diff -r d59bea55db9b -r c445c931472f doc/glpk08.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/doc/glpk08.tex Mon Dec 06 13:09:21 2010 +0100 @@ -0,0 +1,696 @@ +%* glpk08.tex *% + +\chapter{MPS Format} +\label{champs} + +\section{Fixed MPS Format} +\label{secmps} + +The MPS format\footnote{The MPS format was developed in 1960's by IBM +as input format for their mathematical programming system MPS/360. +Today the MPS format is a most widely used format understood by most +mathematical programming packages. This appendix describes only the +features of the MPS format, which are implemented in the GLPK package.} +is intended for coding LP/MIP problem data. This format assumes the +formulation of LP/MIP problem (1.1)---(1.3) (see Section \ref{seclp}, +page \pageref{seclp}). + +{\it MPS file} is a text file, which contains two types of +cards\footnote{In 1960's MPS file was a deck of 80-column punched cards, +so the author decided to keep the word ``card'', which may be understood +as ``line of text file''.}: indicator cards and data cards. + +Indicator cards determine a kind of succeeding data. Each indicator card +has one word in uppercase letters beginning in column 1. + +Data cards contain problem data. Each data card is divided into six +fixed fields: + +\begin{center} +\begin{tabular}{lcccccc} +& Field 1 & Field 2 & Field 3 & Field 4 & Field 5 & Feld 6 \\ +\hline +Columns & 2---3 & 5---12 & 15---22 & 25---36 & 40---47 & 50---61 \\ +Contents & Code & Name & Name & Number & Name & Number \\ +\end{tabular} +\end{center} + +On a particular data card some fields may be optional. + +Names are used to identify rows, columns, and some vectors (see below). + +Aligning the indicator code in the field 1 to the left margin is +optional. + +All names specified in the fields 2, 3, and 5 should contain from 1 up +to 8 arbitrary characters (except control characters). If a name is +placed in the field 3 or 5, its first character should not be the dollar +sign `\verb|$|'. If a name contains spaces, the spaces are ignored. + +All numerical values in the fields 4 and 6 should be coded in the form +$sxx$\verb|E|$syy$, where $s$ is the plus `\verb|+|' or the minus +`\verb|-|' sign, $xx$ is a real number with optional decimal point, +$yy$ is an integer decimal exponent. Any number should contain up to 12 +characters. If the sign $s$ is omitted, the plus sign is assumed. The +exponent part is optional. If a number contains spaces, the spaces are +ignored. + +If a card has the asterisk `\verb|*|' in the column 1, this card is +considered as a comment and ignored. Besides, if the first character in +the field 3 or 5 is the dollar sign `\verb|$|', all characters from the +dollar sign to the end of card are considered as a comment and ignored. + +MPS file should contain cards in the following order: + +$\bullet$ NAME indicator card; + +$\bullet$ ROWS indicator card; + +$\bullet$ data cards specifying rows (constraints); + +$\bullet$ COLUMNS indicator card; + +$\bullet$ data cards specifying columns (structural variables) and +constraint coefficients; + +$\bullet$ RHS indicator card; + +$\bullet$ data cards specifying right-hand sides of constraints; + +$\bullet$ RANGES indicator card; + +$\bullet$ data cards specifying ranges for double-bounded constraints; + +$\bullet$ BOUNDS indicator card; + +$\bullet$ data cards specifying types and bounds of structural +variables; + +$\bullet$ ENDATA indicator card. + +{\it Section} is a group of cards consisting of an indicator card and +data cards succeeding this indicator card. For example, the ROWS section +consists of the ROWS indicator card and data cards specifying rows. + +The sections RHS, RANGES, and BOUNDS are optional and may be omitted. + +\section{Free MPS Format} + +{\it Free MPS format} is an improved version of the standard (fixed) +MPS format described above.\footnote{This format was developed in the +beginning of 1990's by IBM as an alternative to the standard fixed MPS +format for Optimization Subroutine Library (OSL).