Consider a system for which the outputs z are an unknown function f of the system inputs x. The software identifies a function f , i.e., a relationship between the inputs and outputs of the system, that best matches data (pairs of x and corresponding z values) that are collected via simulation or experimentation. ALAMO can:

•build an algebraic model of a simulation or experimental black-box system

•use previously collected data for model building

•call a user-specified function (simulator) to collect measurements

•enforce response variable bounds, physical limits, and boundary conditions

•use a preexisting data set for model validation

•output models in simple algebraic form

The problems addressed by the software have long been studied in the fields of statistics, design of experiments, and machine learning. Whereas existing techniques from this literature can be used to fit data to models, the main challenges addressed by the ALAMO software are in determining where to run the simulations or experiments, what models to fit, and how to determine if the model is accurate and as simple as possible. A distinguishing feature of ALAMO is that it provides models that are accurate yet as simple as possible. Moreover, ALAMO is capable of utilizing theory-driven insights alongside data. The ALAMO models can be used to facilitate subsequent system analysis, optimization, and decision making.

1.1Licensing and software requirements

The code is available for download at http://minlp.com/alamo. The same URL provides infor- mation about licensing the software. Under certain algorithmic options, especially for constrained regression, ALAMO makes calls to the third party software GAMS. If the user intends to apply ALAMO’s constrained regression capability, a separate install and license are required for GAMS and at least one of the mixed-integer quadratic programming solvers under GAMS, preferably GAMS/BARON. More information about GAMS can be found at http://www.gams.com. In the absence of a GAMS license, ALAMO attempts to use enumerative approaches that may be more time consuming or impractical for large problems. For constrained regression only, GAMS/BARON is required.

1.2Installation

Install ALAMO and the ALAMO license file in any directory of your choice and add it to your path. On Windows, ALAMO’s installer will take care of these steps for you. On Linux and OSX systems, unzip the ALAMO download and place the files it contains in a directory in your path. Users should not open the license file in an editor as this may invalidate the license in certain operating systems. For all operating systems, make sure that ALAMO and the ALAMO license file are readable by all intended users on your machine. Do the same for GAMS. Installation of GAMS is optional but recommended.

2Algorithms implemented

ALAMO seeks to identify low-complexity surrogate models using a minimal amount of data for a system that is described by a simulator or experiment. Surrogate models are constructed using a three-step process. In the first step, an initial design of experiments is generated and the simulation is queried at these points. In the second step, an algebraic model is built using this initial training set. The model is built using integer optimization techniques to select the best subset from a collection of potential sets of basis functions that can be used to build up the model. In the third step, an adaptive sampling methodology based on derivative-free optimization techniques is used to identify points where the model is inaccurate. Once these points are added to the training set, execution returns to the second step of the algorithm. The process continues until the third step confirms the accuracy of a previously built model.

Compared to common techniques, such as forward- or backward-regression, that investigate model sensitivities with respect to one basis function at a time, ALAMO’s best subset selection techniques ensure that its model-building steps account for the synergistic effects between differ- ent basis functions. Before ALAMO, best subset selection techniques were considered too time consuming for application to realistic data sets. While developing ALAMO, nonlinear integer programming techniques were devised that rely on the BARON software to solve these models in realistic computing times for many industrially relevant systems. ALAMO is also unique in that it utilizes derivative-free optimization techniques in its adaptive sampling step. These techniques offer a systematic approach to interrogate models, identify weaknesses, and guide experimental design towards parts of the space requiring more attention. Another distinctive feature of ALAMO is its constrained regression feature, which is capable of enforcing theory- driven requirements on response variables, including response variable bounds, thermodynamic limitations, and boundary conditions. To enforce these requirements over the entire domain of input variables, ALAMO relies on BARON to solve semi-infinite nonconvex optimization prob- lems.

The bibliography at the end of this document offers more details of the methodology implemented in ALAMO and demonstrates the advantages of this methodology in comparison to currently utilized approaches, including classical regression and the lasso.

