ndiri

tecar pres.

e

In the classical Gauss-Markov model, the unknown parameters x

where P5weight matrix of observations.

As mentioned, the least squares technique is a powerful math-ematical tool for the estimation of unknown parameters. The onlyassumption that this technique requires to correctly interpret theresults from statistical and physical points of view is that theobservation errors should be random and preferably normally dis-tributed. If these assumptions are violated, that is, if the observa-

for a linear ~linearized! parametric adjustment are determinedbased on the following functional and stochastic models:

l1v5Ax

DTx50 (3)P5Ql215s02Cl21

where vn31 as before is the vector of residuals; ln315vector ofobservations; An3u5rank deficient design matrix; Pn3n5weightmatrix of observations; Du3d5datum matrix of the networkadded to complete the rank deficiency of the design matrix;0d315zero vector; Cl(n3n)5covariance matrix of observations;Ql(n3n)5cofactor matrix; and s025a priori variance factor. Asmentioned, the definition of L1 norm minimization is the estima-tion of parameters, which minimize the weighted sum of the ab-solute residuals. The work presented here originates from theMarshall and Bethel ~1996! paper in which they developed thebasic concepts of L1 norm adjustment and its implementationthrough linear programming and the simplex method. The main

1Dept. of Surveying Engineering, Faculty of Engineering, The Univ.of Isfahan, 81744 Isfahan, Iran. E-mail: amiri@eng.ui.ac.ir; also, Dept. ofMathematical Geodesy and Positioning, Faculty of Civil Engineering andGeosciences, TU Delft, Thijsseweg 11, 2629 JA, Delft, The Netherlands.E-mail: a.r.amiri-simkooei@geo.tudelft.nl

Note. Discussion open until July 1, 2003. Separate discussions mustbe submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on November 1, 2001; approved on May 28, 2002. This paperis part of the Journal of Surveying Engineering, Vol. 129, No. 1, Feb-ruary 1, 2003. ASCE, ISSN 0733-9453/2003/1-3743/$18.00.Formulation of L1 Norm MiMo

AliReza Am

Abstract: L1 norm minimization adjustment is a technique to deimplementation of L1 norm adjustment leads to the solving of a lineminimization for a rank deficient Gauss-Markov model will be pmodels, which confirm the efficiency of the suggested formulation

DOI: 10.1061/~ASCE!0733-9453~2003!129:1~37!

CE Database keywords: Tendons; Creep rupture; Relaxation, mMeasurement; Geodetic surveys.

Introduction

The method of least squares, which gives a best solution by mini-mizing the sum of squares of weighted discrepancies betweenmeasurements, is one of the most frequently used methods toobtain unique estimates for unknown parameters from a set ofredundant measurements. It is important to note that the imple-mentation of the method of least squares does not require theknowledge of the distribution from which the observations aredrawn for the purpose of parameters estimation. It can be shown,however, that if the weight matrix is chosen to be the inverse ofthe covariance matrix of the observations, the least squares esti-mate is an unbiased and minimum variance estimate. If, in addi-tion, the observation errors are normally distributed, the leastsquares method will give an identical solution vector to thosefrom the maximum likelihood method. The weighted leastsquares (L2 norm minimization! method states that the sum of thesquares of the weighted residuals v should be minimal ~Mikhail1976!

vTPv5(i51

n

piv i2min (1)imization in Gauss-Markovels-Simkooei1

t outlier observations in geodetic networks. The usual method forrogramming problem. In this paper, the formulation of the L1 norm

ented. The results have been tested on both linear and non-linear

chanics; Fiber reinforced polymers; Models; Linear programming;

tions are affected by gross errors, the properties of least squaresare no longer valid. In such cases, the gross error of observationsshould be detected and eliminated by using robust techniques, andthen, the least squares adjustment should be applied. An extensivediscussion of robust parameters estimation can be found in Koch~1999!. One of these robust techniques involves the minimizationof the L1 norm of the weighted residuals, that is

pTuvu5(i51

n

piuv iumin (2)

where p5n31 vector, which contains the diagonal elements ofthe matrix P. It should be noted that the L1 norm adjustment is anunbiased estimate like least squares, but other advantages of theleast squares such as minimum variance and maximum likelihoodare no longer valid. The advantage of L1 norm minimization com-pared to the least squares is its robustness, which means that it isless sensitive to outliers. In the following section, the formulationof the L1 norm minimization in a rank deficient Gauss-Markovmodel will be presented.

