may be used for readability around identifiers, parentheses and operation symbols.

Blanks are not allowed within identifiers or numbers, and are significant inside the quotes used to delimit labels. In addition to the simple operations in Table 1 , GAMS also provides the following four indexed operations.

These four operations are performed over one or more controlling indices. The syntax in GAMS for these operations is:.

If there is only one controlling index, the parentheses around it may be removed. The index over which the summation is done, i , is separated from the reserved word sum by a left parenthesis and from the data term capacity i,m by a comma.

The set i is called the controlling index for this operation. The scope of the control is the pair of parentheses that start immediately after the sum.

It is also possible to sum simultaneously over the domain of two or more sets as in the first assignment that follows.

The second assignment demonstrates the use of a less trivial expression than an identifier within the indexed operation.

In the context of sets the sum operator may be used to compute the number of elements in a set and for projections. For details see section Projection and Aggregation of Sets.

The smin and smax operations are used to find the largest and smallest values over the domain of the index set or sets. The index for the smin and smax operators is specified in the same manner as in the index for the sum operator.

In the following example we want to find the largest capacity:. Sometimes it is necessary to restrict the domain of indexed operations.

This may be done with the same techniques as in indexed assignments, see section Restricting the Domain in Assignments for details. See also section Conditional Indexed Operations for more details on conditions in the context of indexed operations.

We now turn to two additional capabilities that are available in GAMS to add power and flexibility to expression calculations: functions and extended range arithmetic.

Functions play an important role in the GAMS language, especially for nonlinear models. Like other programming languages, GAMS provides a number of built-in or intrinsic functions.

GAMS is used in an extremely diverse set of application areas and this creates frequent requests for the addition of new and quite sophisticated or specialized functions.

There is a trade-off between satisfying these requests and avoiding a complexity not needed by most users.

GAMS provides many functions, ranging from commonly used standard functions like exponentiation, logarithms, and trigonometric functions to utility functions for controlling and querying the running GAMS job or process.

For details specific to using these functions in equations, see the section on Expressions in Equation Definitions. Some of the tables that follow contain an endogenous classification column "End.

Note well: functions classified as any are only permitted with exogenous constant arguments. Functions are typically used in assignment statements and equations.

In these cases, they are only evaluated at execution time. Some functions can also be used at compile time, e. Some of the tables below contain a column "Compile Time" indicating which functions can be used at compile time.

A word on the notation in the tables below: for function arguments, lower case indicates that an endogenous variable is allowed. For details on endogenous variables, see section Functions in Equation Definitions.

Upper case function arguments indicate that a constant is required. Arguments in square brackets may be omitted: the default values used in such cases are specified in the function description provided.

Mathematical functions may be used as expressions in assignment statements and in equation definitions. We start with some simple examples to illustrate.

The GAMS function log x returns the natural logarithm of its argument. The assignment above evaluates the logarithm once for each element of the controlling domain j.

The function max returns the maximum of a set of expressions or terms. If the argument DECPL is not specified, it defaults to zero, and the function rounds its argument to the nearest integer value.

For example,. In the first assignment q is rounded to the nearest integer value, while in the second q is rounded to the number of decimals specified by d.

The result of the third assignment is Table 3: Mathematical Functions. GAMS provides some string manipulation capability by extending the card and ord functions to work on strings as well as sets.

In Table 4 the extended behavior is described. In this context, the functions take strings and places as arguments, and the numeric places argument must be a constant.

This behavior only applies to execution-time usage of these functions. Table 4: String Manipulation Functions.

There are four types or sources of strings in this context. A string may be a string literal, i. It may be the symbol text aka the explanatory text associated with any symbol.

Or it may be the labels or text associated with the elements of a set. The following table gives an overview:.

Table 5: String Types. The string types are illustrated in the following example. Note that the directive in the first line activates the end-of-line comment option, see eolCom.

Note that the strings teas. This is a typical usage pattern. Logical functions may be used as expressions in assignment statements as in the following example.

Note that logical functions may also be used in conditions and logic equations, see sections Logical Conditions and Logic Equations respectively.

Most of the logical functions can also be indicated using the familiar operator notation, e. In such cases, the operator notation is allowed at compile time.

The last column in Table 6 indicates if a logical function can be used at compile time yes or if only its alternative notation is available alt. As expected, false becomes 0 and true becomes 1 when converting logical values to numeric, and 0 becomes false and nonzero becomes true when numeric values are converted to logicals.

Table 6: Logical Functions. GAMS offers several functions that relate to time and dates. This serial day number is a real number whose integer part contains a unique number for each day and whose fractional part contains information about hours, minutes, and seconds.

We can think of the serial day number as being a date. The day information extracted from serial day numbers is based on the Gregorian calendar.

All of the functions in Table 7 can be used at compile time. Table 7: Time and Calendar Functions. Apart from the import syntax, the imported functions may be used in the same way as intrinsic functions.

In particular, they may be used in equation definitions. Some function libraries are included with the standard GAMS software distribution, but GAMS users can also create their own libraries using an open programming interface.

For details on using and creating extrinsic function libraries, and on the extrinsic function libraries that are provided with GAMS, see the chapter on Extrinsic Functions.

Up to this point, this section has described and discussed functions without mentioning their derivatives. These derivatives are very important, though: most of the nonlinear solvers integrated with GAMS will require first derivatives to solve models, and many will also use or require second derivatives as well.

It is sometimes useful e. This can be done via function suffixes. Function suffixes can specify the evaluation of a gradient i.

A full list of function suffixes is given in Table 9. For functions whose arguments are constant, the derivatives are zero, so typically func is a mathematical function listed in Table 3 above.

Note that function suffixes are not defined for functions without arguments for example, pi , so specifying something like pi.

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