3Running ALAMO

3.1Running the interactive GUI

The ALAMO GUI allows for a convenient spreadsheet-style input of problem data and algorith- mic options, and offers facilities for visualizing the data and results, and saving your work. On all operating systems, Java is required in order to use the ALAMO GUI. Additionally, the user must have write privileges in the directory where the GUI is invoked.

On Windows systems, the installer will create an ALAMO icon on your desktop. Double click at this icon in order to launch ALAMO’s interactive GUI.

On Linux and OSX, the interactive GUI is available in the download directory and named alamogui.jar. To run this, open a terminal and type the following command at the prompt

options do not call for adaptive sampling to be used, effectively requesting the best possible model that can be derived from the preexisting data set. Finally, the following functions are permitted in the model: linear, logarithmic, exponential, sine, cosine, and monomials with powers 2 and 3.

!Example 1 with data from z = x^2 ninputs 1

noutputs 1 xmin -5 xmax 5

ndata 11

linfcns 1 logfcns 1 expfcns 1 sinfcns 1 cosfcns 1 monomialpower 2 3

BEGIN_DATA

00

11

24

39

416

525

END_DATA

Several additional examples of ALAMO input files accompany the distributed code.

5Input file grammar

The following rules should be followed when preparing an ALAMO input file:

•The name of the input file should include its exact path location if the file is not present in the execute directory.

•The name of the input file should not exceed 1000 characters in length.

•The input is not case sensitive.

•Most options are entered one per line, in the form of ‘keyword’ followed by ‘value’. Certain vector options are entered in multiple lines, starting with ‘BEGIN <keyword>’, followed by the vector input, followed by ‘END <keyword>’.

•Certain options must appear first in the input file. This requirement is discussed explicitly in option descriptions provided below.

•With the exception of arguments involving paths, character-valued options should not contain spaces.

•Blank lines, white space, and lines beginning with *, #, % or ! are skipped. Inline comments that are preceded by #, % or ! are permitted in any line that contains alphanumeric options. Blocks of comment lines are allowed using ‘BEGIN COMMENT’, followed by the block of comment lines, followed by ‘END COMMENT’; these comment blocks are entirely ignored by ALAMO.

6ALAMO data and options specification statements

6.1Required scalar parameters

The following parameters must be specified in the input file in the order listed below.

z.

6.2Required vector parameters

The following parameters must be specified in the input file in the order listed below and only after the scalar required parameters have already been specified.

XMIN and XMAX do not necessarily correspond to the minimum and maximum values of the data provided by the user to ALAMO. Instead, they should reflect the physically meaningful lower and upper bounds for the input variables. These values are used by ALAMO for scaling

purposes as well as for generating samples when no preexisting data set is provided or, in general, when adaptive sampling is done.

6.3Optional data specifications

This section describes optional parameters pertaining to the particular problem being solved.

pute model errors at the validation data points. NVAL- SAMPLE points are sampled randomly and added to the validation data sets. The sampling facility requires that the user provides a SIMULATOR. Validation can rely ex- clusively on preexisting data (through the NVALDATA option), exclusively on sampled data (through the NVAL- SAMPLE option), or on any combination desired by the user. NVALSAMPLE must be an array of NVALSETS nonnegative integers. If NVALSAMPLE is provided and NVALSETS has not already been specified in the input file, ALAMO will assume that NVALSETS equals 1.

Possible SAMPLER values are: 1 (random), 2 (SNOB-

FIT), 3 (Faure).

to read problem, preprocess data, solve optimization sub- problems, and print results.

ger variables, ALAMO’s model will include the rounding of a function to the nearest integer (equivalent to the nint function in Fortran.)

input/output variables. The last NTRANS of the out- put variables are obtained through algebraic transforma- tions of the input variables and/or the first NOUTPUTS- NTRANS output variables; their values should not be pro- vided in any DATA section or calculated by the simulator.

The parser is not case sensitive. For example, output variable labels Z1 and z1 are equivalent. For vector inputs, any items provided in excess of those required will be ignored. For example, if more than NINTPUTS XLABELS are provided, the extra labels are ignored.

