The American Mathematical Monthly pp

ON THE DEVELOPMENT OF OPTIMIZATION THEORY

Andras Prekopa

Intro duction

Farkass famous pap er of b ecame a principal reference for linear inequalities

after the publication of the pap er of Kuhn and Tucker Nonlinear Programming in

In that pap er Farkass fundamental theorem on linear inequalities was used to

derive necessary conditions for optimality for the nonlinear programming problem The

results obtained led to a rapid development of nonlinear optimization theory The work

of John containing similar but weaker results for optimality published in has

b een generally known but it was not until a few years ago that Karushs work of

b ecame widely known although essentially the same result was obtained by Kuhn and

Tucker in

In this pap er we call attention to some imp ortantwork done in the last century and

b efore We show that fundamental ideas ab out the necessary optimality conditions for

nonlinear optimization sub ject to inequality constraints can b e found in pap ers byFourier

Cournot and Farkas as well as by Gauss Ostrogradsky and Hamel

To start to describ e the early development of optimization theory it is very helpful to

lo ok at the rst two sentences in Farkass pap er

The natural and systematic treatment of analytical mechanics has to have as its

background the inequalit y principle of virtual displacements rst formulated byFourier

Andras Prekopa received his PhD in at the University of Budap est under the leadership of

A Renyi He was Assistant Professor and later Asso ciate Professor of Probability Theory at the same

university Since he has held a professorship of mathematics at the Technical University of Budap est

He is also head of the Applied Mathematics Department at the Computer and Automation Institute of the

Hungarian Academy of Sciences He is a corresp onding memb er of the Hungarian Academy of Sciences

the National Academy of Engineering of Mexico memb er of the ISI and Fellow of the Econometric

So ciety His main interests are in Probability Statistics and Op erations Research including applicati on

in engineering design water resources electrical engineering economics and natural sciences This

pap er was prepared while the author was visiting the Mathematics ResearchCenter of the Universityof

Wisconsin Madi son Readers who are not familiar with the sub ject whose history is discussed in this

article should consult Pourciaus article in this Monthly Editors

and later by Gauss The p ossibilityofsuch a treatment requires however some

knowledge of homogeneous linear inequalities that maybesaidtohave b een entirely

missing up to now

We see that Farkas had a denite reason for developing the theory of linear inequalities

He was Professor of Theoretical Physics at the University of Kolozsvar We can certainly

assume that he himself had already applied his inequality theorem to the problem of

mechanical equilibrium In fact he rep orted rst on the applications of the mechanical

principle of Fourier at the session of the Hungarian AcademyheldonDecemb er

This was published later in Hungarian and in German We shall analyze this

pap er in more detail in further sections We present here however a brief summary of

the problem and the principal result

The metho d of Lagrange for nding extrema of functions sub ject to equality constraints

was published in in his famous b o ok Mecanique Analytique as a to ol for nding the

stable equilibrium state of a mechanical system In the case when a p otential exists the

Lagrangian problem can b e formulated in the following manner

minimize g x

sub ject to g x i m

i

where the ob jective function is the p otential The case of inequality constraints was rst

investigated in byFourier If a p otential exists then Fouriers problem is

minimize g x

sub ject to g x i m

i

Assuming that the constraining and the ob jective functions are dierentiable and some

regularity conditions hold the Lagrangian necessary condition for the equilibriu m states

that at the minimizing p oint x the gradient rg x can b e expressed as a suitable linear

combination of the gradients rg x i m

i

For Fouriers problem the necessary condition for equilibrium was proved byFarkas

in and This condition formulated without pro of for sp ecial cases by

Cournot in and for the general case by Ostrogradsky in is that at the

minimizing p oint x the gradient rg x can b e expressed as a linear combination with

nonnegative co ecients of those gradients rg x whose corresp onding constraints hold

i

with equalityatx

Figure illustrates the situation The constraints restricting a mass p ointareve

x plane coincides with the halfspaces Cho osing the co ordinate system so that the x



plane of the ground the p otential corresp onding to the gravitational force is given bythe

wellknown formula mg x provided x is small as compared to the radius of the earth

 

At the equilibrium p oint x four constraints hold with equality and the cone generated

by their negative gradients contains the vector of the gravitational force

x ∗

Figure The mass p oint is in stable equilibrium at x the vector of the gravitational force

is in the cone generated by the negatives of the gradients of the constraining functions at

x

Nowwe need to say a few words ab out those mechanical principles which played an

imp ortant role in the development of optimization theory

On the Principles of Mechanical Equilibr iu m

The principle of virtual work was enunciated by Johann Bernoulli in It app eared

rst in a b o ok byVarignon in the same year Let us quote Lagrange I pp

The principle of virtual velo city can b e given the following general form

Ifasystemofanynumb er of b o dies or p oints on eachofwhichany forces act is in

equilibrium and one gives the system some small displacement on the basis of which

eachpointmoves an innitesimal distance that will express its virtual velo city the

sum of the forces eachmultiplied by the distance moved by the corresp onding p oint

in the direction of the force will always b e equal to zero if innitesimal displacements

in the direction of the forces are taken as p ositive and those in the opp osite direction

negative

John Bernoulli is the rst as far as I know who observed this great general form

of the principle of virtual velo city and its usefulness for the solution of the problems of

