An Interior-point Method for Computing Economic Equilibria

Zoltan Pap Subotica Tech – College of Applied Sciences, Subotica, Serbia

papzoli@vts.su.ac.rs

Abstract— In this paper the possibility of computing equili-brium in static production economies by interior point me-thod is investigated. The performance of the algorithm is tested on examples with known equilibria taken from the literature on general equilibrium models and numerical results are presented. In computing equilibria, economy will be specified by excess demand function.

I. INTRODUCTION Economic equilibria are usually solutions to fixed point

problems [3]. These algorithms, called simplicial algo-rithms, have finite convergence property. However, equi-libria of the large-scale economic problems may be diffi-cult to compute with these algorithms [11]. Alternative methods, such as non-linear programming methods per-form well on high dimensional economic equilibrium problems. The fast development of the computer science and powerful computers enable equilibria computation of extremely large-scale and nonlinear economic models. Unfortunately, powerful computers are not sufficient for solving these models. Inefficient algorithms can increase the computation time and even provide inaccurate results. As one can see, reliable and good computation algorithms are important ingredients in solving economic models. In literature, there are a lot of proposed algorithms which compute equilibria in general equilibrium models. One of these algorithms is based on the interior-point method, which is a technique for nonlinear programming. It is pro-posed in [8]. The main aim of this paper is the analysis of the mentioned algorithm and also some modifications will be proposed.

In order to compute the equilibrium, it is necessary to define conditions that characterize the equilibrium point. Much of the literature on computing equilibria uses the excess demand function : D Dξ

++ℜ → ℜ as characteriza-

tion, where D is finite commodity space. Specification of an economy with excess demand function involves solv-ing the nonlinear complementarity problem (NCP) of the form

( ) ( )0, 0, 0,TF x x Fx x≥ ≥ = (1)

where nx ∈ℜ and : n nF ℜ → ℜ is nonlinear, smooth map-ping. Several items of literature like [1] and [7] have do-cumented the basic theory, algorithms and application of the NCP. One of the significant reasons of why comple-mentarity problems are so pervasive in economics is be-cause the concept of complementarity is synonymous with the notion of system equilibrium. Reference [8] formu-lates NCP as a nonlinear least-squares problem and solves it by an interior-point method. Solving the nonlinear least-squares problem by interior-point method avoids the ill-

conditioning problem which may be present when using traditional techniques for solving the constrained least-squares problem, such as the Gauss-Newton method. The rest of the paper is organized as follows. Section 2 describes the idea of solving the NCP by interior-point method and states the interior-point algorithm. Section 3 is devoted to the basic concept of the equilibrium and it de-scribes how the interior-point algorithm can be applied in order to compute the equilibrium. Section 4 presents nu-merical results obtained applying the algorithm on well-known examples. Section 5 contains the summary and conclusion.

II. THE INTERIOR-POINT ALGORITHM

A. Solving the NCP via Interior-point Method Nowadays there is a wide variety of computational me-

thods for solving complementarity problems. These com-putational methods include the following. − extensions of Newton’s method for nonlinear equations that replace the direction finding routines with comple-mentarity problems; − a path search method that uses a generalization of a line search technique; − quadratic programming-based algorithms that derive extensions of the Gauss-Newton methodology; − differentiable optimization-based descent methods that reformulate the complementarity relationships as a nonli-near equation or program; − projection and proximal methods that extend projected gradient methods; − smoothing techniques that replace the nonsmooth equa-tions with differentiable approximations; − interior-point methods based on removing inequalities by an interior penalty.