Formulation of L1 Norm MinimizationJOURNAL OF SURVEYING ENGINEERING / FEBRUARY 2003 / 37

differences of Marshall and Bethel ~1996! compared to the a 0

present work are mainly based on the following two aspects: InMarshall and Bethel ~1996!, the formulation of the L1 norm mini-mization are presented only for two special examples, and there isno full formulation for a general Gauss-Markov model. The sec-ond advantage of the present paper is the consideration of thedatum problem in the suggested formulation.

For modification of the objective function @Eq. ~2!# and theconstraints @Eq. ~3!#, the usual derivation of the least squares (L2)estimates will not work for L1 estimates, that is, an analog to thewell-known normal equations is not possible for L1 estimation.To transform Eqs. ~2! and ~3! into something we can work with,the usual strategy is to borrow a trick from linear programmingand introduce slack variables that guarantee nonnegativity, andthis allow us to write the objective function without absolutevalue signs ~Marshall and Bethel 1996!.

As mentioned, setting up the L1 estimation problem by a linearprogramming solution requires us to formulate a mathematicalmodel where all variables, both parameters and residuals, are non-negative. The development begins with the familiar parametricEq. ~3! and is then transformed into an L1 estimation problem byadding slack variables. To convert these equations into a formwhere there are nonnegative parameters and nonnegative residu-als, we introduce two slack vectors, a and b, for the parameters,and two slack vectors, u and w, for residuals. The parameters aswell as the residuals may be positive or negative, so we replacethese unknowns and residuals vectors with Marshall and Bethel~1996!

v5u2w, u,w>0

x5a2b, a,b>0 (4)Rewriting the original parametric equations and datum constraints@Eq. ~3!# and the objective function @Eq. ~2!# in terms of slackvariables yields

z5pTuvu5pTuu2wu5pT~u1w!minwhere

ui50 or wi50 (5)subject to

l1u2w5A~a2b!

DT~a2b!50 (6)and

u,w,a,b>0

or equivalently

z5@0T 0T pT pT#F abwu

Gmin (7)subject to

F A 2A I 2IDT 2DT Z Z GF abwu

G5F l0G , a,b,w,u>0 (8)where Z is a zero matrix. Denoting

38 / JOURNAL OF SURVEYING ENGINEERING / FEBRUARY 2003F bwu

G5xO ; F 0ppG5cO (9)

and

F A 2A I 2IDT 2DT Z Z G5AO ; F l0G5bO (10)One gets

z5cOTxOmin (11)subject to

AO xO5bO ; xO>0 (12)This is a special operations research problem that can be solvedby linear programming. The simplex method is a technique de-signed to solve a linear programming problem systematically,moving from corner to corner in the allowed solution space, i.e.,Eq. ~12!, which is usually underdetermined, until the objectivefunction reaches a minimum. For more information refer to theAppendix.

Solving for vector xO will yield the vectors a, b, w, and u,consequently, the solution vector x and the residual vector v canbe obtained. These operations should be iterated for non-linear~linearized! models until the solution vector x converges to zeros.

Numerical Results on Simulated and Real Networks

For verification of the above formulation, three examples are pre-sented. The first example is a simulated linear model; the secondone is a simulated nonlinear model, and the third one is a realnonlinear network. The results presented above were also testedon other simulated and real networks by this author. All the re-sults obtained have been calculated using a computer program inMATLAB software in which the subroutine linprog.m has beenused.