Statics One can see this in one of his Letters to Varignon dated whichVarignon

placed at the b eginning of the ninth Section of his New Mechanics a Section whichis

entirely devoted to showing by means of dierent applications the truth and the use

of the principle in question

This principle was considered by Lagrange as an axiom of Mechanics On page of the

same work we read

In this law lies what is generally called the principle of virtual velo city whichhas

b een recognized for a long time as the fundamental principle of equilibrium as we

have describ ed it in the preceding Section and we can consequentlyconsideritasa

kind of an axiom of Mechanics

For the case of a conservative system of forces ie when the forces are given as nega

tive partial derivatives of a scalar function the principle rst enunciated by Courtivron

applies Concerning this Lagrange writes the following I p

which yields a further principle of Statics namely that out of all congurations

that are successively assumed by the system the one in which it has the greatest or

the smallest kinetic energy is also that in whichitwould have to b e placed in order

for it to remain in equilibrium See Courtivron Memoires de lAcademie des Sciences

of and

Lagrange gave sucient conditions that the p otential takes its minimum As Bertrand

remarked his pro of was incomplete and Dirichlet later gave a correct pro of

The mechanical principle of Fourier was rst published in in his pap er Memoires

sur la Statique This concerns the case of inequality constraints Fark as remarked

p that the declaration of the inequality principle is not the main contribution of

the pap er It has the subtitle Contenantlademonstration du princip e des vitesses

virtuelles This very general pro of however has not b een accepted see eg

thus the main merit of the pap er is still contrary ot Farkass remark the statementofthe

inequality principle We quote from page of Fouriers pap er

As it frequently happ ens that the p oints of the system are only constrained by

xed obstacles without b eing attached to them it is evident that there are p ossible

displacements that do not satisfy the equations of constraint one also sees that for

these displacements the moment of the resulting forces is necessarily p ositive since

the direction of these forces has to b e orthogonal to the resisting surfaces Thus the

sum of the moments of the applied forces is p ositive for all displacements of this kind

but it is imp ossible to displace a solid b o dy that is in equilibrium so that the total

moment of the applied forces will b e negative Finally if one considers the resistances

as forces whichprovide us as weknow with the means of estimating these resistances

the b o dy can b e considered free and the sum of the moments is zero for all p ossible

displacements

The moment of the forces P Q R acting on a mechanical system is dened as the

sum of scalar pro ducts

Pp Q q R r

where p q r are variations of the displacements The Bernoulli principle declares

to b e equal to zero while the Fourier principle declares to b e less than or equal

to zero in case of equilibrium Fourier was using the uxion as it turns out from other

parts of his pap er instead of the displacement these two b eing negatives of each other

This explains whyFourier required to b e nonnegative

If a p otential V exists ie if wehave

V V V

P Q R

p q r

where the derivatives on the right denote vectors then the requirement that is less

than or equal to zero takes the form

V V V

p q r

p q r

On the left side wehave a total dierential A correct mathematical pro of according to

our present standards cannot b e found in Fouriers work We know in particular that

cannot b e derived as a necessary condition that V takes its minimum without some

constraint qualication Fourier remarked that in the case where his principle reduces to

we can nd the equilibrium state by minimizing the function V

In Gauss again enunciated the inequality principle without mentioning Fourier

He added a fo otnote explaining his ideas in more detail

According to the principle of virtual velo cities this equilibrium requires that the

sum of pro ducts of all three factors namelyeach of the masses m m m etc the

line segments cb c b c b etc and the pro jections of the p ossible motions of the

corresp onding p oints in accordance with the constraints should b e equal to zero as

it is ordinarily stated or more exactly that the ab ove sum should never b e p ositive

Fo otnote

The ordinary expression tacitly assumes such constraints as allow the opp osite for

every p ossible motion as eg that a p ointmust remain on a certain surface that the

distance b etween two p oints should b e xed and so on But this is an unnecessary

restriction and do es not always corresp ond to Nature The surface of an imp enetrable

b o dy do es not comp el a material p oint to remain on it but merely prevents it from

going to the other side an inextensible but exible cord connecting twopoints prevents

only the increase but not the decrease of the distance etc Why should we not prefer

to express the lawofvirtualvelo cities from the very b eginning so that it covers all

cases

In Ostrogradsky also enunciated the inequality principle at a session of the

Academie Imperiale des Sciences de SaintPetersb ourg Voss remarks p that

therefore in Russia it is also called Ostrogradskys principle It is interesting to read what

Ostrogradsky wrote ab out the inequality principle

It is very surprising to see that in the new edition of the Mecanique Analytique

published at a time when the full extent of the principle of virtual velo citywas already

known Lagrange not only did not makeany use of the observation that in the equilib

rium of forces the total momentmay acquire a negativevalue but in a way he ruled

it out when it turned up naturally in the pro of he gave for the principle of virtual ve

lo city meanwhile failing to take account of it this great mathematician incompletely

enumerated the p ossible displacements in most of the questions of the rst part of the

Mecanique Analytique and it is easy to recognize that the displacements he neglected

to consider are not forbidden byany condition so that even when all the equations

that he established for the equilibrium are satised equilibrium may not o ccur

In this memoir we prop ose to exp ound the analysis of the use of the principle of

virtual v elo city considered in its utmost generality and to complete the solution of

various questions handled in the rst part of the Mecanique Analytique

Lagranges metho d of multipliers was published in the rst volume of Mecanique A

nalytique pp as a to ol for nding the equilibriu m of a mechanical system using

Bernoullis principle The validity of the metho d was proved by Lagrange entirely on an

algebraic basis This ingenious metho d was unable to attract mathematics students and

teachers for a long time The metho d of presentation used nowadays bymany instructors is

to prove the necessary condition rst in the case of inequality constraints give a geometric

meaning to this and then refer to the case of equality constraints This can makethe

students more enthusiastic ab out this theory We shall see in Sections what happ ened

to Fouriers principle Nowwedevote a few pages to Farkass theorem

The Theorem of Farkas on Linear Inequalities

Gyula Farkas a former memb er of the Hungarian Academy of Sciences con

tributed primarily to theoretical physics In this resp ect his results in Mechanics and