In this paper the NCP will be solved by the interior-point method. The main idea of the algorithm is the re-formulation of the NCP as nonlinear least-squares prob-lem. Reference [5] has documented the basic theory and algorithms for solving the nonlinear least-squares prob-lem. There are two main methods for solving this prob-lem. One is the Gauss-Newton method, which omits the second order information of the objective function. Omit-ting the second order information of the objective function is justified when the nonlinear least-squares problem has the so-called small residual structure. The second method for solving the nonlinear least-squares problem is the Le-venberg-Marquardt method, which avoids one of the weaknesses of the Gauss-Newton method, namely the ill-conditioning of the Jacobian. The interior-point algorithm

SISY 2011 • 2011 IEEE 9th International Symposium on Intelligent Systems and Informatics • September 8-10, 2011, Subotica, Serbia

– 113 –978-1-4577-1974-5/11/$26.00 ©2011 IEEE

studied in this paper will apply the idea of the Gauss-Newton method.

Consider the NCP which is slightly different than (1):

( ) ( )0, 0, 0,Tx x FF x x≤ ≥ = (2)

where nx ∈ℜ and : n nF ℜ → ℜ is nonlinear, smooth map-ping. Problem (2) can be transformed into the system of nonlinear equations by adding nonnegative slack variables

ns∈ℜ . In this way, one obtains system of the form

( ) 0, 0,TF x s x s+ = = (3)

where 0x ≥ , and 0s ≥ . By denoting ( )X diag x= , (3) can be equivalently written as follows:

( ) 0

0·F x s

X s

=

=

+ (4)

In order to simplify notation, system (4) will be denoted by ( ), 0x s = . The problem of solving (4) will be con-sidered as a nonlinear least-squares problem with bound constraints:

( ) 2

20, 0

1,

2min

x sx s

≥ ≥ (5)

Problem (5) is solved by the interior-point method. The log-barrier function of the problem can be written in the following form:

2

2

1( , ; ) ( , ) ln ln

2,

i ii ix s x s x sµ µ µ= − −∑ ∑ (6)

where (0,1]µ ∈ is parameter of the interior-point algo-rithm. The idea of the interior-point algorithm is to solve sequence of unconstraint optimization problems

,( ;in , )m

x sx s µ for decreasing sequence of parame-

ters (0,1]µ ∈ . The first-order optimality conditions for (6) can be written as follows:

1

2

1

2

0

0,

0

T w

w

Xw e

w eS

µ

µ

− =

− =

− =

(7)

where ( ) ( ), ,x s x s= ∇ . In the interior-point algo-rithm the Jacobian is approximated by the central fi-nite-difference formula. Other notations in the (7) are

1 2, , 1, 2, ,i i

i i

w w i nx s

µ µ= = = … . Matrices ( )X diag x=

and ( )S diag s= are diagonal matrices. The vector e

is defined by ( )1,1, ,1 Te = … . The system of the first-order optimality conditions (7) will be denoted by ( )1 2, , , 0x s w w = . The algorithm incorporates the idea of the Gauss-Newton method for solving the nonli-near least-squares problem. It means that the Jacobian ∇ is approximated in such a way that the second order

information of 2·T

ii

+ ∇∑ is neglected. This ap-

proach saves time for computing the individual Hes-sians 2

i∇ , which are needed in the second term. This method is justified when the first term is much more sig-nificant than the second term, like in the case of the small residual problem. In practice, many least-squares prob-lems have small residuals at the solution. One of the weaknesses of the Gauss-Newton method for solving a nonlinear least-squares problem is its poor behavior when the Jacobian is rank-deficient. Reference [8] claims that this weakness is avoided by the interior-point method. The behavior of the algorithm when the Jacobian is ill-conditioned will be studied on numerical examples. The ill-conditioning can cause poor behavior of the algorithm

when calculating the search direction ( )1 2, , ,T

x s w w from the system of linear equations

( ) ( )2

2

1 1,.

,

T I

diag W W diag P

x

S

s

w

w

−= −

(8)

The major computational effort in the algorithm is solving (8). This system is the standard primal-dual system slightly modified since the second order information of ( ),x s is omitted.

B. The Algorithm The interior-point algorithm for solving (2) can be writ-

ten as follows:

Step 1. Set initial values 0 0, nx s++

∈ℜ , and 1 2

0 0,w w for start-ing points. Set the parameters of the algorithm 0ε > ,

_Max It for stopping criteria and 0 1µ = . Set 1k = ,

and 0 0 0, ,k k kx ssx µ µ← ← ← , 1 21

0

2

0 ,k kw ww w← ← . Cal-

culate ( )1 2

2, , ,k k k kx s w w .