Example 1In the first example, a leveling network is assumed. The networkconsists of four points ~P1, P2, P3, and P4! with six height dif-ference observations. The datum of the network is provided byminimum constraints; point P1 has been considered as a fixedstation, with a height of HP15100 m. The degree of freedom ofthe network is d f 53. Two kinds of observations have been con-sidered for this network. Table 1 shows a list of these observa-tions and their observed values ~columns 4 and 7!. In the case ofthe second observations ~column 7!, it is assumed that the ob-served height difference P3 to P1 is erroneous by 5 cm. As seenfrom the table, the accuracy of these observations is 10 mm.

Table 1 also shows the results of adjustment by L1 and L2norm minimization of the residuals. In this simple example, AI inEq. ~12! is a 7320 matrix. Because the model is linear, bothtechniques have been done without iteration. L1 norm results havebeen obtained according to the suggested formulation. The ad-justed residuals of the first group of observations by L1 and L2norm are given in columns 5 and 6 of Table 1. The maximumabsolute values of the residuals for L1 and L2 norms are 21.1 and19.0 mm, respectively. It may imply that the L1 norm minimiza-tion is more sensitive to outliers than L2 . The adjusted residualsof the second observations by L1 and L2 norms are given in

values. Degree of freedom of the network is d f 515. The datumof the network is provided by inner constraints.

Table 1. Two Types of Observations of Leveling Network (Dhi and Dhi) with Their L and L Norm Adjusted ResidualsRe

!

1Each adjustment has been calculated, again, using two tech-niques, i.e., L1 norm minimization and the famous L2 norm mini-mization. In this network, AO in Eq. ~12! is a 31388 matrix. Be-cause Taylor series expansion was used, the final solution isobtained through iterations. The adjusted coordinates of the sta-tions are given in Table 6 after convergence of the corrections tozeros ~after 3 iterations for both techniques!. The initial coordi-nates for both techniques are the same and are given in Table 6.Table 5 also shows the results of adjustment by L1 and L2 normminimization of the residuals. The adjusted residuals of the firstFig. 1. Configuration of the trilateration network in Example 2

JOURNAL OF SURVEYING ENGINEERING / FEBRUARY 2003 / 39columns 8 and 9 of Table 1. The maximum absolute values of theresiduals for L1 and L2 norms are 46.0 and 29.3 mm, respectively.It is obvious that the 5-cm gross error has been detected clearlyby the L1 norm rather than L2 . This means, again, that the L1norm minimization is a powerful technique to detect outlier ob-servations.

Example 2In the second example, a simulated trilateration network is as-sumed. The network consists of 6 points with 30 distance obser-vations. Fig. 1 illustrates the position of the stations as well as thedistance observations. Table 2 shows the list of observations andtheir observed values. The degree of freedom of the network isd f 521. The accuracy of the observations is 10 mm, and thedatum of the network is provided by inner constraints. Two kindsof observations have been considered for this network. These ob-servations are the same except that the first observation is errone-ous by 10 cm.

Each adjustment has been calculated using two techniques,i.e., the L1 norm minimization that was formulated in this paperand the famous L2 norm minimization. In this example, AO in Eq.~12! is a 33384 matrix. Because the Taylor series expansion wasused, the final solution is obtained through iterations. The ad-justed coordinates of the stations are given in Tables 3 and 4 afterconvergence of the corrections to zeros ~after 3 iterations for bothtechniques!. The initial coordinates for both techniques are thesame and are given in Tables 3 and 4. Table 2 also shows the

Observationss ~mm! Dhi(m) ~1!From To L1 ~mm

P1 P2 10 1.0066 0.00P2 P3 10 0.9845 19.0P3 P4 10 0.9697 14.8P4 P1 10 22.9946 0.00P3 P1 10 22.0101 0.00P2 P4 10 2.0091 21.Absolute sum: 54.9results of adjustment by L1 and L2 norm minimization of theresiduals. The adjusted residuals of the first observations by L1and L2 norms are given in columns 5 and 6. In the results of theL1 norm, nine residuals ~equivalent to the minimum observationsfor solving the problem! are zeros. The maximum absolute valuesof the residuals for L1 and L2 norms are 86.82 and 68.65 mm,respectively, which are related to the sixth observation ~distance2-1!. As seen, both techniques resulted in a gross error in thisobservation.