Thermo dynamics are the b est known For further information concerning his life and

work see

Farkass pap er is generally cited for the fundamental theorem on homogeneous

linear inequalities In what follows we shall write vectors in column form and the transp ose

will b e denoted by a prime

Let g g g denote mcomp onentvectors and form the following homogeneous lin

M

ear inequalities



g x i M

i



g x

If holds for every x for which all inequalities in hold then wesay that the

inequality is a consequence of the system of inequalities Farkass theorem is

the following

Theorem The inequality is a consequence of the inequalities if and

only if there exists nonnegative numbers such that

M

g g g

M M

Farkas published this theorem rst in and However the pro of

contains a gap The second pro of that he gavein in Hungarian is also incomplete

Essentially the same pap er app eared in German in The rst complete pro of was

published in Hungarian in and in German in This pro of is included in

his b estknown pap er It is instructive to see what kind of error he made in the rst

pro of We shall quote form

First he shows that wemay supp ose that there are as many linearly indep endent

inequalities in the system of which one consequence is considered as the number of

variables Then he represents the co ecientvector of the consequence inequalityasa

linear combination of the other co ecientvectors Now if for the sake of simplicitythe

rst n inequalities are linearly indep endent then we can express the multipliers b elonging

to these in terms of the others in the following manner

I I I

 n  n

K K K

  n  n

n

What Farkas then writes can b e summarized as follows a necessary condition that it is

imp ossible for all values simultaneously to assume nonnegativevalues is that in

or in one of its equivalent forms that can b e obtained using subsequent eliminations at

least one righthand side has the two prop erties that its rst terms is negative and the

co ecients of the remaining terms are either negative or zero

Farkas seems to have had in mind a pro cedure similar to what is used in the dual

simplex metho d We know however that cycling is p ossible there We can correct

the pro of by applying the lexicographic dual metho d a short presentation is given in

Let us summarize the whole corrected pro of briey

Proof of Farkass theorem First weremarkthatifwe apply a linear transfor

mation y Bx where B is a nonsingular square matrix to the inequalities

then the new inequality will b e consequence of the new system Furthermore

proving Farkass theorem for the new inequalities we see that is satised with the

same nonnegativemultipliers

M

Let us assume rst that the system containsasmany linearly indep endent relations

as the number of variables Consider the linear programming problem

minimize

M

sub ject to g g g

M M m

Starting from any basis and applying the lexicographic dual metho d at the end wereach

a system of the form where either I K orthereexistsarow in which

 

the rst term on the righthand side is negativeandallvariables there have p ositiveor

zero co ecients The second case cannot o ccur In fact the lexicographic dual metho d

guarantees that the column vectors of are obtained from the vectors g g g by

M

a nonsingular linear transformation thus the system of linear inequalities in the variables

y y

n

y I y K y



y I y K y

   

y

n

has the consequence

I y K y

  

Now if eg we had I I I then the vector of comp onents y

 

y y would satisfy and would not satisfy

 n

If in wehave hM linearly indep endent relations and for the sake of simplicity

g g are linearly indep endent then wecho ose ncomp onentvectors d d so

h nh

that B g g d d is a nonsingular matrix and apply the transformation

h nh

y Bx for the inequalities in and Then will dep end only on y y

h

and since the inequality in is a consequence of the former ones y y will have

h n

zero co ecients there also Nowwe can forget ab out y y and apply the ab ove

h n

reasoning to the system of variables y y At the end we can reestablish y y

h h n

everywhere with zero co ecients and reach the original inequalities by setting y Bx

Thus wehave proved Farkass theorem Wehavealsoproved

Theorem If is not a consequence of the system then there is a linear

transformation y Bx with nonsingular square matrix B such that for at least one i

the variable y has negative coecients in the new inequality and has nonnegative

i

coecients in the new system

Necessary Condition for Equilibr i um

n

We consider a mechanical system the state of which is describ ed by the vector x R

which is sub ject to the following constraints

g x i m

i

n

where the functions g R R i m are dierentiable

i

Denote by X X the comp onents of the forces acting on the system and supp ose

n

that equilibrium is reached at x Then byFouriers principle the inequality

X x X x

n n

is satised where x x are variations of the co ordinates x x ie small

n n

quantities with the prop erty that the vector of comp onents

x x x x

n

n

satises In what follows we shall use general terms but our statements can b e made

exact under not very restrictive mathematical conditions

The inactive constraints in ie those for whichwehave strict inequalityatthe

p oint x do not restrict small changes in the co ordinates Thus to obtain the conditions

on small changes we only have to consider the constraints activeatx Let us assume that

these are the rst M constraints Then writing dx instead of x the Fourier principle

requires that

X dx X dx

n n

and the increments are sub ject to the inequalities

g x g x

i i

dx dx i M

n

x x

n

Nowwe can forget ab out the order of magnitude of the quantities dx dx satisfying

n

the ab ove inequalities In fact if dx dx satisfy a homogeneous linear inequality

n

then the same holds for tdx tdx where t is any nonnegative constant

n

Let X denote the vector with comp onents X X We shall write it in rowform

n

and also use the convention that gradients are rowvectors

If x is an equilibrium p oint then using Fouriers principle we nd that the linear in