Step 2. While ( )1 2

2, , ,k k k kx s w w ε> and _k Max It<

− Solve (8) in order to obtain search direc-

tion ( )1 2, , ,k k k

T

kx s w w .

− Determine feasible step-lengths 1kw

α and 2kw

α for 1 2,k kw w

which ensure positiveness of the 1 2

1 1,k kw w+ +

by the formu-

la 1 2

1 2

1 20.95·min , 0.95·min

k kw w

k k

k k

w w

w wα α= =

.

− Consider the merit function which is the same as the log-barrier function (6) and define linear function

:m++

ℜ → ℜ in the next manner: ( ) ( ),0 ;xm s µ= ,

and ( ) ( )· , · ;x x s sm α α α µ+ += . Perform the back-tracking algorithm on the merit function for (0,1]α ∈ in order to obtain the step lengthα which satisfies the Wolfe conditions. The backtracking algorithm performs halving

Z. Pap • An Interior-point Method for Computing Economic Equilibria

– 114 –

of the step length until the Armijo condition is not satis-fied. − Update variables 1 1,k k kk k kx x s sx sα α

++← + ← + ,

1 2

1 1 2 2

1

1 2

1· ·,k k

k k kk w k k ww w w ww wα α

+ +← ←+ + and the algo-

rithm parameter 1 2

1 1 1 11 0.1·

2

T T

k k k kk

x w s w

nµ + + + +

+

+= which de-

creases in every iteration.

− Calculate ( )1 2

2, , ,k k k kx s w w and set 1k k← + .

Step 3. Go back to Step 2. The interior-point parameter µ is updated in terms of

the average values of the complementarity products 1

1 1

T

k kx w+ +

and 2

1 1

T

k ks w+ +

.

III. APPLICATION OF THE ALGORITHM TO THE COMPUTATION OF THE EQUILIBRIUM

This section discusses how the interior point-algorithm can solve the problem of determining equilibrium point in the static production economic.

Consider a static exchange economy with I consumers and D commodities. Every commodity d has its own positive price , 1, 2 ,,dp d D= … . The price vector is de-

noted by Dp∈ℜ . Each consumer has their own prefe-rence defined by the utility function :i iu X →ℜ ,

1, 2, ,i I= … , where D

iX ⊆ ℜ is the choice set of the

consumer, and Dℜ is the commodity space. Vector , 1, 2, ,ii X ix I∈ = … denotes the consumption bundle of

the i -th consumer. Each consumer is endowed with a vector , 1, 2, ,i

D iw I∈ = … that is strictly positive. This vector represents the wealth level of the consumer.

An equilibrium of this economy is a price vector * Dp

++∈ℜ and an allocation of commodi-

ties ( )1 2ˆ ˆ ˆ, , , Ix x x… , where ˆ D

ix+

∈ℜ , such that:

1. ˆ , 1, 2, ,ix i I= … maximizes the preference of the con-sumer

( )* *

ma

.

0

x ,

.

i

T T

i

u x

s pt p wx

x

≤

≥

(9)

2. total demand cannot exceed the total endowment

1 1

ˆI

ii

I

ii

x w= =

≤∑ ∑ .

The solution of the (9) depends from the price vector p . It is called demand function. The demand function is map-ping ˆ : D Dx

++ +ℜ → ℜ with property of homogeneity 0 , i.e.

for all 0λ > it holds that ( ) ( )ˆ ˆx p x pλ = . It is assumed that the demand function is continuously differentiable.