} Note that all +changes in free MPS format concern only the coding of data while the +structure of data is the same for both fixed and free versions of the +MPS format. + +In free MPS format indicator and data records\footnote{{\it Record} in +free MPS format has the same meaning as {\it card} in fixed MPS format.} +may have arbitrary length not limited to 80 characters. Fields of data +records have no predefined positions, i.e. the fields may begin in any +position, except position 1, which must be blank, and must be separated +from each other by one or more blanks. However, the fields must appear +in the same order as in fixed MPS format. + +Symbolic names in fields 2, 3, and 5 may be longer than 8 +characters\footnote{GLPK allows symbolic names having up to 255 +characters.} +and must not contain embedded blanks. + +Numeric values in fields 4 and 6 are limited to 12 characters and must +not contain embedded blanks. + +Only six fields on each data record are used. Any other fields are +ignored. + +If the first character of any field (not necessarily fields 3 and 5) +is the dollar sign (\$), all characters from the dollar sign to the end +of record are considered as a comment and ignored. + +\section{NAME indicator card} + +The NAME indicator card should be the first card in the MPS file (except +optional comment cards, which may precede the NAME card). This card +should contain the word \verb|NAME| in the columns 1---4 and the problem +name in the field 3. The problem name is optional and may be omitted. + +\section{ROWS section} +\label{secrows} + +The ROWS section should start with the indicator card, which contains +the word \verb|ROWS| in the columns 1---4. + +Each data card in the ROWS section specifies one row (constraint) of the +problem. All these data cards have the following format. + +`\verb|N|' in the field 1 means that the row is free (unbounded): +$$-\infty < x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} +< +\infty;$$ + +`\verb|L|' in the field 1 means that the row is of ``less than or equal +to'' type: +$$-\infty < x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} +\leq b_i;$$ + +`\verb|G|' in the field 1 means that the row is of ``greater than or +equal to'' type: +$$b_i \leq x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} +< +\infty;$$ + +`\verb|E|' in the field 1 means that the row is of ``equal to'' type: +$$x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} \leq +b_i,$$ +where $b_i$ is a right-hand side. Note that each constraint has a +corresponding implictly defined auxiliary variable ($x_i$ above), whose +value is a value of the corresponding linear form, therefore row bounds +can be considered as bounds of such auxiliary variable. + +The filed 2 specifies a row name (which is considered as the name of +the corresponding auxiliary variable). + +The fields 3, 4, 5, and 6 are not used and should be empty. + +Numerical values of all non-zero right-hand sides $b_i$ should be +specified in the RHS section (see below). All double-bounded (ranged) +constraints should be specified in the RANGES section (see below). + +\section{COLUMNS section} + +The COLUMNS section should start with the indicator card, which contains +the word \verb|COLUMNS| in the columns 1---7. + +Each data card in the COLUMNS section specifies one or two constraint +coefficients $a_{ij}$ and also introduces names of columns, i.e. names +of structural variables. All these data cards have the following format. + +The field 1 is not used and should be empty. + +The field 2 specifies a column name. If this field is empty, the column +name from the immediately preceeding data card is assumed. + +The field 3 specifies a row name defined in the ROWS section. + +The field 4 specifies a numerical value of the constraint coefficient +$a_{ij}$, which is placed in the corresponding row and column. + +The fields 5 and 6 are optional. If they are used, they should contain +a second pair ``row name---constraint coefficient'' for the same column. + +Elements of the constraint matrix (i.e. constraint coefficients) should +be enumerated in the column wise manner: all elements for the current +column should be specified before elements for the next column. However, +the order of rows in the COLUMNS section