The residuals of the second adjustment by L1 and L2 normsare given in columns 7 and 8. The maximum absolute values ofthe residuals for L1 and L2 norms are 108.50 and 63.52 mm,respectively, which are related to the first observation. As is ob-vious, a 10-cm gross error has been detected clearly by the L1norm rather than L2 . Another point that is clear from these resultsis that the residual of observation 6 in the L1 norm adjustment hasnot been changed from its initial value ~286.82! in the first ad-justment. This means that the error can be detected in the networkjust as it had been detected previously. But such a situation doesnot exist for the L2 norm residual of the sixth observation. It hasbeen reduced from 268.65 to 241.84 mm. This means that thisresidual may not detect the error of its observation. The aboveresults may be summarized as follows: Implementation of the L1norm estimation as a powerful procedure for outliers detectionproves to be very effective. The efficiency of this technique ismore straightforward when more than one observation have er-rors.

Example 3In the third example, a real 2D-trilateration network was obtainedfrom the Department of Surveying Engineering ~1993! for an area~Baghbahadoran! in Isfahan Province in Iran. This network wasestablished for educational purposes. The network consists of 8points with 28 distance observations with nominal standard de-viation of 3 mm12 ppm using EDM DI1600 Wild. Fig. 2 illus-trates the position of the stations as well as the distance observa-tions. Table 5 gives a list of observations and their observed

1 2

sidualsDhi(m) ~2!

ResidualsL2 ~mm! L1 ~mm! L2 ~mm!

20.50 1.0031 27.20 219.618.5 1.0142 218.2 222.417.4 1.0033 0.00 6.9

21.60 22.9952 0.00 9.71.10 21.996010.05 46.0 29.319.0 1.9993 0.00 2.858.0 71.3 90.7

Table 2. Two Types of Observations of Trilateration Network with Their L1 and L2 Norm Adjusted Residuals

1 2 5 128.0688 0.00 28.68 0.00 216.2310 2 6 169.9940

24.58 29.76 24.58 212.1024 5 4 206.1591 0.00 27.27 0.00 210.32observations by L1 and L2 norms are given in columns 6 and 7 ofTable 5. In the results of the L1 norm, 13 residuals ~equivalent tothe minimum observations for solving the problem! are zeros. Themaximum absolute values of the residuals for the L1 norm are17.16, 16.01, and 10.87 mm. The maximum absolute values ofthe residuals for the L2 norm are 8.85, 9.95, and 6.57 mm. Theyall, both L1 and L2 residuals, are related to the same observations,i.e., distances 24, 35, and 46. This means that both tech-niques resulted in gross errors in these observations. One pointwhich is, again, obvious from these results is that the L1 normresiduals of the errors are more than the L2 norm residuals. Thismeans that the L2 norm criterion tends to distribute the errors into

good quality observations. This can be implied by looking at theresidual of the 23rd observation, which equals 6.38 mm.

Conclusions

L1 norm adjustment is a technique used in geodetic networks todetect errors in observations. In this paper, the formulation andimplementation of L1 norm minimization were presented for rankdeficient Gauss-Markov models, which leads to the solving a lin-ear programming problem. For verification of the suggested for-mulation, three examples were presented on linear and nonlinear

25 5 6 250.0094 211.80 219.03 211.80 219.8226 6 1 70.6895 9.80 3.38 9.80 27.4827 6 2 169.9936 0.43 12.14 0.43 25.2528 6 3 158.1068 0.00 0.14 0.00 22.4029 6 4 70.7005 0.00 0.28 0.00 20.7030 6 5 249.9982 20.53 27.76 20.53 28.54

Table 3. L1 and L2 Norm Adjusted Coordinates for First Observations in Example 2

Point NumberInitial Coordinates Adjusted Coordinates (L1 norm! Adjusted Coordinates (L2 norm!