Alternatively, the above described economy can be cha-racterized by the excess demand function

: D Dξ++

ℜ → ℜ defined by

( )1 1

ˆ .I I

i ii i

p x wξ= =

= −∑ ∑ (10)

It is assumed that the excess demand function is conti-nuous and homogeneous with degree zero. In this charac-terization of the static exchange economy, price * Dp

+∈ℜ is

equilibrium if and only if it holds

( ) ( )* * * *0, 0, 0.Tp p p pξ ξ≤ ≥ = (11)

The last equation in (11) is known as Walras’ law. Prob-lem (11) can be considered as NCP (2) and the interior-point algorithm can be directly applied.

Consider a static production economy by adding pro-duction technology to static exchange economy consi-dered earlier. It is assumed that production technology has a constant return to scale. The production set Y exhibits constant return to scale if y Y∈ implies that y Yα ∈ for any scalar 0α ≥ . Geometrically, production setY is a cone. Vector ( )1 2, , , Jyy y y= … denotes elementary activ-ities and the production technology is described by an activity analysis matrix ,[ ]i jA a= , 1, 2 ,,i D= … ,

1, 2, ,j J= … ,where ( ), ,0, 0ii jja a≥ ≤ denotes output (input). The production defined in this way is called the linear activity model. A price vector * Dp

+∈ℜ and a vector

of activity levels * Jy+

∈ℜ constitute a general equilibrium if:

1. ( )* * 0p Ayξ − ≤ ;

2. 0TA p ≤ ;

3. ( )( )** * 0T pp Ayξ − = ;

4. ( )* 0TTA p y = and

5. ** 0, 0p y≥ ≥ . Finding general equilibrium which is defined in terms of the above-mentioned definition involves solving NCP problem. Denoting variables p and y by ( ), Tx p y= and

( ) ( )( ),TTp A pF yx Aξ −= , problem of determining the

equilibrium can be considered as NCP (2) and it can be solved by the interior-point algorithm described above.

IV. NUMERICAL RESULTS In this section the performance of the interior-point al-

gorithm will be studied on well-known test-problems. Test-problems are examples of pure exchange economy and economy with linear production technology. The per-formance of the interior-point algorithm will be tested with different starting points. During the testing of the performance, the number of the required iteration for meeting stopping rule, ill-conditioning of the Jacobian, and accuracy will be recorded. The ill-conditioning of the Jacobian is one of the weaknesses of the Gauss-Newton

SISY 2011 • 2011 IEEE 9th International Symposium on Intelligent Systems and Informatics • September 8-10, 2011, Subotica, Serbia

– 115 –

method for solving nonlinear least-squares problem. The accuracy of the obtained solution is in sense of

2 .

As a consequence of the homogeneity of degree zero of the demand function in the prices, if *p represents equili-

brium prices, so does *pλ for any scalar 0λ > . Arguing in this way, one can, without loss of generality, define prices on the positive simplex

{ }1

1 .D

D

ii

p p+ +

=

= ∈ℜ =∑

The interior point-algorithm searches equilibrium prices in the D

+ℜ space. The obtained vector of prices will be norma-

lized on the positive simplex+ .

The stopping criteria of the interior-point algorithm are given by 410ε −= and the maximum number of the itera-tion is given by 6_ 0Max It = .

The initial values for 1 2,w w and for s will be in all ex-amples appropriate dimensional vector with elements one. Example 1. (static exchange economy [11]) Consider a static exchange economy with two consumers and two items of goods. The demand function of the i -th consumer is given by

( )

215

1

11 1 45 5 5

1

ˆ ,ij k ik

kij

j ik kk

a p wx p

p a p

=

=

=∑

∑

where 1024ija = if i j= and 1ija = if i j≠ . Initial endow-

ments of the consumers are 12ijw = if i j= and 1ijw = if i j≠ . The economy defined in this way is example of the economy with several equilibria. The interior-point algorithm will be applied to this problem with several initial points 0p .

The statistic of the performance of the interior-point al-gorithm is summarized in Table II.