equality is a consequence of the system of linear inequalities hence byFarkass

theorem there exist nonnegativenumbers such that

M

X rg x rg x

M M

If the system of forces is conservative ie there exists a p otential V x so that at every

p oint of the state space wehave

V

i n X

i

x

i

then b ecomes

V V

dx dx

n

x x

n

Here on the lefthand side wehave a total dierential Having a p otential we can start

from by applying Courtivrons principle which states that if a mechanical system is

in stable equilibrium then the p otential has a lo cal minimum and observing that except

for pathological cases the total dierential is nonnegative at the minimum p ointofthe

function

We can ask Who if anyb o dy deduced equation from the inequality princi

ple for mechanical equilibrium To answer this question we can start by analyzing the

pap ers cited byFarkas in and we should also lo ok at the volumes of Enzyklopadie

der Mathematischen Wissenschaften published at the b eginning of this centuryInthe

four volumes on mechanics there are two pap ers Voss and Stackel that mention

theories of statics and dynamics under inequality constraints These authors and Farkas

cite ab out thirty b o oks and pap ers some of them are unfortunately not available to the

other author but Farkas Voss and Stackel together very likely give a go o d picture of the

history of the sub ject Voss writes p

This case not considered by Lagrange was rst considered indep endently of

Fourier by Gauss and by Ostrogradsky therefore in Russia the Fourier princi

ple is also called Ostrogradskys principle In France the Fourier principle has b een

less often neglected A A Cournot develop ed Ostrogradskys equations as early as

One pap er by Ostrogradsky is also cited byFarkas Stackel mentions also

Voss do es not refer to Farkas but Stackel cites the pap ers We can forgive

Stackel for not recognizing the imp ortance of Farkass work b ecause the sub ject of his

pap er is Elementare Dynamik

Dynamical problems sub ject to inequality constraints were considered by Gibbs in

Nonnegativity of the multipliers in some sp ecial cases are proved byMayer

and Zermelo Farkas gave a more general form of the theory in

Fourier Cournot OstrogradskyandFarkas seem to b e the principal contributors to

the present form of the necessary condition of static equilibriu m Cournot and later

Ostrogradsky presented the equations in the form of a conjecture as wemaysay

now Farkas proved by relying on Fouriers work concerning the rst part of the

theorem considering a conservative system of forces where as it is obvious to day but

did not seem necessary for those in the last century dealing with mechanical problems a

constraint qualication is needed Wehave to add to these that the work of Cournot was

based to a great extentonthework of Poinsot

Fourier and F arkas b oth contributed further imp ortant ideas to optimization theory

Let us summarize briey their principal results in this resp ect

Fourier

anticipated the formulation of the linear programming problem in II

p

formulated the inequality principle for the mechanical equilibriu m in

initiated the parametric solution of homogeneous linear inequalities in

Farkas

proved the basic theorem concerning homogeneous linear inequalities rst men

tioned in and rst complete pro of in

gave a rigorous pro of for the dual form of Fouriers mechanical inequality

principle rst in

gaveanelegant parametric representation for the solutions of homogeneous

linear inequalities rst in

The Work of Cournot on the Problem of Equilib ri um

AntoineAugustin Cournot a famous p olyhistor was one of the greatest

mathematical economists The work of his that we analyze here was done as a contri

bution to mechanics It is interesting to remark that though optimization theory is an

imp ortant to ol of mathematical economics in his famous b o ok Recherches sur les Principes

Mathematiques de la Theorie des Richesses no application can b e found of his previous

investigation of the problem of mechanical equilibriu m Since the rst publication of

this b o ok in several editions have app eared eg with the critiques of Walras

Bertrand and Pareto and intro duction and biographical notes by Lutfalla and with

the notes of Irving Fisher

Cournot b ecame docteur es sciences in in Paris His thesis contributed to

dynamics where he applied his earlier result published in One year earlier he

published an elementary pap er on inequalities there is no anticipation in that pap er

of any form of Farkass theorem

In the pap er that is at present most imp ortant for us Cournot do es not refer to

the work of Fourier He seems to have b een unawareoftheenunciation of the inequality

principle byFourier in Cournot rediscovered the principle but also derived the

necessary conditions for equilibrium We quote from his short pap er p

It often happ ens that the constraints of the system cannot b e expressed by equa

tions from these constraints there result conditions of equilibrium that have always

b een considered as not b eing derivable from the principle of virtual velo cit y and b eing

outside of this principle Mecanique de M Poisson tom I p Our purp ose here

is to show that when the constraints of the system can b e expressed algebraically by

inequality signs the principle of virtual velo city suitably mo died still applies and

provides the conditions of equilibrium by a uniform metho d

Let us assume that a system is sub ject to a certain numb er of similar constraints

expressed by the inequalities

I I etc J J etc

the signs and always b eing supp osed not to exclude the case of the equality

One cannot in general dierentiate an inequality like an equation or derivea

relation b etween the increments of the variables but when equilibrium conditions are

sought only two cases can app ear Either the nature of the system is such that one

could suppress the constraints without changing its state as happ ens if cords that

connect some of its p ointsarenottautorifthepoints to which certain surfaces form

imp enetrable obstacles do not touch these surfaces in this case equilibrium must o ccur

indep endently of the constraints and it is unnecessary to taketheminto account or

the system is so situated that the constraints pro duce their eect and can decrease the

numb er of conditions necessary for equilibrium then one has the present co ordinate

values I J etc and if one varies these values their increments will not

be entirely arbitrary on the basis of the primitive conditions they must satisfy the