In deciding whether to deactivate printing to the screen or file, users should consider that model coefficients are printed with two significant digits to the screen and with 23 digits to the listing file.

If the parameter NDATA is set, then a data section must follow subsequently in the input file with precisely NDATA rows, one for each data point (pair of x and z values) specified in the following form:

BEGIN_DATA

.

.

.

END_DATA

If the parameter NPREDATA is set, then a data section must follow subsequently in the input file with precisely NPREDATA rows, each containing precisely NINPUTS values, thus corresponding to a point in the x-space, specified in the following form:

BEGIN_XPREDATA

.

.

.

END_XPREDATA

If the parameter NVALDATA is set, a similar data section must be provided using a similar construct:

BEGIN_VALDATA

.

.

.

END_VALDATA

If the parameter NCUSTOMBAS is set, user-specified basis functions must be provided using the construct:

BEGIN_CUSTOMBAS

.

.

.

END_CUSTOMBAS

where basis functions are provided one per line. The parser is not case sensitive and allows for Fortran functional expression in terms of the XLABELS. The following functions are currently accepted by the parser: addition, subtraction, multiplication, division, power (∗∗ and ^), abs, exp, log, log10, sqrt, sinh, cosh, tanh, sin, cos, tan, asin, acos, and atan. Other functions may be expressed in terms of the preceding operators and functions, e.g., min(a, b) = (a+b)/2−|a−b|/2.

If the parameters MONO, MULTI2, MULTI3, or RATIOS are set, the corresponding powers must also be specified as row vectors of corresponding length in the following way:

Parameter Description

MONOMIALPOWERRow vector of monomial powers considered in basis func- tions; powers of 0 or 1 are not allowed. If MONO is pro-

The entries of the above vectors must be space separated. As stated above, the user is not obligated to specify the parameters MONO, MULTI2, MULTI3, or RATIOS. If any of the cor- responding power options are provided, ALAMO will count them and infer the total number of powers specified by the user.

If the parameter NTRANS is set to a positive entry, NTRANS functions must be provided using the construct:

BEGIN_TRANSFORMS

.

.

.

END_TRANSFORMS

where transformation functions are provided one per line. Line k of a TRANSFORMS section provides an algebraic transformation that determines transformed output k which, in turn, cor- responds to output variable NOUTPUTS-NTRANS+k. The parser is not case sensitive and allows for Fortran functional expression in terms of the XLABELS and ZLABELS. For the types of functions supported, see the discussion under BEGIN CUSTOMBAS above.

6.4Constrained regression

This section describes ALAMO’s constrained regression capabilities. There are two distinct constrained regression capabilities implemented currently in ALAMO:

•The ability to enforce constraints, such as bounds, on the response function.

•The ability to enforce combinatorial constraints on the types of basis functions utilized, including constraints on groups of basis functions.

The primary options that control application of constrained regression to ALAMO’s response function are:

In this construct, each group is uniquely associated with a Group-id ranging from 1 to NGROUPS. Each line must contain three required parameters (Group-id, Member-type, Member-indices); the fourth parameter (Powers) is required only in the context of basis functions that involve powers. The syntax of this section must obey the following rules:

•Each line pertains to a single group.

•A group may be described over several lines, with each line restricted to describing a single type of component of the group.

•Group-id is a nonnegative integer between 1 and NGROUPS that denotes the numerical id (index) of a group described (at least partly) in a line.

•Member-type is an attribute for the member(s) of the group described in the specific line and can take anyone of the values LIN, LOG, EXP, SIN, COS, MONO, MULTI2, MULTI3, RATIO, RBF, CUST, and CONST corresponding to different types of basis functions. In addition, Member-type may be set equal to GRP if it is desired to specify a group of groups.