Equilibria obtained by the interior-point algorithm are consistent with the results given in [11]. The third equili-brium differs from the equilibrium obtained in [11] in third and fourth decimal place. Example 2. (static exchange economy [2]) Consider a static exchange economy with three consumers and three

goods. The market excess demand function is given by

( )

( )

( )

321

1 2 1 3

312

1 2 2 3

2 13

2 3 1 3

;

;

.

ppp

p p p p

ppp

p p p p

p pp

p p p p

ξ

ξ

ξ

=−

++ +

= −+ +

= −+ +

Equilibrium of the economy defined in this way is diffi-cult to find, because the problem is locally completely unstable. There is a small region around the equilibrium

point ( )* 1 1 1, ,3 3 3p = such that if the initial price is

in the region, the prices will eventually leave this region and stay out. The motion of the prices will be in concen-tric curves about the equilibrium point. Some algorithms for computing equilibria are not so suitable for solve these kinds of unstable problems. The performance of the inte-rior-point algorithm is tested on several initial prices. The attained results are summarized in Table III.

The performance of the algorithm is shown in Table IV.

The interior-point algorithm reached the equilibrium in few iteration taking different initial prices. For the first and second starting price derived equilibrium differs from the one given in [2] in fourth decimal place. Example 3. (static exchange economy [3]) Consider a static exchange economy with five consumers and ten

TABLE I. INITIAL POINTS AND EQUILIBRIA OF THE EXAMPLE 1.

Initial point Normalized equilibrium

1. ( )1,1 ( )0.5, 0.5

2. ( )0.01, 0.99 ( )0.1229, 0.8871

3. ( )0.9, 0.1 ( )0.8870, 0.1130

TABLE II. PERFORMANCE OF THE ALGORITHM IN THE EXAMPLE 1.

Iteration No. Accuracy Convergence Ill-conditioning

1. 7 5

·4 1110.−

yes no

2. 13 5

·3 5310.−

yes no

3. 10 5

·2 8110.−

yes no

TABLE III. INITIAL POINTS AND EQUILIBRIA OF THE EXAMPLE 2.

Initial point Normalized equilibrium

1. ( )0.1, 0.2, 0.7 ( )0.3333, 0.3334, 0.3333

2. ( )0.45, 0.30, 0.25 ( )0.3333, 0.3333, 0.3334

3. ( )0.6, 0.3, 0.1 ( )0.3333, 0.3333, 0.3333

TABLE IV. PERFORMANCE OF THE ALGORITHM IN THE EXAMPLE 2.

Iteration No. Accuracy Convergence Ill-conditioning

1. 11 5

·2 79 10.−

yes no

2. 9 5

·2 6310.−

yes no

3. 10 5

·2 8110.−

yes no

Z. Pap • An Interior-point Method for Computing Economic Equilibria

– 116 –

goods. The i -th consumer has an initial endowment iw . The consumer has a demand function of the form

( )10

1

1

ˆ , 1, , 5, 1, ,10,i i

T

i iij

b b

j ik kk

p wx p i j

p p

α

α −

=

= = … = …

∑

where ikα is the demand share parameter for

er i and good k and ib is the elasticity of substitution for consumer i . These parameters are given below.

1 1 3 0.1 0.1 1.2 2 1 1 0.7

1 1 1 1 1 1 1 1 1 1

9.9 0.1 5 0.2 6 0.2 8 1 1 0.2

1 2 3 4 5 6 7 8 9 10

1 13 11 9 4 0.9 8 1 2 10

α =

( )2,1.3, 3, 0.2, 0.6 Tb = , and endowments are given by

0.6 0.2 0.2 20 0.1 2 9 5 5 15

0.2 11 12 13 14 15 16 5 5 9

0.4 9 8 7 6 5 4 5 7 12

1 5 5 5 5 5 5 8 3 17

8 1 22 10 0.3 0.9 5.1 0.1 6.2 11

This example is problem of economic equilibrium of higher dimension. As initial price vector 0 1Tp = is taken. The algorithm converged in18 iteration to the equilibrium price

* 0.1873, 0.1094, 0.0989, 0.0432, 0.1169,

0.0770, 0.1170, 0.1024, 0.0987, 0.0494).