relations

a I I etc J J etc

In addition when the system is sub ject to constraints of the kind that weare

considering here it is clear that one could suppress them provided that forces were

applied to pro duce the same eect thus the resistance of a surface can b e replaced

by the application of a suitably oriented force normal to the surface the constraint

of a taut cord by the application of a force oriented in the direction of the tension

in the cord etc Let F F b e the forces applied directly to the system in the

directions f f andP P the auxiliary forces that take the place of obstacles and

are directed along lines p p on the basis of the principle of virtual velo citywe

will have the fundamental equation

Ff F f etc Pp P p etc

Now it is easy to see that all virtual movements in which P and p P and p etc

would have opp osite signs would tend to overcome the obstacles that the surfaces

cord etc opp ose to the dierentpoints of the system and are consequently incom

patible with the constraints of the system expressed by the inequalities a The only

movements compatible with the constraints are those for which P and p P and p

have the same sign and for which as a consequence of this the quantities Pp P p

etc are essentially p ositive

Thus disregarding the auxiliary forces P P etc and the terms containing them

in the equation of the virtual velo cities for the constraints that have to b e satised

by the system wehave

b Ff F f etc

The ab ove pro of given by Cournot concerns the validity of the Fourier principle

In the pap er Cournot then derives the nonnegativity of the Lagrangian multipliers

for sp ecial cases similar to some of those investigated by Lagrange in the rst part of

Mecanique Analytique in case of equality constraints When dealing with the case of a

ts lying on planes parallel to the xy plane Cournot refers to the work of system of p oin

Poinsot rst published in In fact for this sp ecial case Poinsot already obtained

the conditions of equilibrium

The pap ers of Cournot seem to b e less well known than his other works They

are not listed in the bibliography of his works published in

The Work of Ostrogradsky on the Problem of Equilibr i um

Mikhail Vasilevich Ostrogradsky contributed imp ortant results to mechanics

among other sub jects He was a studentinParis and attended the courses of Fourier

Poisson Cauchy and other wellknown French mathematicians He returned to Russia

St Petersburg in In he b ecame an extraordinary memb er and in an

ordinary memb er of the AcademyofStPetersburg

We should mention two pap ers of Ostrogradsky concerning the problem of mechanical

equilibriu m Farkas refers only to presented in b efore the Academy but the

other is an improved version of his theory In the earlier pap er four applications are

mentioned a a p oint that is allowed to move in one part of the space sub divided by

a surface b the pivoted p olygon c the exible cord d the incompressible liquid

Further problems are mentioned concerning dynamics

In Ostrogradsky refers to what wenow call Farkass theorem as an obvious

algebraic fact and derives the equation of equilibrium We repro duce here part of

pages and of

Assume that the quantities s s s s b elong not only to those displace

ments of the system of which the reaction forces are capable but also to all other

displacements whether p ossible or not or rather consider s s s s tobe

entirely arbitraryWehave to express that the reaction forces R R R R are

not capable of pro ducing any displacement for the system that satises the conditions

L L L L

 

the sign do es not exclude equality

Nowwe know that a system of forces is capable of every displacement that con

tributes a p ositivevalue to the total moment and of none of those that corresp onds

to negativeorzerovalues of the total moment Thus for the reaction forces to b e

incapable of pro ducing any displacements satisfying the conditions it is neces

sary that their moment is negative or zero for these displacementsinotherwords it

is necessary that the function

R s cos R s cos R s cos R s cos

 

in which designate the angles R s R s R s R s re

sp ectively and which consequently represents the moment of the forces R R R R

is negative or zero whenever s s s s satisfy

The solution of the question of making the function

R cos R s cos R s cos R s cos

negative or zero whenever the functions of the same kind L L L L are

 

p ositive or zero b elongs to the most elementary algebra It is necessary and sucient

that R s cos R s cos R s cos R s cos can b e reduced to a

linear function of L L L L with negative co ecients Thus we only have

 

to make for arbitrary s s s s

R s cos R s cos R s cos R s cos

L L L L

   

and add the condition that all s b e negative Or if one wants to avoid considering

negative s one can make

R s cos R s cos R s cos R s cos

 L L L L

   

then all s will b e p ositive By the last equation and the preceding one it is evident

that the moment R s cos R s cos R s cos R s cos will

b e negative or zero whenever the functions L L L L will b e p ositiveor

 

zero

If we transp ose all terms to the same side the equation of equilibrium of the

reaction forces b ecomes

R s cos R s cos R s cos R s cos

L L L L

   

It has to hold for all s s s s of arbitrary magnitude and direction But

one should not forget to add to the equation the inequalities

 

Ostrogradsky made twoerrors One consists of asserting Farkass theorem without

pro of Ostrogradskys authoritywas so strong that many authors to ok the theorem for

granted The second error was discovered by Study as communicated

byMayer p On page of Ostrogradsky infers that some of his inequalities

should reduce to equations of equilibriu m This would have made it p ossible to determine

the multipliers in the equation at equilibriu m He thought that those multipliers should

b e equal to zero whichwould turn out to b e negativeifwe did not have the inequality

constraints

The Work of Farkas on the Problem of Equilibr iu m

Since the two pap ers are essentially the same we quote from the German version

In the intro ductory part of the pap er we read the following

The purp ose of this pap er is to show that with a suitable mo dication the metho d

of multipliers of Lagrange can b e carried over to the Fourier principle also

In the rst section of the pap er Farkas deals with his inequality theorem His pro of is

incomplete howeveraswe p ointed out In the second section the necessary condition of

