In this section, we provide two numerical examples to compare different algorithms. All programs are written in Matlab version 7.0 and performed on a desktop PC with Intel(R) Core(TM) i5-4200U CPU @ 2.30 GHz, RAM 4.00 GB.
Example 5.1
First, we consider an equilibrium-optimization model which was investigated by Yen et al. [
37]. This model can be regarded as an extension of a Nash–Cournot oligopolistic equilibrium model in electricity markets. The latter model has been investigated in some research papers (see, for example, [
10,
27]).
In this equilibrium model, it is assumed that there are
n companies. Let
x denote the vector whose entry
\(x_{i}\) stands for the power generated by company
i. Following Contreras et al. [
10], we suppose that the price
\(p_{i}(s)\) is a decreasing affine function of
s with
\(s = \sum_{i=1}^{n} x_{i}\), that is,
$$ p_{i}(s) = \alpha-\beta_{i} s. $$
Then the profit made by company
i is given by
$$ f_{i}(x) = p_{i}(s)x_{i}-c_{i}(x_{i}), $$
where
\(c_{i}(x_{i})\) is the cost for generating
\(x_{i}\) by the company
i.
Suppose that \(C_{i}\) is the strategy set of company i, that is, condition \(x_{i}\in C_{i}\) must be satisfied for each i. Then the strategy set of the model is \(C:=C_{1}\times C_{2}\times\cdots \times C_{n}\).
Actually, each company seeks to maximize its profit by choosing the corresponding production level under the presumption that the production of the other companies are parametric input. A commonly used approach to this model is based upon the famous Nash equilibrium concept.
Now, we recall that a point
\(x^{*}\in C = C_{1}\times C_{2}\times \cdots\times C_{n}\) is an
equilibrium point of the model if
$$ f_{i}\bigl(x^{*}\bigr)\geq f_{i} \bigl(x^{*}[x_{i}]\bigr) $$
for all
\(x_{i}\in C_{i}\) and
\(i=1,2,\dots,n\), where
\(x^{*}[x_{i}]\) stands for the vector obtained from
\(x^{*}\) by replacing
\(x_{i}^{*}\) with
\(x_{i}\). By taking
$$ f(x,y):=\varPsi(x,y)-\varPsi(x,x) $$
with
$$ \varPsi(x,y):=-\sum_{i=1}^{n}f_{i} \bigl(x[y_{i}]\bigr), $$
(22)
the problem of finding a Nash equilibrium point of the model can be formulated as follows:
$$ \text{Find } x^{*}\in C \text{ such that } f\bigl(x^{*},x\bigr)\geq0 \text{ for all } x\in C. $$
(EP)
In [
37], Yen et al. extended this equilibrium model by additionally assuming that the companies use some materials to produce electricity.
Let \(a_{l,i}\) denote the quantity of material l
\((l = 1,\dots, m)\) for producing one unit of electricity by company i
\((i=1, \dots,n)\). Let A be the matrix whose entries are \(a_{l,i}\). Then entry l of the vector Ax is the quantity of material l for producing x. Using materials for production may cause environmental pollution, for which companies have to pay a fee. Suppose that \(g(Ax)\) is the total environmental fee for producing x.
The task now is to find a production
\(x^{*}\) such that it is a Nash equilibrium point with a minimum environmental fee, while the quantity of the materials satisfies constraint
Q. This problem can be formulated as the
split feasibility problem of the following form:
$$\begin{aligned} &\text{Find } x^{*}\in C \text{ such that } f\bigl(x^{*},x\bigr)\geq0 \text{ for all } x\in C \\ &\quad\text{and } g \bigl(Ax^{*}\bigr)\leq g(Ax) \text{ for all } Ax\in Q. \end{aligned}$$
(SEP)
Suppose that, for every i, cost \(c_{i}\) for production and environmental fee g are increasing convex functions. The convexity assumption here means that both the cost and fee for producing a unit of product increase as the quantity of the product gets larger.