(p =

Starting from the16 -th iteration, warning about ill-conditioning of the Jacobian appeared but it did not affect the performance of the algorithm. The norm of the Jaco-bian of the first-order optimality condition in equilibrium is 55.1·10 − . The performance of the algorithm was studied with several random, normalized initial price vectors and it always converged to the equilibrium, although warn-ings about ill-conditioning appeared. The iteration num-ber for reaching the equilibrium for different initial points was between18 and 33 . Example 4. (production economy [11]) Consider a pro-duction economy with two consumers and four items of goods. The demand function is given by the correspon-dence

( )

4

1

4

1

ˆ , 1, 2, 1, , 4ij k ik

kij

j ikk

a p wx p i j

p a

=

=

= = =∑

∑

where initial endowments and parameter a are given by matrices

0 0 10 0 0.8 0.2 0 0

,0 0 0 20 0. 0.9 0 0

.w a= =

The production technology is specified by an activity matrix A

1 0 0 0 3 5 1 1

0 1 0 0 1 1 5 5

0 0 1 0 1 1 1 4

0 0 0 1 1 4 3 1

.

− − −

− − −

− − − − −

− − − − −

The problem of finding the equilibrium is an example of economy with equilibrium which is not unique. Reference [11] pointed out that this problem has three equilibria. The first equilibrium is ( )* 0.25, 0.25, 0.25, 0.25p = with activ-

ity level ( )* 0, 0, 0, 0, 5, 0, 5, 0y = . The second equilibrium

is ( )* 0.2500, 0.2222, 0.3611, 0.1667p = with the activity

level ( )* 0, 0, 0, 0, 5.1806, 0.3611, 4.4583, 0y = and the third equilibrium point is

( )* 0.2500, 0.2708, 0.1667, 0.1190p = with alloca-

tion ( )* 0, 0, 0, 0, 4.3690, 0, 5.1548, 0.1190y = . Taking

0 1Tp = as initial point, the algorithm reached the second equilibrium in 34 iteration, but starting from the 28 -th iteration, warnings about bad scaling of the Jacobian ap-peared. The interior-point algorithm was tested with sev-eral random normalized initial points. The first and third equilibrium was not reached during testing, but for some random, normalized initial points the algorithm ob-tained ( )0.3000, 0.2556, 0.1778, 0.2667p = and the ac-

tivity level ( )0, 0, 0, 0, 0, 2.1986, 3.3655,1.1090y = which is not mentioned in [11] as equilibrium. This point is pos-sibly obtained because it is the local minimum. Bad scal-ing of the Jacobian could also be the reason for obtaining a false equilibrium. Example 5. (production economy [6]) Consider a static production economy with one consumer, one producer and three items of goods. The consumer has a demand function of the form

( ) ( )2 2 3 3ˆ , 1, 2, 3,i

i

ia w p w px p i

p

⋅ + ⋅= =

and initial endowment ( )0 ,5, 3w = . The firm has tech-

nology matrix ( )1, 1, 1A = − − . The parameter of the de-

mand function is given by ( )0.9, 0.1, 0a = . The algorithm has been tested with several starting points and always converged to the normalized equilibrium alloca-tion ( )* 0.5000, 0.0833, 0.4167p = and to the activity lev-

SISY 2011 • 2011 IEEE 9th International Symposium on Intelligent Systems and Informatics • September 8-10, 2011, Subotica, Serbia

– 117 –

el * 3y = . There were no warnings about ill-conditioning of the Jacobian. The iteration number needed for reaching the equilibrium was between 14 and 20 depending on the initial point. Example 6. (production economy [3]) Consider a static production economy with five consumers, eight activity sectors and six items of goods. Each consumer has an initial endowment iw . The consumer has a demand func-tion of the form

( )6

1

1

ˆ , 1, , 5, 1, , 6,i i

T

ij i

ijb b

j ik kk

a p wx p i j

p a p −

=

= = =

∑

where a the demand is share parameter for consumer i , and ib is the elasticity of substitution for consumer i . The values of parameters and endowment are given in [3]. The equilibrium allocation is given by normalized price vec-tor ( )* 0.2203, 0.2511, 0.1610, 0.0549, 0.1061, 0.2066p =

and ( )* 1 60 , 0.4635, 0, 3.9392, 0.0060, 0, 0, 0.4383, 0y ×= . The interior-point algorithm reached this equilibrium with several initial points in 24 48− iteration. Warnings about bad scaling of the Jacobian appeared, but it did not affect the convergence of the algorithm.