X Y X Y X Y

1 100 100 99.9875 99.9993 99.9816 100.00242 200 70 200.0000 70.0025 200.0101 69.99583 200 200 200.0031 199.9915 200.0035 199.99384 100 200 99.9981 199.9970 99.9992 199.99765 300 150 300.0045 150.0071 299.9980 150.00726 50 150 50.0068 150.0026 50.0076 150.0032

40 / JOURNAL OF SURVEYING ENGINEERING / FEBRUARY 20030.00 11.70 0.00 24.820.00 3.90 0.00 7.880.00 8.95 0.00 7.984.19 3.41 4.19 0.92

14.32 9.14 14.32 6.8022.11 21.96 22.11 24.5226.99 29.37 26.99 214.090.00 11.29 0.00 19.07

27.31 28.08 27.31 210.5810.10 2.83 10.10 20.2113.07 13.35 13.07 12.3630.68 29.45 30.68 41.5324.20 212.88 24.20 220.43Observation NumberDistances

Observed values ~m!Residuals Residuals

From To L1 ~mm! L2 ~mm! L1 ~mm! L2 ~mm!

1 1 2 104.4226 28.50 9.66 108.50 63.522 1 3 141.4264 0.46 4.37 0.46 8.353 1 4 100.0186 221.01 223.39 221.01 228.124 1 5 206.1519 21.76 20.53 21.76 32.615 1 6 70.6993 0.00 26.42 0.00 217.286 2 1 104.5010 86.82 68.65 86.82 41.847 2 3 130.0119 222.90 213.95 222.90 214.918 2 4 164.0010 7.95 19.24 7.95 27.0211 3 1 141.426912 3 2 129.989013 3 4 100.000914 3 5 111.783415 3 6 158.109016 4 1 100.004617 4 2 164.009018 4 3 100.012419 4 5 206.149020 4 6 70.687421 5 1 206.143022 5 2 128.073023 5 3 111.8023

Appendix: Linear Programming Problemx1>0,x2>0, . . . ,xn>0 (14)

and simultaneously subject to m additional constraints of the form

Table 4. L1 and L2 Norm Adjusted Coordinates for Second Observations in Example 2Fig. 2. Configuration of the real trilateration network in Example 3

JOURNAL OF SURVEYING ENGINEERING / FEBRUARY 2003 / 41ai1x11ai2x21 . . .1ainxn5bi ; i51,2,...,m (15)Here, z5the objective function to be minimized. The coefficientsc1 , c2 ,. . . , cn are the ~known! cost coefficients, and x1 , x2 ,. . . , xnare the decision variables to be determined. The coefficients ai jfor i51,2, . . . ,m and j51,2, . . . ,n are called the technologicalcoefficients. The constraints in Eq. ~14! are the nonnegativity con-straints. A set of variables x1 , x2 ,. . . , xn satisfying all the con-straints are called a feasible point or a feasible vector. The set ofall such points constitutes the feasible region or the feasible~both real and simulated! models. The results showed that L1norm minimization is more sensitive than least squares for outlierdetection. L1 norm minimization is more efficient than the L2norm when there are more errors in observations.

It seems that the lack of attention paid to L1 norm minimiza-tion in geodetic applications has been mainly due to the relativecomplexity of its implementation compared to least squares, butthis complexity is not important in the presence of modern com-puting techniques.

Standard Format of LP

The subject of linear programming, sometimes called linear opti-mization, in standard form is concerned with the following prob-lem: For n independent variables x1 ,x2 ,. . . ,xn , minimize thefunction

z5c1x11c2x21 . . . 1cnxn (13)subject to the primary constraints

Point NumberInitial Coordinates Adjusted Coordinates (L1 norm! Adjusted Coordinates (L2 norm!