may differ from the order of +rows in the ROWS section. + +Constraint coefficients not specified in the COLUMNS section are +considered as zeros. Therefore zero coefficients may be omitted, +although it is allowed to explicitly specify them. + +\section{RHS section} + +The RHS section should start with the indicator card, which contains the +word \verb|RHS| in the columns 1---3. + +Each data card in the RHS section specifies one or two right-hand sides +$b_i$ (see Section \ref{secrows}, page \pageref{secrows}). All these +data cards have the following format. + +The field 1 is not used and should be empty. + +The field 2 specifies a name of the right-hand side (RHS) +vector\footnote{This feature allows the user to specify several RHS +vectors in the same MPS file. However, before solving the problem a +particular RHS vector should be chosen.}. If this field is empty, the +RHS vector name from the immediately preceeding data card is assumed. + +The field 3 specifies a row name defined in the ROWS section. + +The field 4 specifies a right-hand side $b_i$ for the row, whose name is +specified in the field 3. Depending on the row type $b_i$ is a lower +bound (for the row of \verb|G| type), an upper bound (for the row of +\verb|L| type), or a fixed value (for the row of \verb|E| +type).\footnote{If the row is of {\tt N} type, $b_i$ is considered as +a constant term of the corresponding linear form. Should note, however, +this convention is non-standard.} + +The fields 5 and 6 are optional. If they are used, they should contain +a second pair ``row name---right-hand side'' for the same RHS vector. + +All right-hand sides for the current RHS vector should be specified +before right-hand sides for the next RHS vector. However, the order of +rows in the RHS section may differ from the order of rows in the ROWS +section. + +Right-hand sides not specified in the RHS section are considered as +zeros. Therefore zero right-hand sides may be omitted, although it is +allowed to explicitly specify them. + +\section{RANGES section} + +The RANGES section should start with the indicator card, which contains +the word \verb|RANGES| in the columns 1---6. + +Each data card in the RANGES section specifies one or two ranges for +double-side constraints, i.e. for constraints that are of the types +\verb|L| and \verb|G| at the same time: +$$l_i \leq x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} +\leq u_i,$$ +where $l_i$ is a lower bound, $u_i$ is an upper bound. All these data +cards have the following format. + +The field 1 is not used and should be empty. + +The field 2 specifies a name of the range vector\footnote{This feature +allows the user to specify several range vectors in the same MPS file. +However, before solving the problem a particular range vector should be +chosen.}. If this field is empty, the range vector name from the +immediately preceeding data card is assumed. + +The field 3 specifies a row name defined in the ROWS section. + +The field 4 specifies a range value $r_i$ (see the table below) for the +row, whose name is specified in the field 3. + +The fields 5 and 6 are optional. If they are used, they should contain +a second pair ``row name---range value'' for the same range vector. + +All range values for the current range vector should be specified before +range values for the next range vector. However, the order of rows in +the RANGES section may differ from the order of rows in the ROWS +section. + +For each double-side constraint specified in the RANGES section its +lower and upper bounds are determined as follows: + +\begin{center} +\begin{tabular}{cccc} +Row type & Sign of $r_i$ & Lower bound & Upper bound \\ +\hline +{\tt G} & $+$ or $-$ & $b_i$ & $b_i + |r_i|$ \\ +{\tt L} & $+$ or $-$ & $b_i - |r_i|$ & $b_i$ \\ +{\tt E} & $+$ & $b_i$ & $b_i + |r_i|$ \\ +{\tt E} & $-$ & $b_i - |r_i|$ & $b_i$ \\ +\end{tabular} +\end{center} + +\noindent +where $b_i$ is a right-hand side specified in the RHS section (if $b_i$ +is not specified, it is considered as zero), $r_i$ is a range value +specified in the RANGES section. + +\section{BOUNDS section} +\label{secbounds} + +The BOUNDS