Package lintools. July 30, 2018

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1 Package lintools July 30, 2018 Maintainer Mark van der Loo License GPL-3 Title Manipulation of Linear Systems of (in)equalities LazyData no Type Package LazyLoad yes Variale elimination (Gaussian elimination, Fourier-Motzkin elimination), Moore-Penrose pseudoinverse, reduction to reduced row echelon form, value sustitution, projecting a vector on the convex polytope descried y a system of (in)equations, simplify systems y removing spurious columns and rows and collapse implied equalities, test if a matrix is totally unimodular, compute variale ranges implied y linear (in)equalities. Version URL BugReports Imports utils Suggests testthat, knitr VignetteBuilder knitr RoxygenNote NeedsCompilation yes uthor Mark van der Loo [aut, cre], Edwin de Jonge [aut] Repository CRN Date/Pulication :10:05 UTC R topics documented: lock_index compact

2 2 lock_index echelon eliminate is_feasile is_totally_unimodular lintools normalize pinv project ranges sparse_constraints sparse_project sust_value Index 18 lock_index Find independent locks of equations. Find independent locks of equations. lock_index(, eps = 1e-08) eps [numeric] Matrix [numeric] Coefficients with asolute value < eps are treated as zero. list containing numeric vectors, each vector indexing an independent lock of rows in the system x <=. <- matrix(c( 1,0,2,0,0, 3,0,4,0,0, 0,5,0,6,7, 0,8,0,0,9 ),yrow=true,nrow=4) <- rep(0,4) i <- lock_index() lapply(i,function(ii) compact([ii,,drop=flse],=[ii])$)

3 compact 3 compact Remove spurious variales and restrictions system of linear (in)equations can e compactified y removing zero-rows and zero-columns (=variales). Such rows and columns may arise after sustitution (see sust_value) or eliminaton of a variale (see eliminate). compact(,, x = NULL, neq = nrow(), nleq = 0, eps = 1e-08, remove_columns = TRUE, remove_rows = TRUE, deduplicate = TRUE, implied_equations = TRUE) x neq nleq eps [numeric] matrix [numeric] vector [numeric] vector [numeric] The first neq rows in and are treated as linear equalities. [numeric] The nleq rows after neq are treated as inequations of the form a.x<=. ll remaining rows are treated as strict inequations of the form a.x<. [numeric] nything with asolute value < eps is considered zero. remove_columns [logical] Toggle remove spurious columns from and variales from x remove_rows [logical] Toggle remove spurious rows deduplicate [logical] Toggle remove duplicate rows implied_equations [logical] replace cases of a.x<= and a.x>= with a.x==. list with the following elements. : The compactified version of input : The compactified version of input x: The compactified version of input x neq: numer of equations in new system nleq: numer of inequations of the form a.x<= in the new system cols_removed: [logical] indicates what elements of x (columns of ) have een removed It is assumend that the system of equations is in normalized form (see link{normalize}).

4 4 echelon echelon Reduced row echelon form Transform the equalities in a system of linear (in)equations or Reduced Row Echelon form (RRE) echelon(,, neq = nrow(), nleq = 0, eps = 1e-08) neq nleq eps [numeric] matrix [numeric] vector [numeric] The first neq rows of, are treated as equations. [numeric] The nleq rows after neq are treated as inequations of the form a.x<=. ll remaining rows are treated as strict inequations of the form a.x<. [numeric] s of magnitude less than eps are considered zero (for the purpose of handling machine rounding errors). list with the following components: : the matrix with equalities transformed to RRE form. : the constant vector corresponding to neq: the numer of equalities in the resulting system. nleq: the numer of inequalities of the form a.x <=. This will only e passed to the output. The parameters, and neq descrie a system of the form x<=, where the first neq rows are equalities. The equalities are transformed to RRE form. system of equations is in reduced row echelon form when ll zero rows are elow the nonzero rows For every row, the leading coefficient (first nonzero from the left) is always right of the leading coefficient of the row aove it. The leading coefficient equals 1, and is the only nonzero coefficient in its column.