X Y X Y X Y

1 100 100 99.9875 99.9993 99.9682 100.00592 200 70 200.0000 70.0025 200.0229 69.99333 200 200 200.0031 199.9915 200.0028 199.99024 100 200 99.9981 199.9970 100.0010 199.99645 300 150 300.0045 150.0071 299.9974 150.00926 50 150 50.0068 150.0026 50.0078 150.0050Linear programming ~LP! is a branch of operations research. Thismethod deals with the problem of minimizing or maximizing ~op-timizing! a linear function in the presence of linear equalityand/or inequality constraints. This appendix describes the prob-lem of LP. The main concepts of this optimization problem arepresented. To solve the problem, some effective solution algo-rithms will be introduced. For details of these solution methods,some references are suggested. The discussion will be started byformulating a particular type ~standard format! of linear program-ming. Any general linear programming may be manipulated intothis form.

Table 5. List of Observations of Real Geodetic Network with Their L1 and L2 Norm Adjusted Residuals

11 2 6 325.765 3.07 0 1.2312 2 7 140.336 3.01 22.03 0.76

25 5 8 140.562 3.01 1.08 3.5526 6 7 216.929 3.03 0 1.96space. Using this terminology, the linear programming problemcan be stated as follows: Among all feasible vectors, find one thatminimizes the objective function.

A linear programming problem can be stated in a more conve-nient form using a matrix notation. Consider the following col-umn vectors x, c, and b, and the m3n matrix A

x5F x1x2]xn

G , c5F c1c2]cn

G , b5F b1b2]bmG

A5F a11 a12 fl a1na21 a22 fl a2n] ] ]am1 am2 fl amn

G (16)Then, the standard form of the LP problem can be written asfollows:

Minimize cTx, (17)subject to Ax5b, (18)

27 6 8 383.912 3.10 0 2.0628 7 8 173.118 3.02 0.26 20.52

Table 6. L1 and L2 Norm Adjusted Coordinates of Real Geodetic Network in Example 3

Point NumberInitial Coordinates Adjusted Coordinates (L1 Norm! Adjusted Coordinates (L2 Norm!

X ~m! Y ~m! X ~m! Y ~m! X ~m! Y ~m!

1 1,000 1,000 1,000.000 1,000.001 999.999 1,000.0012 818.516 812.010 818.511 812.010 818.511 812.0133 677.672 688.732 677.674 688.732 677.673 688.7344 877.679 488.325 877.680 488.326 877.677 488.3225 951.401 584.753 951.398 584.759 951.401 584.7586 1,143.558 833.680 1,143.555 833.676 1,143.556 833.6797 943.863 748.931 943.866 748.925 943.867 748.9248 818.243 629.803 818.248 629.804 818.248 629.804

42 / JOURNAL OF SURVEYING ENGINEERING / FEBRUARY 200313 2 8 182.20714 3 4 283.13515 3 5 292.81716 3 6 487.91317 3 7 272.91318 3 8 152.42319 4 5 121.38220 4 6 435.85621 4 7 268.87122 4 8 153.45423 5 6 314.45424 5 7 164.339Observation NumberDistances

Observed value Accuracy ~mm!Residuals

From To L1 norm ~mm! L2 norm ~mm!

1 1 2 261.298 3.05 4.66 0.992 1 3 448.088 3.13 0 22.043 1 4 526.093 3.18 0 3.64 1 5 418.081 3.11 23.94 23.725 1 6 219.709 3.03 0 20.876 1 7 257.271 3.04 3.69 3.887 1 8 412.409 3.11 20.87 22.548 2 3 187.167 3.02 2.84 4.49 2 4 329.064 3.07 16.01 9.9510 2 5 263.253 3.05 0 5.023.02 0 2.753.05 0 1.963.06 10.87 6.573.15 25.64 23.223.05 0 1.123.02 2.99 4.813.01 0 5.733.12 17.16 8.853.05 1.68 5.173.02 0 2.163.07 3.82 6.383.02 0 0.03

x>0

Another form of the LP problem is called a canonical form. Aminimization problem is in canonical form if all variables arenonnegative, and all the constraints in Eqs. ~15! and ~18! are ofthe > type. By simple manipulations, this problem can be trans-formed to the standard form. For example, an inequality can beeasily transformed into an equation. To illustrate, consider theconstraint given by (ai jx j>bi . This constraint can be put in anequation form by subtracting the nonnegative slack variable xn1i ,leading to (ai jx j2xn1i5bi and xn1i>0.