•Member-indices is used to specify the composition of a group in terms of indices of input variables and groups. The keyword CONST should not be followed by any Member-indices (as there is only one constant in the model). The keywords MONO, EXP, LOG, SIN and COS must be followed by exactly one index that has a value between 1 and NINPUTS corresponding to the input variable involved in the basis function; alternatively, a value of −1 may be used to denote that all input variables should be considered (with a specific power in the case of MONO). The keywords MULTI2, MULTI3 and RATIO must be followed by two, three, and two indices, respectively, indicating the input variables involved in the corresponding basis function; alternatively, an index of −1 may be used to denote that all possible input variable combinations in any of these basis functions should be considered. The keyword LIN may be followed by as many as NINPUTS Member-indices and specifies which linear terms of the model are included in the group; alternatively, an index of −1 may be used to denote that all possible linear terms are included in the group. Similarly, the keywords CUST and RBF may be followed by as many as NCUSTOMBAS and NDATA Member-indices and an index of −1 may be used to denote all possible custom basis functions and RBFs in a group. As many as NGROUPS−1 indices may follow the keyword GRP in order to specify which groups form a group.

•When Member-type is one of MONO, MULTI2, MULTI3, or RATIO, in addition to Member-indices, the input line must specify the power(s) involved in the group; if power equals −1111, all powers are considered.

•Membership in a group is non-exclusive; a basis function or group may belong to multiple groups.

Once the number of groups has been specified and each group has been described through the GROUPS construct, group constraints can be specified through the GROUPCON section:

BEGIN_GROUPCON

.

.

.

END_GROUPCON

ALAMO permits different group constraints to be imposed on different output variables. Each line of the GROUPCON section is dedicated to a group-output variable combination and has the following information:

Group-id Output-id Constraint-type Integer-parameter

The rules for completing this section are as follows:

•Each line describes a single group constraint.

•Group-id is a nonnegative integer between 1 and NGROUPS that denotes the numerical id (index) of the primary (and sometimes only) group involved in the constraint.

•Output-id is a nonnegative integer between 1 and NOUTPUTS for which the constraint will be imposed; a value of −1 can be used in this entry to denote that the constraint should be enforced for all output variables.

•Constraint-type is a string descriptor that can take anyone of the following values:

–NMT: to denote a no-more-than constraint, i.e., require that no more than Integer- parameter members of this group should be selected in the model.

–ATL: to denote an at-least constraint, i.e., require that at least Integer-parameter members of this group should be selected in the model.

–REQ: to require that, if the primary group is selected, then the group with id equal to Integer-parameter should also be selected.

–XCL: to require that, if the primary group is selected, then the group with id equal to Integer-parameter should not be selected.

•A group may appear in more than one constraint. This flexibility coupled with the fact that input variables and groups may belong to multiple groups allows us to enforce sparsity constraints on model attributes within groups, between groups, and groups organized in clusters, trees, or any other structure.

7ALAMO output

7.1ALAMO screen output

The screen output below is obtained for problem e1.alm.

***************************************************************************

ALAMO version 2021.5.8. Built: WIN-64 Sat May 8 17:51:31 EDT 2021

If you use this software, please cite:

22ALAMO user manual and installation guide v. 2021.5.8

Cozad, A., N. V. Sahinidis and D. C. Miller,

Automatic Learning of Algebraic Models for Optimization,

AIChE Journal, 60, 2211-2227, 2014.

ALAMO is powered by the BARON software from http://www.minlp.com/

***************************************************************************

Licensee: Nick Sahinidis at The Optimization Firm, LLC, niksah@minlp.com.

***************************************************************************

Reading input data

Checking input consistency and initializing data structures Warning: eliminating basis log(X1)

Step 0: Initializing data set

User provided an initial data set of 11 data points We will sample no more data points at this stage

***************************************************************************

Iteration 1 (Approx. elapsed time 0.16E-01 s) Step 1: Model building using BIC

Model building for variable Z

----

BIC = -0.100E+31 with Z = X1^2

Calculating quality metrics on observed data set.