5 eliminate 5 echelon( = matrix(c( 1,3,1, 2,7,3, 1,5,3, 1,2,0), yrow=true, nrow=4), = c(4,-9,1,8), neq=4 ) eliminate Eliminate a variale from a set of edit rules Eliminating a variale amounts to deriving all (non-redundant) linear (in)equations not containing that variale. Geometrically, it can e interpreted as a projection of the solution space (vectors satisfying all equations) along the eliminated variale s axis. eliminate(,, neq = nrow(), nleq = 0, variale, H = NULL, h = 0, eps = 1e-08) neq nleq variale H h eps [numeric] Matrix [numeric] vector [numeric] The first neq rows in and are treated as linear equalities. [numeric] The nleq rows after neq are treated as inequations of the form a.x<=. ll remaining rows are treated as strict inequations of the form a.x<. [numeric logical character] Index in columns of, representing the variale to eliminate. [numeric] (optional) Matrix indicating how linear inequalities have een derived. [numeric] (optional) numer indicating how many variales have een eliminated from the original system using Fourier-Motzkin elimination. [numeric] Coefficients with asolute value <= eps are treated as zero.

6 6 is_feasile list with the folowing components : the corresponding to the system with variales eliminated. : the constant vector corresponding to the resulting system neq: the numer of equations H: The memory matrix storing how each row was derived h: The numer of variales eliminated from the original system. For equalities Gaussian elimination is applied. If inequalities are involved, Fourier-Motzkin elimination is used. In principle, FM-elimination can generate a large numer of redundant inequations, especially when applied recursively. Redundancies can e recognized y recording how new inequations have een derived from the original set. This is stored in the H matrix when multiple variales are to e eliminated (Kohler, 1967). References D.. Kohler (1967) Projections of convex polyhedral sets, Operational Research Center Report, ORC 67-29, University of California, Berkely. H.P. Williams (1986) Fourier s method of linear programming and its dual. merican Mathematical Monthly 93, pp # Example from Williams (1986) <- matrix(c( 4, -5, -3, 1, -1, 1, -1, 0, 1, 1, 2, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0),yrow=TRUE,nrow=6) <- c(0,2,3,0,0,0) L <- eliminate(=, =, neq=0, nleq=6, variale=1) is_feasile Check feasiility of a system of linear (in)equations Check feasiility of a system of linear (in)equations

7 is_totally_unimodular 7 is_feasile(,, neq = nrow(), nleq = 0, eps = 1e-08, method = "elimination") neq nleq eps method [numeric] matrix [numeric] vector [numeric] The first neq rows in and are treated as linear equalities. [numeric] The nleq rows after neq are treated as inequations of the form a.x<=. ll remaining rows are treated as strict inequations of the form a.x<. [numeric] solute values < eps are treated as zero. [character] t the moment, only the elimination method is implemented. # n infeasile system: # x + y == 0 # x > 0 # y > 0 <- matrix(c(1,1,1,0,0,1),yrow=true,nrow=3) <- rep(0,3) is_feasile(=,=,neq=1,nleq=0) # feasile system: # x + y == 0 # x >= 0 # y >= 0 <- matrix(c(1,1,1,0,0,1),yrow=true,nrow=3) <- rep(0,3) is_feasile(=,=,neq=1,nleq=2) is_totally_unimodular Test for total unimodularity of a matrix. Check wether a matrix is totally unimodular. is_totally_unimodular() n oject of class matrix.