V. SUMMARY AND CONCLUSION There are many methods for computation of equilibria

in static exchange and static production equilibrium mod-els. When the functional form of the excess demand func-tion is available, definition of the equilibrium can be con-sidered as NCP. This article is devoted to the analysis of the interior-algorithm algorithm proposed in [8]. The main idea is solving the NCP as a least-squares problem via interior-point methods. As in the Gauss-Newton method, this algorithm exploits the special structure of the least-squares problem, omitting the second order information of the system of equation to solve. This is a great advantage in terms of cost. Literature for the least-squares problems points out, that the Gauss-Newton method has a disadvan-tage, namely ill-conditioning. The problem of ill-

conditioning is also present in the analyzed algorithm and its influence is studied through numerical examples. The algorithm performed well on most of the numerical exam-ples, and it seems that ill-conditioning is less harmful than in the Gauss-Newton method. As the algorithm did not perform well on Example 4, some attention needs to be paid to avoiding ill-conditioning.

The interior-point algorithm has great potential, but it can and should be improved in terms of bad scaling. Ref-erence [4] is devoted to interior-point algorithm for solv-ing the constrained nonlinear least-squares problem, and it deals with bad scaling of the Jacobian. Reference [9] also deals with the nonlinear least-squares problem. Methods used in this literature can be applied to the algorithm pre-sented in this article in order to improve its performance.

REFERENCES [1] G. Isac, Complementarity problems. Lecture Notes in Mathemat-

ics, Springer-Verlag, New York, 1992 [2] H. E. Scarf, “Some examples of global instability of the competi-

tive equilibrium”, International of Economic Review, vol. 1, no. 3, pp. 157-171, 1960

[3] H. E. Scarf, “The computation of equilibrium prices”, in Hand-book of Mathematical Economics, vol. II, K. J. Arrow, M. D. Intri-ligator, Eds. Amsterdam, North-Holland, 1982

[4] J. Herskovits, et. al., “Interior point algorithms for nonlinear con-strained least squares problems”, Inverse Problems in Science and Engineering, vol. 12, no. 2, 2004, pp. 211-223

[5] J. Nocedal and S. J. Wright, Numerical Optimization. Springer Series in Operations Research, New York: Springer-Verlag, 1999

[6] L. Mathiesen, “An algorithm based on a sequence of linear com-plementarity problems applied to a Walrasian equilibrium model: an example”, Mathematical programming, vol 37, pp. 1-18

[7] M. C. Ferris and J. S. Pang, “Engineering and economic applica-tion of complementarity problems”, SIAM Review, vol. 39, No. 4, pp. 669-713, December 1997

[8] M. Esteban-Bravo, “Computing equilibria in general equilibrium models via interior-point methods”, Computational Economics, vol. 23, pp. 147-171, 2004

[9] P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem”, SIAM Journal on Numerical Analysis, vol. 15, no. 5, 1978, pp. 977-992

[10] R. J. Vanderbei and D. F. Shanno, “An interior-point algorithm for nonconvex nonlinear programming”, Computational Optimization and Application, vol. 13, 1999, pp. 231-252

[11] T. J. Kehoe, “Computation and multiplicity of equilibria”, in Handbook of Mathematical Economics, vol. IV, W. Hildebrand, H. Sonnenschein, Eds. Elsevier Science Publishers B. V., 1991

Z. Pap • An Interior-point Method for Computing Economic Equilibria

– 118 –