Under this convexity assumption, it is not hard to see (see also Quoc et al. [
27]) that problem (
EP) with
f given by (
22) can be formulated as follows:
$$ \text{Find } x^{*}\in C \text{ such that } f\bigl(x^{*},x\bigr):=\bigl\langle \tilde{B}_{1}x^{*}-\bar{a}, x-x^{*}\bigr\rangle + \varphi(x)-\varphi\bigl(x^{*}\bigr)\geq0 \text{ for all } x\in C, $$
(23)
where
(24)
Note that, when
\(c_{i}\) is differentiable and convex for each
i, problem (
23) is equivalent to the following
variational inequality problem:
$$ \text{Find } x^{*}\in C \text{ such that } \bigl\langle \tilde{B}_{1}x^{*}-\bar{a}+ \nabla\varphi\bigl(x^{*}\bigr), x-x^{*} \bigr\rangle \geq0 \text{ for all } x\in C. $$
We tested the proposed algorithm with the cost function given by
$$ c_{i}(x_{i})=\frac{1}{2}p_{i}x_{i}^{2}+q_{i}x_{i}, $$
(25)
where
\(p_{i}\geq0\). In [
37], Yen et al. showed that function
\(f(x,y)\) defined by (
23), (
24) and (
25) satisfies Assumptions (A1), (A2) and (A4).
In [
37], the author denoted by
\(g(z)\) the total environmental fee. It is unreasonable. Firstly, the total environmental fee should be included in the cost, that is, it is a part of
\(c_{i}(x_{i})\). Secondly, it is supposed that the companies behave as players in an oligopolistic market, but at the same time they are subordinated to the centralized planning decision in order to minimize the total environmental fee for the whole system. That is, the model is not concordant with the real system behavior.
It may be reasonable to denote by \(g(z)\) the restriction for the emission of contaminants. To protect the environment, governments generally adopt policies to restrict emissions of contaminants.
Assume that the production of electricity brings p contaminants and governments require that the quantity of contaminants brought by the production of one unit of electricity is in a given region. We use a set \(K\subset\mathbb{R}^{p}\) to denote this region.
Let \(b_{k,l}\) denote the quantity of the contaminant k
\((k = 1, \dots, p)\) for consuming one unit of material l
\((l = 1,\dots, m)\). Let B be the matrix whose entries are \(b_{k,l}\). Then entry k of vector Bz is the quantity of contaminant k for consuming one unit of material \(z_{l}\)
\((l = 1,\dots, m)\). So, the quantity of contaminant k
\((k = 1,\dots, p)\) for producing one unit of electricity is entry k of \(BAx\), and \(BAx\) should be in the set K, i.e., \(BAx\in K\). We get \(Bz\in K\) when letting \(z=Ax\).
Therefore we define function
\(g(z)\) as follows:
$$ g(z)=\frac{1}{2} \bigl\Vert Bz-P_{K}(Bz) \bigr\Vert ^{2}, $$
(26)
which is differentiable and
\(\nabla g(z)=B^{T}(I-P_{K})(Bz)\) (see, e.g., [
6]).
Take the sequences
\(\{\beta_{k}\}\),
\(\{\epsilon_{k}\}\),
\(\{\delta_{k} \}\) of the parameters as follows:
$$ \beta_{k}=\frac{4}{k+1}, \qquad \epsilon_{k}=0,\qquad \delta_{k}=3, \qquad \gamma_{k}=\max\bigl\{ 3, \Vert {{\eta_{k}}} \Vert \bigr\} $$
for each
\(k\geq1\) and take
\(\nu=\frac{1.99}{\|B\|}\). The entries of matrix
A were randomly generated in the interval [0, 5]. In the bifunction
\(f(x,y)\) defined by (
23), (
24) and (
25), the parameters
\(\alpha=0.5\) and
\(b_{i}\),
\(p_{i}\) and
\(q_{i}\) for each
\(i = 1,\dots, n\) were generated randomly in the interval
\((0, 1]\),
\([1, 3]\), and
\([1, 3]\), respectively. In the function
\(g(z)\), we take
\(B\in\mathbb{R}^{p}\times\mathbb{R}^{m}\), and its elements are generated randomly in
\((0,1)\).
Since function
\(g(z)\) is differentiable, we use Algorithm
4.1 to solve Problem
4.1 and compare it with Algorithms
1.1 and
3.1. In Algorithms
1.1 and
3.1 we substitute
\(\operatorname{prox}_{\lambda g}\) with
\(I-\nu\nabla g\) and do not consider the constraint set
Q.
According to the condition of the convergence of Halpern-type algorithm, we assume that \(\lim_{k\rightarrow\infty} \tau_{k}=0\) and \(\sum_{k=1}^{\infty}\tau_{k}=\infty\).