Quality metrics for output Z

----------------------------

Normal termination

***************************************************************************

The software first reports the version, platform, and compilation date of the executable, followed by credits. Then, after reading the input data, a consistency check is run on the problem data and, if passed, the data structures are initialized. In this specific example, a warning is issued that logarithmic basis functions are not considered since the input variable is allowed to take negative values. Subsequently, information is provided for all algorithmic steps. During initialization (Step 0), it is reported that 11 data points are used for sampling and that no simulator is called in addition to using the preexisting data set. In Step 1, the model is built in stages. Earlier in the search, the perfect model z = x2 is identified. Since there is no simulator provided, there is no adaptive sampling and execution terminates here after reporting a detailed breakdown of CPU times for the different algorithmic steps, including the number of calls to the optimizer (0 in this example) and the simulator (0 in this example). There are no calls to an optimizer in this example because the problem is small enough to be solved faster by complete enumeration.

8Termination conditions and error messages

Errors in the input file are reported on the screen and/or the listing file in the form of “warnings” and “errors.” ALAMO attempts to continue execution despite warnings. If the errors are severe, the program execution is stopped and the line where the fatal error occurred is displayed. The input file should be checked even if the warnings are not severe, as the problem might have been parsed in a way other than it was intended to be. Detailed error messages are provided in that case.

If execution terminates normally, ALAMO prints ‘Normal termination.’ If there is an error, the message on the screen or file is ‘ALAMO terminated with termination code,’ followed by one of the following error codes, all of which are self-explanatory:

1.ALAMO must be called with one or two command line arguments.

2.ALAMO input file name must be no longer than 1000 characters.

3.ALAMO input file not found.

4.ALAMO input file cannot be opened.

5.Keyword not recognized in input file.

6.Keyword too long in input file.

7.Incomplete input file.

8.Input value in error in input file.

9.Number of input variables (NINPUT) must be specified before specifying XMIN values.

10.Number of input variables (NINPUT) must be specified before specifying XMAX values.

11.Number of input variables (NINPUT) must be specified before specifying XFACTOR val- ues.

12.Number of input variables (NINPUT) must be specified before specifying XLABELS.

13.Number of output variables (NOUTPUT) must be specified before specifying ZLABELS.

14.MONOMIALPOWER values have been set already. Multiple declarations are not allowed.

15.Number of input variables (NINPUT) must be specified before the DATA section of the input file.

16.Number of output variables (NOUTPUT) must be specified before the DATA section of the input file.

17.Number of data points (NDATA) must be specified before the DATA section of the input file.

18.Number of input variables (NINPUT) must be specified before the XDATA section of the input file.

19.Number of data points (NDATA) must be specified before the XDATA section of the input file.

20.Number of output variables (NOUTPUT) must be specified before the ZDATA section of the input file.

21.Number of data points (NDATA) must be specified before the ZDATA section of the input file.

22.Input data file missing required keyword(s).

23.END DATA missing or incomplete DATA section.

24.END XDATA missing or incomplete XDATA section.

25.END ZDATA missing or incomplete ZDATA section.

26.Only one of XDATA and DATA sections is allowed.

27.Only one of ZDATA and DATA sections is allowed.

28.MULTI2POWER values have been set already. Multiple declarations are not allowed.

29.MULTI3POWER values have been set already. Multiple declarations are not allowed.

30.Unable to open output file.

31.Maximum number of iterations reached.

32.RATIOPOWER values have been set already. Multiple declarations are not allowed.

33.Error while trying to use the optimizer to solve the MIP for best subset.

34.Error while attempting to access the ALAMO execution directory.

35.Error while attempting to access the ALAMO scratch directory.

36.Error while attempting to access the external simulator.

37.Error while attempting to write the external simulator input file.

38.Error while attempting to read the external simulator output file.

39.Scaling by zero is not allowed.

40.XMAX−XMIN for all input variables must be positive.

41.XDATA must be in the range [XMIN, XMAX].

42.Simulator should not return NaN for input variable values.

43.Simulator should not return NaN for output variable values. For any variable that the simulator cannot compute, return the value of PRESET.

44.Input file is missing XMIN values.

45.Input file is missing XMAX values.

46.MONOMIALPOWERS must be specified if MONO is used.

47.MULTI2POWER must be specified if MULTI2 is used.