8 8 is_totally_unimodular matrix for which the determinant of every square sumatrix equals 1, 0 or 1 is called totally unimodular. This function tests wether a matrix with coefficients in { 1, 0, 1} is totally unimodular. It tries to reduce the matrix using the reduction method descried in Scholtus (2008). Next, a test ased on Heller and Tompkins (1956) or Raghavachari is performed. logical References Heller I and Tompkins CB (1956). n extension of a theorem of Danttzig s In kuhn HW and Tucker W (eds.), pp Princeton University Press. Raghavachari M (1976). constructive method to recognize the total unimodularity of a matrix. _Zeitschrift fur operations research_, *20*, pp Scholtus S (2008). lgorithms for correcting some ovious inconsistencies and rounding errors in usiness survey data. Technical Report 08015, Netherlands. # Totally unimodular matrix, reduces to nothing <- matrix(c( 1,1,0,0,0, -1,0,0,1,0, 0,0,01,1,0, 0,0,0,-1,1),nrow=5) is_totally_unimodular() # Totally unimodular matrix, y Heller-Tompson criterium <- matrix(c( 1,1,0,0, 0,0,1,1, 1,0,1,0, 0,1,0,1),nrow=4) is_totally_unimodular() # Totally unimodular matrix, y Raghavachani recursive criterium <- matrix(c( 1,1,1, 1,1,0, 1,0,1)) is_totally_unimodular()

9 lintools 9 lintools Tools for manipulating linear systems of (in)equations Tools for manipulating linear systems of (in)equations This package offers a asic and consistent interface to a numer of operations on linear systems of (in)equations not availale in ase R. Except for the projection on the convex polytope, operations are currently supported for dense matrices only. The following operations are implemented. Split matrices in independent locks Remove spurious rows and columns from a system of (in)equations Rewrite equalities in reduced row echelon form Eliminate variales through Gaussian or Fourier-Motzkin elimination Determine the feasiility of a system of linear (in)equations Compute Moore-Penrose Pseudoinverse Project a vector onto the convec polytope descried y a set of linear (in)equations Simplify a system y sustituting values Most functions assume a system of (in)equations to e stored in a standard form. The normalize function can ring any system of equations to this form. normalize Bring a system of (in)equalities in a standard form Bring a system of (in)equalities in a standard form normalize(,, operators, unit = 0) [numeric] Matrix [numeric] vector operators [character] operators in {<,<=,==,>=,>}. unit [numeric] (nonnegative) Your unit of measurement. This is used to replace strict inequations of the form a.x < with a.x <= -unit. Typically, unit is related to the units in which your data is measured. If unit is 0, inequations are not replaced.

10 10 pinv list with the folowing components : the corresponding to the normalized sytem. : the constant vector corresponding to the normalized system neq: the numer of equations nleq: the numer of non-strict inequations (<=) order: the index vector used to permute the original rows of. For this package, a set of equations is in normal form when The first neq rows represent linear equalities. The next nleq rows represent inequalities of the form a.x <= ll other rows are strict inequalities of the form a.x < If unit>0, the strict inequalities a.x < are replaced with inequations of the form a.x <= -unit, where unit represents the precision of measurement. <- matrix(1:12,nrow=4) <- 1:4 ops <- c("<=","==","==","<") normalize(,,ops) normalize(,,ops,unit=0.1) pinv Moore-Penrose pseudoinverse Compute the pseudoinverse of a matrix using the SVD-construction pinv(, eps = 1e-08) eps [numeric] matrix [numeric] tolerance for determining zero singular values

11 project 11 The Moore-Penrose pseudoinverse (sometimes called the generalized inverse) + of a matrix has the property that + + =. It can e constructed as follows. Compute the singular value decomposition = UDV T Replace diagonal elements in D of which the asolute values are larger than some limit eps with their reciprocal values Compute + = UDV T References S Lipshutz and M Lipson (2009) Linear lgera. In: Schuam s outlines. McGraw-Hill <- matrix(c( 1, 1, -1, 2, 2, 2, -1, 3, -1, -1, 2, -3 ),yrow=true,nrow=3) # multiply y 55 to retrieve whole numers pinv() * 55 project Project a vector on the order of the region defined y a set of linear (in)equality restrictions. Compute a vector, closest to x in the Euclidean sense, satisfying a set of linear (in)equality restrictions. project(x,,, neq = length(), w = rep(1, length(x)), eps = 0.01, maxiter = 1000L) x neq w eps maxiter [numeric] Vector that needs to satisfy the linear restrictions. [matrix] Coefficient matrix for linear restrictions. [numeric] Right hand side of linear restrictions. [numeric] The first neq rows in and are treated as linear equalities. The others as Linear inequalities of the form x <=. [numeric] Optional weight vector of the same length as x. Must e positive. The maximum allowed deviation from the constraints (see details). maximum numer of iterations