equilibriu m is derived

the constraint expressions go over to the following

P P

Fq G q

P P

Sq  Tq 

and the fundamental inequalityis

X X

Q q  or  Q q 

The constraint equations have to b e expressed in the form of inequalities so that

the system of constraint conditions app ears as follows

P P

Fq  G q 

P P

 Fq   G q 

P P

Sq  Tq 

The Fourier principle requires that the inequality is satised by all systems of val

ues q that satisfy It will b e shown that this happ ens only when there are p ositive

multipliers such that the co ecients Q can b e represen ted as homogeneous linear func

tions of the co ecients FGF GST Let

denote these p ositivemultipliers Wemust have

Q  F  G S T

The dierences   may also take negativevalues consequently

the Fourier principle will b e satised by those Q values that can b e determined from

equations suchas

Q F G S T

where have the same values in the expressions for each Q and

moreover are completely arbitrarybut are arbitrary nonnegative

quantities Conversely the fundamental inequality follows from the system

by using and simple pro cedures

In a further section of the pap er the twomaintyp es of application are presented

a the equilibrium equation for tangential solid b o dies b the equilibriu m equation for

nonsolid b o dies

On the Constraint Qualication

The constraint qualication is of fundamental imp ortance not only from the p ointof

view of nonlinear optimization but also from the p oint ofviewofmechanics So on we

shall see why

At the b eginning of this century the axiomatic foundation of the mathematical and

physical sciences was an imp ortant activity In his famous pap er Hilb ert urges mathe

maticians to axiomatize twophysical discipline s probability theory and mechanics The

pap er of Hamel of is an attempt in the direction of the axiomatic foundation

of classical mechanics The Fourier inequality principle was unfortunately not included

An improved version of his axiomatics is contained in his pap er which app eared in

There the inequality principle of Fourier is already mentioned as one of the axioms

Axiom I Ik on page On page concerning Das Energieprinzip Axiom I Ic de

clares denoting the p otential by U that U is a necessary condition for equilibriu m

Hamel had to establish consistency b etween the two axioms and this could only b e done

by a constraint qualication in fact we nd it in Axiom I Ic on page It requires

that to every mass particle there corresp ond a scalar function u such that the following

equality holds

X

U dmru r

where dm denotes the mass and r the state of a particle If instead of we require

U rUr

where r denotes the state of the whole system of particles then we obtain essentially the

KarushKuhnTucker constraint qualication Equation implies b ecause wecan

generate the u functions for the purp ose of satisfying in suchawa y that in U we

subsequently x all variables except for those b elonging to one particle

Unfortunately Hamel was unaware of the existence of Farkass theorem Even in his

Theoretische Mechanik rst published in this theorem is not referred to though

part of the pages are devoted to the Fourier inequality principle There

we also see that his constraint qualication is not derived rigorously and his reasoning

concerning the Fourier principle is more heuristic

Miscellaneous Remarks Concerning Linear Inequalities

Farkass most imp ortant results concerning the Fourier principle and the theory of linear

inequalities are summarized in his pap ers

The parametric representation of the solutions of linear inequalities was initiated by

Fourier Minkowski gave the representation of all solutions using extremal rays In

the parametric representation given byFarkas it is particularly simple to generate

the rays whose convex combinations constitute all solutions of the linear inequalities

Writing z z instead of dx dx in and we can represent all solutions

n n

of in a parametric form insert it into and if the number of variables and the

active constraints in is not large wecansolve the equilibrium problem in some

cases For this wehave to know which are the active constraints in This metho d

was recommended byFarkas The application of the Farkas parametric representation

technique for linear programming is p erhaps b est done as in the pap er by Uzawa He

completes one inequality to equality if necessary eliminates the nonzero constants on the

righthand side from the other constraints inserts the parametric form of the solutions

of these into the remaining equality and the ob jective function and nds the optimal

solution

The pap ers in Hungarian and German resp ectivelyhave the same content

Farkas gave a further mechanical application of his theorem on linear inequalities This

concerns the decomp osition of the forces of reaction in a mechanical system into sho cks

and others satisfying the negatives of the constraints given for the displacements Voss

mentions in p that this kind of distinction b etween the forces is attributed to

Painleve The mathematical results of the pap er are repro duced in The b o ok

in Hungarian do es not contain new results as compared to his earlier pap ers and to the

pap er

Between and Farkas did not publish on linear inequalities In after

Haar generalized Farkass theorem for the inhomogoneous case Farkas returned to the

area and published further pap ers on system of linear inequalities The

pap ers are also partially relevant is the German version of

Haar wrote three pap ers on linear inequalities The rst two are essen

tially Hungarian and German versions of the same pap er He gave the following general

ization for Farkass theorem we shall use the notation of Section

Theorem If the linear inequality



g x b

is a consequence of the linear inequalities



g x b i M

i

i

ie if every x satisfying also satises thenthere exist nonnegative constants

n

such that for every x R we have

M

M

X

 

g x b g x b

i i

i

i

This is the same statement referred to by Kuhn and Tucker in a fo otnote supplied to von