section should start with the indicator card, which contains +the word \verb|BOUNDS| in the columns 1---6. + +Each data card in the BOUNDS section specifies one (lower or upper) +bound for one structural variable (column). All these data cards have +the following format. + +The indicator in the field 1 specifies the bound type: + +\begin{tabular}{@{}ll} +\verb|LO| & lower bound; \\ +\verb|UP| & upper bound; \\ +\verb|FX| & fixed variable (lower and upper bounds are equal); \\ +\verb|FR| & free variable (no bounds); \\ +\verb|MI| & no lower bound (lower bound is ``minus infinity''); \\ +\verb|PL| & no upper bound (upper bound is ``plus infinity''); \\ +\end{tabular} + +The field 2 specifies a name of the bound vector\footnote{This feature +allows the user to specify several bound vectors in the same MPS file. +However, before solving the problem a particular bound vector should be +chosen.}. If this field is empty, the bound vector name from the +immediately preceeding data card is assumed. + +The field 3 specifies a column name defined in the COLUMNS section. + +The field 4 specifies a bound value. If the bound type in the field 1 +differs from \verb|LO|, \verb|UP|, and \verb|FX|, the value in the field +4 is ignored and may be omitted. + +The fields 5 and 6 are not used and should be empty. + +All bound values for the current bound vector should be specified before +bound values for the next bound vector. However, the order of columns in +the BOUNDS section may differ from the order of columns in the COLUMNS +section. Specification of a lower bound should precede specification of +an upper bound for the same column (if both the lower and upper bounds +are explicitly specified). + +By default, all columns (structural variables) are non-negative, i.e. +have zero lower bound and no upper bound. Lower ($l_j$) and upper +($u_j$) bounds of some column (structural variable $x_j$) are set in the +following way, where $s_j$ is a corresponding bound value explicitly +specified in the BOUNDS section: + +\begin{tabular}{@{}ll} +\verb|LO| & sets $l_j$ to $s_j$; \\ +\verb|UP| & sets $u_j$ to $s_j$; \\ +\verb|FX| & sets both $l_j$ and $u_j$ to $s_j$; \\ +\verb|FR| & sets $l_j$ to $-\infty$ and $u_j$ to $+\infty$; \\ +\verb|MI| & sets $l_j$ to $-\infty$; \\ +\verb|PL| & sets $u_j$ to $+\infty$. \\ +\end{tabular} + +\section{ENDATA indicator card} + +The ENDATA indicator card should be the last card of MPS file (except +optional comment cards, which may follow the ENDATA card). This card +should contain the word \verb|ENDATA| in the columns 1---6. + +\section{Specifying objective function} + +It is impossible to explicitly specify the objective function and +optimization direction in the MPS file. However, the following implicit +rule is used by default: the first row of \verb|N| type is considered +as a row of the objective function (i.e. the objective function is the +corresponding auxiliary variable), which should be {\it minimized}. + +GLPK also allows specifying a constant term of the objective function +as a right-hand side of the corresponding row in the RHS section. + +\section{Example of MPS file} +\label{secmpsex} + +In order to illustrate what the MPS format is, consider the following +example of LP problem: + +\medskip +\noindent minimize +$$ +value = .03\ bin_1 + .08\ bin_2 + .17\ bin_3 + .12\ bin_4 + .15\ bin_5 ++ .21\ al + .38\ si +$$ + +\noindent subject to linear constraints +$$ +\begin{array}{@{}l@{\:}l@{}} +yield &= \ \ \ \ \;bin_1 + \ \ \ \ \;bin_2 + \ \ \ \ \;bin_3 + + \ \ \ \ \;bin_4 + \ \ \ \ \;bin_5 + \ \ \ \ \;al + + \ \ \ \ \;si \\ +FE &= .15\ bin_1 + .04\ bin_2 + .02\ bin_3 + .04\ bin_4 + .02\ bin_5 + + .01\ al + .03\ si \\ +CU &= .03\ bin_1 + .05\ bin_2 + .08\ bin_3 + .02\ bin_4 + .06\ bin_5 + + .01\ al \\ +MN &= .02\ bin_1 + .04\ bin_2 + .01\ bin_3 + .02\ bin_4 + .02\ bin_5 + \\ +MG &= .02\ bin_1 + .03\ bin_2 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + .01\ bin_5 \\ +AL &= .70\ bin_1 + .75\ bin_2 + .80\ bin_3 + .75\ bin_4 + .80\ bin_5 + + .97\ al \\ +SI &= .02\ bin_1 + .06\ bin_2 + .08\ bin_3 + .12\ bin_4 + .02\ bin_5 + + .01\ al + .97\ si \\ +\end{array} +$$ +and bounds of (auxiliary and structural) variables +$$ +\begin{array}{r@{\ }l@{\ }l@{\ }l@{\ }rcr@{\ }l@{\ }l@{\ }l@{\ }r} +&&yield&=&2000&&0&\leq&bin_1&\leq&200\\ +-\infty&<&FE&\leq&60&&0&\leq&bin_2&\leq&2500\\ +-\infty&<&CU&\leq&100&&400&\leq&bin_3&\leq&800\\ +-\infty&<&MN&\leq&40&&100&\leq&bin_4&\leq&700\\ +-\infty&<&MG&\leq&30&&0&\leq&bin_5&\leq&1500\\ +1500&\leq&AL&<&+\infty&&0&\leq&al&<&+\infty\\ +250&\leq&SI&\leq&300&&0&\leq&si&<&+\infty\\ +\end{array} +$$ + +A complete MPS file which specifies data for this example is shown +below (the first two comment lines show card positions). + +\begin{verbatim} +*000000001111111111222222222233333333334444444444555555555566 +*234567890123456789012345678901234567890123456789012345678901 +NAME PLAN +ROWS + N VALUE + E YIELD + L FE + L CU + L MN + L MG + G AL + L SI +COLUMNS + BIN1 VALUE .03000 YIELD 1.00000 + FE .15000 CU .03000 + MN .02000 MG .02000 + AL .70000 SI .02000 + BIN2 VALUE .08000 YIELD 1.00000 + FE .04000 CU .05000 + MN .04000 MG .03000 + AL .75000 SI .06000 + BIN3 VALUE .17000 YIELD 1.00000 + FE .02000 CU .08000 + MN .01000 AL .80000 + SI .08000 + BIN4 VALUE .12000 YIELD 1.00000 + FE .04000 CU .02000 + MN .02000 AL .75000 + SI .12000 + BIN5 VALUE .15000 YIELD 1.00000 + FE .02000 CU .06000 + MN .02000 MG .01000 + AL .80000 SI .02000 + ALUM VALUE .21000 YIELD 1.00000 + FE .01000 CU .01000 + AL .97000 SI .01000 + SILICON VALUE .38000 YIELD 1.00000 + FE .03000 SI .97000 +RHS + RHS1 YIELD 2000.00000 FE 60.00000 + CU 100.00000 MN 40.00000 + SI 300.00000 + MG 30.00000 AL 1500.00000 +RANGES + RNG1 SI 50.00000 +BOUNDS + UP BND1 BIN1 200.00000 + UP BIN2 2500.00000 + LO BIN3 400.00000 + UP BIN3 800.00000 + LO BIN4 100.00000 + UP BIN4 700.00000 + UP BIN5 1500.00000 +ENDATA +\end{verbatim} + +\section{MIP features} + +The MPS format provides two ways for introducing integer variables into +the problem. + +The first way is most general and based on using special marker cards +INTORG and INTEND. These marker cards are placed in the COLUMNS section. +The INTORG card indicates the start of a group of integer variables +(columns), and the card INTEND indicates the end of the group. The MPS +file may contain arbitrary number of the marker cards. + +The marker cards have the same format as the data cards (see Section +\ref{secmps}, page \pageref{secmps}). + +The fields 1, 2, and 6 are not used and should be empty. + +The field 2 should contain a marker name. This name may be arbitrary. + +The field 3 should contain the word \verb|'MARKER'| (including +apostrophes). + +The field 5 should contain either the word \verb|'INTORG'| (including +apostrophes) for the marker card, which begins a group of integer +columns, or the word \verb|'INTEND'| (including apostrophes) for the +marker card, which ends the group. + +The second way is less general but more convenient in some cases. It +allows the user declaring integer columns using three additional types +of bounds, which are specified in the field 1 of data cards in the +BOUNDS section (see Section \ref{secbounds}, page \pageref{secbounds}): + +\begin{tabular}{@{}lp{112.3mm}@{}} +\verb|LI| & lower integer. This bound type specifies that the +corresponding column (structural variable), whose name is specified in +field 3, is of integer kind. In this case an lower bound of the +column should be specified in field 4 (like in the case of \verb|LO| +bound type). \\ +\verb|UI| & upper integer. This bound type specifies that the +corresponding column (structural variable), whose name is specified in +field 3, is of integer kind. In this case an upper bound of the +column should be specified in field 4 (like in the case of \verb|UP| +bound type). \\ +\end{tabular} + +\pagebreak + +\begin{tabular}{@{}lp{112.3mm}@{}} +\verb|BV| & binary variable. This bound type specifies that the +corresponding column (structural variable), whose name is specified in +the field 3, is of integer kind, its lower bound is zero, and its upper +bound is one (thus, such variable being of integer kind can have only +two values zero and one). In this case a numeric value specified in the +field 4 is ignored and may be omitted.