Fundamental Theorems of Linear Programming

Simplex MethodBecause the extreme points may be enumerated by algebraicallyenumerating all basic feasible solutions that are bounded by

S nm Done may think of simply listing all basic feasible solutions andpicking the one with the minimal objective value. This is unsat-isfactory, however, for a number of reasons. First, the number ofbasic feasible solutions is bounded by

S nm Dwhich is large, even for moderate values of m and n. Second, thissimple approach does not tell us if the problem has an unboundedConsider the system Ax5b and x>0, where A is an m3n ma-

trix, and b is an m31 vector. Suppose that rank (A,b)5rank(A)5m . After possibly rearranging the columns of A, let A5@B,N# where B is an m3m invertible matrix, and N is an m3(n2m) matrix. The solution is x5(

xN

xB) to equations Ax5b,where

xB5B21b; and xN50 (19)is called a basic solution of the system. If xB>0, then x is calleda basic feasible solution of the system. Here B is called the basicmatrix ~or simply basis!, and N is called the nonbasic matrix. Thecomponents of xB are called basic variables and the componentsof xN are called nonbasic variables. If xB.0, then x is called anondegenerate basic feasible solution, and if at least one compo-nent of xB is zero, then x is called a degenerate basic feasiblesolution.

Considering the above definitions, some theorems on linearprogramming can be summarized as follows: Theorem 1. Every basic feasible solution is equivalent to the

extreme point of the nonempty feasible region and vice versa. Theorem 2. If an optimal solution exists, then an optimal ex-

treme point ~or equivalently, an optimal basic feasible solu-tion! exists that is the optimal solution.

Theorem 3. For every extreme point ~basic feasible solution!there corresponds a basis ~not necessarily unique!, and, con-versely, for every basis there corresponds a unique extremepoint. Moreover, if an extreme point has more than one basisrepresenting it, then it is degenerate. Conversely, if a degener-ate extreme point has more than one basis representing it, thenit is degenerate. Conversely, a degenerate extreme point hasmore than one basis representing it if and only if the systemAx5b itself does not imply that the degenerate basic variablescorresponding to an associated basis are identically zero.solution that may occur if the feasible region is unbounded.Thirdly, if the feasible region is empty, we shall discover that onlyafter all possible ways.

There are different ways to approach the solution of a linearprogramming problem; the simplex method by Dantzig is recom-mended first. The simplex method is a clever procedure thatmoves from an extreme point to another extreme point, with abetter ~at least not worse! objective. It also discovers whether thefeasible region is empty and whether the optimal solution is un-bounded. In practice, the method only enumerates a small portionof the extreme points of the feasible region. The interested readercan find more explanations in, Dantzig ~1963! and Bazaraa et al.~1990!.

Acknowledgments

The writer would like to thank R. C. Burtch, the editor, and thetwo anonymous reviewers for their valued and constructive com-ments on this paper.

References

Bazaraa, M. S., Jarvis, J. J., and Sherali, H. D. ~1990!. Linear program-ming and network flows, 2nd Ed., Wiley, New York.

Dantzig, G. B. ~1963!. Linear programming and extensions, PrincetonUniversity Press, Princeton, N.J.

Department of Surveying Engineering ~1993!. Micro-geodesy networkfor educational purposes in Baghbahadoran. The Univ. of Isfahan,81744, Isfahan, Iran.

Koch, K. R. ~1999!. Parameter estimation and hypothesis testing in lin-ear models, Springer, Berlin.

Marshall, J., and Bethel, J., ~1996!. Basic concepts of L1 norm minimi-zation for surveying applications. J. Surv. Eng., 122~4!, 168179.

Mikhail, E. M. ~1976!. Observations and least squares, IEP-A dun-Donnelley, Chicago.

JOURNAL OF SURVEYING ENGINEERING / FEBRUARY 2003 / 43