2. Polyhedral complementarity problem

The basic scheme of the considered approach is the polyhedral complementarity. We consider polyhedrons in R<sup>n</sup>. Let two polyhedral complexes ω and ξ with the same number of cells r be given. Let <sup>R</sup><sup>⊂</sup> <sup>ω</sup> � <sup>ξ</sup> be a one-to-one correspondence: <sup>R</sup> <sup>¼</sup> <sup>Ω</sup><sup>i</sup> f g ð Þ ;Ξ<sup>i</sup> <sup>r</sup> <sup>i</sup>¼<sup>1</sup> with <sup>Ω</sup><sup>i</sup> <sup>∈</sup> <sup>ω</sup>, <sup>Ξ</sup><sup>i</sup> <sup>∈</sup> <sup>ξ</sup>.

3. Classical linear exchange model

I ¼ f g 1;…; m be the index sets of commodities and consumers.

modities is realized with respect to some nonnegative prices pj

The consumer i∈I has to choose a consumption vector x<sup>i</sup> ∈ Rn

; xi ! max, p; xi ⩽ p; wi , xi ⩾0:

; xi under budget restriction:

ci

Let x~<sup>i</sup> be a vector xi that solves this program.

individual optimization problems such that

Each consumer i ∈I possesses a vector of initial endowments w<sup>i</sup> ∈R<sup>n</sup>

 

A price vector <sup>~</sup><sup>p</sup> 6¼ 0 is an equilibrium price vector if there exist solutions <sup>x</sup>~<sup>i</sup>

well-known description [13].

Figure 1. Polyhedral complementarity.

function ci

We demonstrate the main idea of the approach on the classical linear exchange model in the

Consider a model with n commodities (goods) and m consumers. Let J ¼ f g 1;…; n and

¼) The problem of consumer <sup>i</sup>:

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

http://dx.doi.org/10.5772/intechopen.77206

29

þ. The exchange of com-

, i ¼ 1, …, m, for the

þ.

, forming a price vector p ∈R<sup>n</sup>

<sup>þ</sup> maximizing his linear utility

We say that the complexes ω and ξ are in duality by R if the subordination of cells in ω and the subordination of the corresponding cells in ξ are opposite to each other:

$$
\Omega\_i \prec \Omega\_j \iff \Xi\_i \succ \Xi\_j.
$$

The polyhedral complementarity problem is to find a point that belongs to both cells of some pair Ω<sup>i</sup> ð Þ ; Ξ<sup>i</sup> :

$$p^\* \text{ is the solution} \iff p^\* \in \Omega\_i \cap \Sigma\_i \text{ for some } i.$$

This is natural generalization of linear complementarity, where (in nonsingular case) the complexes are formed by all faces of two simplex cones.

Figure 1 shows an example of the polyhedral complementarity problem. Each of two complexes has seven cells. There is a unique solution of the problem—the point x<sup>∗</sup> that belongs to Ω<sup>6</sup> and Ξ6.

The polyhedral complementarity problem can be reformulated as a fixed point one. To do this the associated mapping is introduced as follows:

$$G(p) = \Xi\_i \quad \forall p \in \Omega\_i^\* \ \_{\prime}$$

where Ω<sup>∘</sup> <sup>i</sup> is the relative interior of Ωi.

Now <sup>p</sup><sup>∗</sup> is the solution of complementarity problem if <sup>p</sup><sup>∗</sup> <sup>∈</sup> G p<sup>∗</sup> ð Þ.

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models http://dx.doi.org/10.5772/intechopen.77206 29

Figure 1. Polyhedral complementarity.

a model with fixed budgets, known more as Fisher's problem. The convex programming reduction of it, given by Eisenberg and Gale [6], is well known. This result has been used by many authors to study computational aspects of the problem. Some reviews of that can be found in [7]. The polyhedral complementarity approach gives an alternative reduction of the Fisher's problem to a convex program [2, 8]. Only the well-known elements of transportation problem algorithms are used in the procedures obtained by this way [9]. These simple pro-

The mathematical fundamental base of the approach is a special class of piecewise constant multivalued mappings on the simplex in R<sup>n</sup>, which possesses some monotonicity property (decreasing mappings). The problem is to find a fixed point of the mapping. The mappings in the Fisher's model proved to be potential ones. This makes it possible to reduce a fixed point problem to two optimization problems which are in duality similarly to dual linear programming problems. The obtained algorithms are based on the ideas of suboptimization [11]. The mapping for the general exchange model is not potential. The proposed finite algorithm can be considered as an analogue of the Lemke's method for linear complementarity problem with

The basic scheme of the considered approach is the polyhedral complementarity. We consider polyhedrons in R<sup>n</sup>. Let two polyhedral complexes ω and ξ with the same number of cells r be

We say that the complexes ω and ξ are in duality by R if the subordination of cells in ω and the

Ω<sup>i</sup> ≺ Ω<sup>j</sup> () Ξ<sup>i</sup> ≻Ξj: The polyhedral complementarity problem is to find a point that belongs to both cells of some

<sup>p</sup><sup>∗</sup> is the solution () <sup>p</sup><sup>∗</sup> <sup>∈</sup> <sup>Ω</sup><sup>i</sup> <sup>∩</sup>Ξ<sup>i</sup> for some <sup>i</sup>:

This is natural generalization of linear complementarity, where (in nonsingular case) the

Figure 1 shows an example of the polyhedral complementarity problem. Each of two complexes has seven cells. There is a unique solution of the problem—the point x<sup>∗</sup> that belongs to Ω<sup>6</sup> and Ξ6. The polyhedral complementarity problem can be reformulated as a fixed point one. To do this

G pð Þ¼ <sup>Ξ</sup><sup>i</sup> <sup>∀</sup><sup>p</sup> <sup>∈</sup> <sup>Ω</sup><sup>∘</sup>

i ,

<sup>i</sup>¼<sup>1</sup> with <sup>Ω</sup><sup>i</sup> <sup>∈</sup> <sup>ω</sup>, <sup>Ξ</sup><sup>i</sup> <sup>∈</sup> <sup>ξ</sup>.

cedures can be used for getting iterative methods for more complicate models [5, 10].

positive principal minors of the restriction matrix (class P) [12].

given. Let <sup>R</sup><sup>⊂</sup> <sup>ω</sup> � <sup>ξ</sup> be a one-to-one correspondence: <sup>R</sup> <sup>¼</sup> <sup>Ω</sup><sup>i</sup> f g ð Þ ;Ξ<sup>i</sup> <sup>r</sup>

subordination of the corresponding cells in ξ are opposite to each other:

2. Polyhedral complementarity problem

28 Optimization Algorithms - Examples

complexes are formed by all faces of two simplex cones.

the associated mapping is introduced as follows:

<sup>i</sup> is the relative interior of Ωi.

Now <sup>p</sup><sup>∗</sup> is the solution of complementarity problem if <sup>p</sup><sup>∗</sup> <sup>∈</sup> G p<sup>∗</sup> ð Þ.

pair Ω<sup>i</sup> ð Þ ; Ξ<sup>i</sup> :

where Ω<sup>∘</sup>