This concerns the duality theorem of linear programming Neumanns manuscript

proved by Gale Kuhn and Tucker

Haar remarked that the theory of linear inequalities was develop ed byFarkas

and Minkowski The famous b o ok of Minkowski alsocontains results on linear in

equalities pages in b oth editions He considered a nite system of homogeneous

linear inequalities from twopoints of view to represent them in a parametric form and

to discover those which are sup eruous ie can b e omitted without changing the set

of solutions This latter problem led him to prove a theorem closely related to Farkass

theorem mentioned in Section For the sake of completeness we present here the exact

statement of Minkowskis theorem

We assume that in the system the numb er of linearly indep endent relations equals

n the numb er of comp onents of x Thiscausesnolossofgenerality as Minkowski re

marked We also assume that there exists at least one x for which holds with all

strict inequalities Suchanx is called essential An essential x is said to b e an extremal

ray of the cone if x is not the sum of two nonzero vectors which satisfy and

neither of them is a constantmultiple of the other



Theorem Among the linear forms g x i M those which are

i

for m linearly independent extremal solutions have the property that each of them

is essential and al l other forms can be expressedaslinear combinations of them with

nonnegative weights

For further references to early pap ers on linear inequalities see

References

Remarks In the list given b elow the name of Farkas app ears with the initials J or Gy

Up to the b eginning of this centuryitwas customary to translate the given name into

the language of the pap er J refers to the German name Julius Gy refers to the

Hungarian name Gyula and these two names were supp osed to b e the same

Volumes of pap ers presented at an academy frequently have double yearassignments

One indicates the years when the pap ers in the volume were presented b efore the

academy whereas the other indicates the year when the publisher published the volume

In the reference list b elow the latter date is always given

K J Arrow L Hurwicz and H Uzawa eds Studies in Linear and Nonlinear Program

ming Stanford University Press Stanford Calif

E M L Beale Cycling in the dual simplex algorithm Naval Research Logistics Quarterly

A Cournot Sur le calcul des conditions dinegalite annonce par M Fourier Bull Sci

Math Premiere Section du Bulletin Universel des Sciences et de lIndustrie publiesousla

dir de Ferussac

A Cournot Extension du princip e des vitesses virtuelles au cas ou les conditions de liaison

du systeme sont exprimees par des inegalites Bull Sci Math Premiere Section du Bulletin

Universel des Sciences et de lIndustrie publie sous la dir de Ferussac

A CournotMemoire sur le mouvement dun corps rigide soutenu par un plan xe J Reine

Angew Math

A Cournot Du mouvement dun corps sur un plan xe quand on a egard alaresistance

du frottement J Reine Angew Math

A Cournot Recherches sur les Princip es Mathematiques de la Theorie des Richesses Nou

velle Edition Marcel Riviere Paris

A Cournot Researches into the Mathematical Principles of the Theory of Wealth R D Ir

win Homewo o d Ill

L L Dines and N H McCoy On linear inequalities Pro c and Trans of the Royal So c

of Canada rd Series Section Mathematical Physical and Chemical Sciences

L Dirichlet Sur un moyen general de verier lexpression du p otential relatif a une masse

quelconque homogene ou heterogene J Reine Angew Math

L Dirichlet Ub er die Stabilitat des Gleichgewichts J Reine Angew Math

Encyklopadie der Mathematischen Wissenschaften Teubner Leipzig Vierter

Band Mechanik Erster Teilband

Gy FarkasAFourierfele mechanikai elv alkalmazasai Applications of the mechanical

principle of Fourier Matematikai es Termeszettudomanyi Ertesto

J Farkas Ub er die Anwendungen des mechanischen Princips von Fourier Mathematische

und Naturwissenschaftliche Berichte aus Ungarn

Gy FarkasAFourierfele mechanikai elv alkalmazasainak algebrai alap jarol The algebraic

basis of the applications of the mechanical principle of Fourier Mathematikai es Physikai

Lap ok

J Farkas Die algebraischen Grundlagen der Anwendungen des Fourierschen Princips in der

Mechanik Mathematische und Naturwissenschaftliche Berichte aus Ungarn

Gy FarkasAFourierfele mechanikai elv alkalmazasanak algebrai alap ja The alge

braic basis of the application of the mechanical principle of Fourier Mathematikai es

Termeszettudomanyi Ertesto

J Farkas Die algebraische Grundlage der Anwendungen des mechanischen Princips von

Fourier Mathematische und Naturwissenschaftliche Berichte aus Ungarn

Gy FarkasParameteres modszer Fourier mechanikai elvehez Metho d of parameters in the

mechanical principle of Fourier Mathematikai es Physikai Lap ok

Gy FarkasVectortan es az egyszeru inaequatiok tana Theory of vectors and simple in

equalities Kolozsvar

Gy Farkas Altalanos mechanikai elvek az aether szamara General mechanical principles

o of the ether Mathematikai es Termeszettudomanyi Ertest

J Farkas Allgemeine Principien fur die Mechanik des Aethers Archives Neerlandaises Serie

Livre jubilaire dedieaHALorentz

J Farkas Theorie der einfachen Ungleichungen J Reine Angew Math

J Farkas Beitrage zu den Grundlagen der analytischen Mechanik J Reine Angew Math

Gy Farkas Biztos egyensuly p otencial nelkul Stable equilibrium without p otential Math

ematikai es Termeszettudomanyi Ertesto

Gy Farkas Nemvonalas egyenlotlensegek vonalassatetele Conversion of nonlinear inequal

ities into linear ones Mathematikai es Termeszettudomanyi Ertesto

Gy Farkas Multiplicatoros modszer negyzetes alakokhoz Metho d of multipliers for

quadratic forms Mathematikai es Termeszettudomanyi Ertesto

Gy Farkas Egyenlotlensegek alkalmazasanakuj modjai New ways to apply inequalities

Mathematikai es Termeszettudomanyi Ertesto

Gy Farkas A linearis egyenlotlensegek kovetkezmenyei Consequences of linear inequali

ties Mathematikai es Termeszettudomanyi Ertesto

J Farkas Stabiles GleichgewichtohnePotential Mathematische und Naturwissenschaftliche