12 12 project list with the following entries: x: the adjusted vector status: Exit status: 0: success 1: could not allocate enough memory (space for approximately 2(m + n) doules is necessary). 2: divergence detected (set of restrictions may e contradictory) 3: maximum numer of iterations reached eps: The tolerance achieved after optimizing (see ). iterations: The numer of iterations performed. duration: the time it took to compute the adjusted vector ojective: The (weighted) Euclidean distance etween the initial and the adjusted vector The tolerance eps is defined as the maximum asolute value of the difference vector x for equalities. For inequalities, the difference vector is set to zero when it s value is lesser than zero (i.e. when the restriction is satisfied). The algorithm iterates until either the tolerance is met, the numer of allowed iterations is exceeded or divergence is detected. See lso sparse_project # the system # x + y = 10 # -x <= 0 # ==> x > 0 # -y <= 0 # ==> y > 0 # <- matrix(c( 1,1, -1,0, 0,-1), yrow=true, nrow=3 ) <- c(10,0,0) # x and y will e adjusted y the same amount project(x=c(4,5), =, =, neq=1) # One of the inequalies violated project(x=c(-1,5), =, =, neq=1) # Weighted distances: 'heavy' variales change less

13 ranges 13 project(x=c(4,5), =, =, neq=1, w=c(100,1)) # if w=1/x0, the ratio etween coefficients of x0 stay the same (to first order) x0 <- c(x=4,y=5) x1 <- project(x=x0,=,=,neq=1,w=1/x0) x0[1]/x0[2] x1$x[1] / x1$x[2] ranges Derive variale ranges from linear restrictions Gaussian and/or Fourier-Motzkin elimination is used to derive upper and lower limits implied y a system of (in)equations. ranges(,, neq = nrow(), nleq = 0, eps = 1e-08) neq nleq eps [numeric] Matrix [numeric] vector [numeric] The first neq rows in and are treated as linear equalities. [numeric] The nleq rows after neq are treated as inequations of the form a.x<=. ll remaining rows are treated as strict inequations of the form a.x<. [numeric] Coefficients with asolute value <= eps are treated as zero. using Fourier-Motzkin elimination. sparse_constraints Generate sparse set of constraints. Generate a constraint set to e used y sparse_project Read sparse constraints from a data.frame Print sparse_constraints oject

14 14 sparse_constraints sparse_constraints(oject,...) sparseconstraints(oject,...) ## S3 method for class 'data.frame' sparse_constraints(oject,, neq = length(), ase = 1L, sorted = FLSE,...) ## S3 method for class 'sparse_constraints' print(x, range = 1L:10L,...) oject R oject to e translated to sparse_constraints format.... options to e passed to other methods Constant vector neq The first new equations are interpreted as equality constraints, the rest as <= ase are the indices in oject[,1:2] ase 0 or ase 1? sorted x Note range is oject sorted y the first column? an oject of class sparse_constraints integer vector stating which constraints to print Oject of class sparse_constraints (see details). s of version , sparseconstraints is deprecated. Use sparse_constraints instead. The sparse_constraints ojects holds coefficients of and of the system x in sparse format, outside of R s memory. It can e reused to find solutions for vectors to adjust. In R, it is a reference oject. In particular, it is meaningless to Copy the oject. You only will only generate a pointer to physically the same oject. Save the oject. The physical oject is destroyed when R closes, or when R s garage collector cleans up a removed sparse_constraints oject. The $project method Once a sparse_constraints oject sc is created, you can reuse it to optimize several vectors y calling sc$project() with the following parameters: x: [numeric] the vector to e optimized