48.MULTI3POWER must be specified if MULTI3 is used.

49.RATIOPOWER must be specified if RATIOS is used.

50.DATA section must be specified when NDATA is nonzero.

51.Insufficient memory to allocate data structures.

52.Number of validation data points (NVALDATA) must be specified before the VALDATA section of the input file.

53.VALDATA section must be specified when NVALDATA is nonzero.

54.VALDATA section must be specified when NVALSECTIONS is nonzero.

55.Premature end of input file.

56.Number of custom constraints (CRNCUSTOM) must be specified before specifying CUS- TOMCON section.

57.END ZMIN missing or incomplete ZMIN section.

58.END ZMAX missing or incomplete ZMAX section.

59.Number of input variables (NINPUT) must be specified before specifying EXTRAPXMIN values.

60.Number of input variables (NINPUT) must be specified before specifying EXTRAPXMAX values.

61.END CUSTOMCON missing or incomplete CUSTOMCON section.

62.Number of output variables (NOUTPUT) must be specified before specifying ZMIN values.

63.Unable to open trace file TRACEFNAME.

64.No keyword may be specified more than once.

65.Variable index is out of range.

66.Error while trying to run SNOBFIT.

67.Error while trying to solve ordinary least squares regression subproblem with the optimizer.

68.Maximum CPU time (MAXTIME) exceeded.

69.Error while trying to write in the ALAMO scratch directory.

70.Number of output variables (NOUTPUT) must be specified before specifying TOLMEAN- ERROR values.

71.A least squares subproblem failed during enumeration and no optimizer is available.

72.Licensing error. A valid license is required in order to run this software.

73.Error while trying to use the optimizer to solve the constrained regression model.

74.Error while trying to copy file to disk.

75.CUSTOMCON section must be specified when CRNCUSTOM is nonzero.

76.All output variables ignored by user. No point in calling ALAMO.

77.END CUSTOMBAS missing or incomplete CUSTOMBAS section.

78.Number of custom basis functions (NCUSTOMBAS) must be specified before the CUS- TOMBAS section of the input file.

79.Syntax error in custom basis function.

80.All variable labels must begin with an alphabetical character (A-Z or a-z).

81.Variable labels may only contain alphanumerical characters (A-Z, a-z, 0-9) or underscores.

82.All variable labels must be distinct.

83.All CRCUSTOMIND values must range from 1 to NOUTPUTS.

84.Each custom constraint must be expressed in terms of the labels of input variables and a single output variable.

85.Each line of the input file must contain no more than 10000 characters. Longer data records may be split into multiple lines using & at the end of a line to signify continuation of the record in the next line.

86.Syntax error in input file.

87.Inline comments must be preceded by ! or #.

88.Inconsistent use of NDATA and INITIALPOINTS.

89.A least squares subproblem failed during model buildup and no optimizer is available.

90.Number of output variables (NOUTPUT) must be specified before specifying MAXTERMS values.

91.Number of output variables (NOUTPUT) must be specified before specifying TOLREL- METRIC values.

92.Number of output variables (NOUTPUT) must be specified before specifying TOLABS- METRIC values.

93.END TRANSFORMS missing or incomplete TRANSFORMS section.

94.Number of transformed output variables (NTRANS) must be specified before the TRANS- FORMS section of the input file.

95.Syntax error in output transformation function.

96.Number of transformed output variables (NTRANS) cannot exceed total number of outputs (NOUTPUTS).

97.Number of transformed output variables (NTRANS) must be specified after specifying total number of outputs (NOUTPUTS).

98.Number of transformed output variables (NTRANS) must be specified before providing output data section (DATA or ZDATA).

99.Number of output variables (NOUTPUT) must be specified before specifying ZISINT val- ues.

100.Number of prediction points (NPREDATA) must be specified before the PREDATA section of the input file.

101.END XPREDATA missing or incomplete XPREDATA section.

102.Number of input variables (NINPUT) must be specified before the XPREDATA section of the input file.