Berichte aus Ungarn

Gy Farkas Alapvetes az egyszeru egyenlotlensegek vektorelmeletehez Foundation of the

vectoriel theory of simple inequalities Mathematikai es Termeszettudomanyi Ertesto

I Fenyes Meg jegyzesek es kiegesztesek a mechanika elveinek Farkas Gyulafele

targyalasmodjahoz Remarks on and extensions of the treatmentofmechanical principles

byFarkas Fizikai Szemle

L FilepFarkas Gyula Termeszet Vilaga

J FourierMemoire sur la statique Journal de lEcole Polytechnique in vol

J Fourier Solution dune question particuliere du calcul des inegalites Nouveau Bulletin

des Sciences par la So ciete Philomatique de Paris in vol

J Fourier Oeuvres GauthierVillars Paris

D Gale H W Kuhn and A W Tucker Linear programming and the theory of games

in

C F Gauss Ub er ein neues allgemeines Grundgesetz der Mechanik J Reine Angew Math

J W Gibbs On the fundamental formulae of dynamics Amer J Math

anGuinness Joseph Fouriers anticipation of linear programming Op erational I Gratt

Research Quarterly

A T GrigorianOcherki Istorii Mechaniki v Rossii Outline of the History of Mechanics

in Russia Akad Nauk SSSR Moskva

A Haar A linearis egyenlotlensegekrol On linear inequalities Mathematikai es

Termeszettudomanyi Ertesto

A Haar Die Minkowskische Geometrie und die Annaherung an stetige Funktionen Math

Ann

A Haar Ub er lineare Ungleichungen Acta Sci Math

A Haar Gesammelte Arb eiten Akademiai Kiado Budap est

G Hamel Ub er die Grundlagen der Mechanik Math Ann

G Hamel Die Axiome der Mechanik in

Handbuch der Physik H Geiger and K Scheel ed Band Grundlagen der Mechanik

Mechanik der Punkte und Starren Korp er Springer Berlin

D Hilbert Mathematische Probleme Vortrag gehalten auf dem internationalen

MathematikerKongress zu Paris Nachrichten von der Konigl Gesselschaft der Wis

senschaften zu Gottingen Mathematischphysikalische Klasse

F John Extremum problems with inequalities as subsidiary conditions in

W Karush Minima of Functions of Several Variables with Inequalities as Side Conditions

Masters Thesis Dept of Math University of Chicago

T C Koopmans ed Activity Analysis of Pro duction and Allo cation Wiley New York

H W Kuhn and A W Tucker Nonlinear programming in

H W Kuhn and A W Tucker Nonlinear programming A historical view in

J L LagrangeMecanique Analytique II I Paris

J L Lagrange Oeuvres GauthierVillars Paris

C E Lemke The dual metho d for solving the linear programming problem Naval Research

Logistics Quarterly

A Mayer Ub er die Aufstellung der Dierentialgleichungen der Bewegung fur reibungslose

Punktsysteme die Bedingungsungleichungen unterworfen sind Berichteub er die Verhand

lungen der koniglichsachsischen Gessellschaft der Wissenschaften zu Leipzig Mathematisch

Physische Classe

A Mayer Zur Regulierung der Stosse in reibungslosen Punktsystemen die dem Zwange

von Bedingungsungleichungen unterliegen Berichteub er die Verhandlungen der koniglich

sachsischen Gesellschaft der Wissenschaften zu Leipzig MathematischPhysische Classe

H Minkowski Geometrie der Zahlen Teubner Leipzig and Berlin st ed nd ed

J von Neumann Discussion of a maximum principle Collected Works vol

J Neyman ed Pro c of the Second Berkeley Symp osium on Mathematical Statistics and

Probability University of California Press Berkeley Calif

Nonlinear Programming SIAMAMS Pro ceedings vol American Mathematical So ciety

Providence R I

L Nordheim Die Prinzip e der Dynamik in

anyos muk odese The scientic work of Gyula Farkas R OrtvayFarkas Gyula tudom

Matematikai es Fizikai Lap ok

R OrtvayFarkas Gyula emlekezete In memory of Gyula Farkas Az MTATag jai Felett

Tartott Emlekb eszedek Obituaries of Memb ers of the Hungarian Academy of Sciences

M Ostrogradsky Considerations generales sur les moments des forces Memoires de

lAcademie Imperiale des Sciences de SaintPetersb ourg Sixieme Serie

M OstrogradskyMemoire sur les deplacements instantanes des systemes assujettis ades

conditions variables Memoires de lAcademie Imperiale des Sciences de SaintPetersb ourg

Sixieme Serie

L Poinsot Elements de Statique Douzieme Ed GauthierVillars Paris rst pub

lished in

A PrekopaLinearis programozas Bolyai Janos Mathematikai Tarsulat Budap est

P Stackel Elementare Dynamik der Punktsysteme und starren Korp er in

Studies and Essays presented to Courant on his th birthdayInterscience New York

B Szenassy A Magyarorszagi MatematikaTortenete History of Mathematics in Hungary

Akademiai Kiado Budap est

H Uzawa An elementary metho d for linear programmingin

A Voss Die Prinzipien rationellen Mechanik in

E Zermelo Ub er die Bewegung eines Punktsystemes b ei Bedingungsungleichungen

Nachrichten von der konigl Gessellschaft der Wissenschaften zu Gottingen Mathematisch

physikalische Klasse

Computer and Automation Institute of the Hungarian Academy of Sciences Kende u

Budap est Hungary