15 sparse_project 15 w: [numeric] the weight vector (of length(x)). By default all weights equal 1. eps: [numeric] desired tolerance. By default 10 2 maxiter: [integer] maximum numer of iterations. By default The return value of $spa is the same as that of sparse_project. See lso sparse_project, project # The following system of constraints, stored in # row-column-coefficient format # # x1 + x8 == 950, # x3 + x4 == 950, # x6 + x7 == x8, # x4 > 0 # <- data.frame( row = c( 1, 1, 2, 2, 3, 3, 3, 4), col = c( 1, 2, 3, 4, 2, 5, 6, 4), coef = c(-1,-1,-1,-1, 1,-1,-1,-1) ) <- c(-950, -950, 0,0) sc <- sparse_constraints(,, neq=3) # djust the 0-vector minimally so all constraints are met: sc$project(x=rep(0,8)) # Use the same oject to adjust the 100*1-vector sc$project(x=rep(100,8)) # use the same oject to adjust the 0-vector, ut with different weights sc$project(x=rep(0,8),w=1:8) sparse_project Successive projections with sparsely defined restrictions Compute a vector, closest to x satisfying a set of linear (in)equality restrictions.

16 16 sparse_project sparse_project(x,,, neq = length(), w = rep(1, length(x)), eps = 0.01, maxiter = 1000L,...) x [numeric] Vector to optimize, starting point. [data.frame] Coeffiencient matrix in [row,column,coefficient] format. [numeric] Constant vector of the system x neq [integer] Numer of equalities w [numeric] weight vector of same length of x eps maximally allowed tolerance maxiter maximally allowed numer of iterations.... extra parameters passed to sparse_constraints list with the following entries: x: the adjusted vector status: Exit status: 0: success 1: could not allocate enough memory (space for approximately 2(m + n) doules is necessary). 2: divergence detected (set of restrictions may e contradictory) 3: maximum numer of iterations reached eps: The tolerance achieved after optimizing (see ). iterations: The numer of iterations performed. duration: the time it took to compute the adjusted vector ojective: The (weighted) Euclidean distance etween the initial and the adjusted vector The tolerance eps is defined as the maximum asolute value of the difference vector x for equalities. For inequalities, the difference vector is set to zero when it s value is lesser than zero (i.e. when the restriction is satisfied). The algorithm iterates until either the tolerance is met, the numer of allowed iterations is exceeded or divergence is detected. See lso project, sparse_constraints

17 sust_value 17 # the system # x + y = 10 # -x <= 0 # ==> x > 0 # -y <= 0 # ==> y > 0 # Defined in the row-column-coefficient form: <- data.frame( row = c(1,1,2,3), col = c(1,2,1,2), coef= c(1,1,-1,-1) ) <- c(10,0,0) sparse_project(x=c(4,5),=,=) sust_value Sustitute a value in a system of linear (in)equations Sustitute a value in a system of linear (in)equations sust_value(,, variales, values, remove_columns = FLSE) [numeric] matrix [numeric] vector variales [numeric logical character] vector of column indices in values [numeric] vecor of values to sustitute. remove_columns [logical] Remove spurious columns when sustituting? list with the following components: : the corresponding to the simplified sytem. : the constant vector corresponding to the new system system of the form x <= can e simplified if one or more of the x[i] values is fixed.

18 Index lock_index, 2 compact, 3 echelon, 4 eliminate, 3, 5 is_feasile, 6 is_totally_unimodular, 7 lintools, 9 lintools-package (lintools), 9 matrix, 7 normalize, 9, 9 pinv, 10 print.sparse_constraints (sparse_constraints), 13 project, 11, 15, 16 ranges, 13 sparse_constraints, 13, 16 sparse_project, 12, 13, 15, 15 sparseconstraints (sparse_constraints), 13 sust_value, 3, 17 18