\\ +\end{tabular} + +Consider the following example of MIP problem: + +\medskip + +\noindent +\hspace{1in} minimize +$$Z = 3 x_1 + 7 x_2 - x_3 + x4$$ +\hspace{1in} subject to linear constraints +$$ +\begin{array}{c} +\nonumber r_1 = 2 x_1 - \ \ x_2 + \ \ x_3 - \ \;x_4 \\ +\nonumber r_2 = \ \;x_1 - \ \;x_2 - 6 x_3 + 4 x_4 \\ +\nonumber r_3 = 5 x_1 + 3 x_2 \ \ \ \ \ \ \ \ \ + \ \ x_4 \\ +\end{array} +$$ +\hspace{1in} and bound of variables +$$ +\begin{array}{cccl} +\nonumber 1 \leq r_1 < +\infty && 0 \leq x_1 \leq 4 &{\rm(continuous)}\\ +\nonumber 8 \leq r_2 < +\infty && 2 \leq x_2 \leq 5 &{\rm(integer)} \\ +\nonumber 5 \leq r_3 < +\infty && 0 \leq x_3 \leq 1 &{\rm(integer)} \\ +\nonumber && 3 \leq x_4 \leq 8 &{\rm(continuous)}\\ +\end{array} +$$ + +The corresponding MPS file may look like the following: + +\begin{verbatim} +NAME SAMP1 +ROWS + N Z + G R1 + G R2 + G R3 +COLUMNS + X1 R1 2.0 R2 1.0 + X1 R3 5.0 Z 3.0 + MARK0001 'MARKER' 'INTORG' + X2 R1 -1.0 R2 -1.0 + X2 R3 3.0 Z 7.0 + X3 R1 1.0 R2 -6.0 + X3 Z -1.0 + MARK0002 'MARKER' 'INTEND' + X4 R1 -1.0 R2 4.0 + X4 R3 1.0 Z 1.0 +RHS + RHS1 R1 1.0 + RHS1 R2 8.0 + RHS1 R3 5.0 +BOUNDS + UP BND1 X1 4.0 + LO BND1 X2 2.0 + UP BND1 X2 5.0 + UP BND1 X3 1.0 + LO BND1 X4 3.0 + UP BND1 X4 8.0 +ENDATA +\end{verbatim} + +The same example may be coded without INTORG/INTEND markers using the +bound type UI for the variable $x_2$ and the bound type BV for the +variable $x_3$: + +\begin{verbatim} +NAME SAMP2 +ROWS + N Z + G R1 + G R2 + G R3 +COLUMNS + X1 R1 2.0 R2 1.0 + X1 R3 5.0 Z 3.0 + X2 R1 -1.0 R2 -1.0 + X2 R3 3.0 Z 7.0 + X3 R1 1.0 R2 -6.0 + X3 Z -1.0 + X4 R1 -1.0 R2 4.0 + X4 R3 1.0 Z 1.0 +RHS + RHS1 R1 1.0 + RHS1 R2 8.0 + RHS1 R3 5.0 +BOUNDS + UP BND1 X1 4.0 + LO BND1 X2 2.0 + UI BND1 X2 5.0 + BV BND1 X3 + LO BND1 X4 3.0 + UP BND1 X4 8.0 +ENDATA +\end{verbatim} + +%\section{Specifying predefined basis} +%\label{secbas} +% +%The MPS format can also be used to specify some predefined basis for an +%LP problem, i.e. to specify which rows and columns are basic and which +%are non-basic. +% +%The order of a basis file in the MPS format is: +% +%$\bullet$ NAME indicator card; +% +%$\bullet$ data cards (can appear in arbitrary order); +% +%$\bullet$ ENDATA indicator card. +% +%Each data card specifies either a pair "basic column---non-basic row" +%or a non-basic column. All the data cards have the following format. +% +%`\verb|XL|' in the field 1 means that a column, whose name is given in +%the field 2, is basic, and a row, whose name is given in the field 3, +%is non-basic and placed on its lower bound. +% +%`\verb|XU|' in the field 1 means that a column, whose name is given in +%the field 2, is basic, and a row, whose name is given in the field 3, +%is non-basic and placed on its upper bound. +% +%`\verb|LL|' in the field 1 means that a column, whose name is given in +%the field 3, is non-basic and placed on its lower bound. +% +%`\verb|UL|' in the field 1 means that a column, whose name is given in +%the field 3, is non-basic and placed on its upper bound. +% +%The field 2 contains a column name. +% +%If the indicator given in the field 1 is `\verb|XL|' or `\verb|XU|', +%the field 3 contains a row name. Otherwise, if the indicator is +%`\verb|LL|' or `\verb|UL|', the field 3 is not used and should be +%empty. +% +%The field 4, 5, and 6 are not used and should be empty. +% +%A basis file in the MPS format acts like a patch: it doesn't specify +%a basis completely, instead that it is just shows in what a given basis +%differs from the "standard" basis, where all rows (auxiliary variables) +%are assumed to be basic and all columns (structural variables) are +%assumed to be non-basic. +% +%As an example here is a basis file that specifies an optimal basis +%for the example LP problem given in Section \ref{secmpsex}, +%Page \pageref{secmpsex}: +% +%\pagebreak +% +%\begin{verbatim} +%*000000001111111111222222222233333333334444444444555555555566 +%*234567890123456789012345678901234567890123456789012345678901 +%NAME PLAN +% XL BIN2 YIELD +% XL BIN3 FE +% XL BIN4 MN +% XL ALUM AL +% XL SILICON SI +% LL BIN1 +% LL BIN5 +%ENDATA +%\end{verbatim} + +%* eof *%