103.XPREDATA section must be specified when NPREDATA is nonzero.

104.A GROUPS section is allowed only if NGROUPS is positive.

105.A GROUPS section is allowed only after NINPUTS has been defined.

106.A GROUPS section is allowed only after NOUTPUTS has been defined.

107.A GROUPS section is allowed only after NDATA has been defined.

108.Group-ids must be integers between 1 and NGROUPS.

109.Member-type must be one of LIN, LOG, EXP, SIN, COS, MONO, MULTI2, MULTI3, RATIO, RBF, CUST, and CONST.

110.All powers in group definitions must appear in user-specified basis functions.

111.NGROUPS has been specified but a smaller number of groups has been described in the GROUPS section or the GROUPS section is entirely missing.

112.A GROUPCON section is allowed only if NGROUPS is positive.

113.A GROUPCON section is allowed only after NOUTPUTS has been defined.

114.Member-indices for input variables must be integers between 1 and NINPUTS.

115.Member-indices for radial basis functions must be integers between 1 and number of data points (NDATA).

116.Member-indices for custom basis functions must be integers between 1 and NCUSTOMBAS.

117.Member-indices for groups must be integers between 1 and NGROUPS.

118.Output variable indices must be integers between 1 and NOUTPUTS.

119.Constraint-type must be one of NMT, ATL, REQ, and XCL.

120.Integer-parameters for REQ and XCL group constraints must be integers between 1 and

NGROUPS.

121.Number of input variables (NINPUT) must be specified before specifying EXCLUDE val- ues.

122.Number of output variables (NOUTPUT) must be specified before specifying IGNORE values.

123.Unable to find the external simulator.

124.Simulator failed MAXSIM times.

125.TRANSFORMS section must be specified when NTRANS is nonzero.

126.Evaluation error with transformation function. Try a different transformation.

127.Error while trying to write file to disk.

128.Number of output variables (NOUTPUT) must be specified before specifying ZMAX values.

129.Powers for polynomial basis functions have been set already. Multiple declarations are not allowed.

130.Number of input variables (NINPUT) must be specified before specifying XISINT values.

131.Number of output variables (NOUTPUT) must be specified before specifying TOLMAX- ERROR values.

132.A MIP solver is required to proceed with the current set of options but one is not avail- ablefound.

9Compatibility with previous versions of ALAMO

Starting with ALAMO v. 2013.10.0, the input format was changed. Input requirements of earlier versions were maintained with two exceptions:

•Previous versions required that ALAMO options be specified in a separate file than preex- isting data. All ALAMO input must now be entered in a single file.

•Preexisting data can now be entered in a format that combines input and output measure- ments in a column wise fashion.

For compatibility with early versions of ALAMO, the following keywords are also acceptable in ALAMO v. 2013.10.0 and beyond:

Starting with ALAMO v. 2019.7.30, the keyword REGULARIZER was replaced by the more appropriate SCREENER. The keyword REGULARIZER is still acceptable and the possible values of SCREENER are backwards compatible with those of REGULARIZER.

10 Bibliography

The following is a partial list of ALAMO-related publications that describe the algorithms im- plemented in the software, the theory behind them, and some related applications.

1.A. Cozad, N. V. Sahinidis, and D. C. Miller. Learning surrogate models for simulation- based optimization. AIChE Journal, 60, 2211–2227, 2014.

2.A. Cozad, N. V. Sahinidis, and D. C. Miller. A combined first-principles and data-driven approach to model building. Computers & Chemical Engineering, 73, 116–127, 2015.

3.Z. T. Wilson and N. V. Sahinidis. The ALAMO approach to machine learning. Computers & Chemical Engineering, 106, 785-795, 2017.

4.K. Lindqvist, Z. T. Wilson, E. Næss and N. V. Sahinidis. A machine learning approach to correlation development applied to fin-tube bundle heat exchangers Energies, 11(12), 3450, 2018.

5.Z. Wilson and N. V. Sahinidis. Automated learning of chemical reaction networks. Com- puters & Chemical Engineering, 127, 88-98, 2019.