Nash equilibrium points and their finding for nonsmooth case

TL;DR

This paper introduces a numerical method for locating Nash equilibrium points in nonsmooth convex models across various fields, utilizing Steklov averages to handle nonsmoothness and proving convergence of the method.

Contribution

The paper develops a novel numerical approach for finding Nash equilibria in nonsmooth convex models, extending existing methods to more complex nonsmooth cases.

Findings

The method converges to equilibrium points under certain conditions.

Limit points of the method are proven to be equilibrium points.

Convergence rate estimates are provided using the Kantorovich theorem.

Abstract

The purpose of this paper is to develop a numerical method for finding an equilibrium point in a model, in which the loss function of each object (subject) is described by a convex function with respect to one of its variables. Such models are found in medicine, economics, game theory, and biology. For the more complex case, with nonsmooth functions describing the state of each element of the system as damage, loss, or gain, the Steklov average integrals are used that turn nonsmooth functions into smooth ones. Numerical methods for finding equilibrium points in the more general non-smooth case are constructed. In the process of optimization, the diameters of the sets, over which the averaging takes place, are decreased in accordance with the optimization steps. All limit points are proved to be equilibrium points. Under some conditions, the convergence rate can be estimated using the Kantorovich theorem. The necessity to develop new methods for finding Nash equilibrium points in the nonsmooth case is concluded.

Equations

$$f_j(x_1^{\ast },x_2^{\ast },\dots ,x_j^{\ast },\dots ,x_m^{\ast })\le f_j(x_1^{\ast },x_2^{\ast },\dots ,x_{j-1}^{\ast },x_j,x_{j+1}^{\ast },\dots ,x_m^{\ast })$$

$$\Theta (x)=\left(\begin{array}{c}{f}_{1,x_1}^{\prime }(x)\\ f_{2,x_2}^{\prime }(x)\\ \dots \\ f_{m,x_m}^{\prime }(x)\end{array}\right)$$

$$\Theta (x+\Delta x)=\Theta (x)+{\Theta }^{\prime }(x)\Delta x+o(\Delta x),$$

$$\underset{\Delta x\to 0}{\lim }\frac{o(\Delta x)}{\Vert \Delta x\Vert }=0,$$

$$\Theta (x+\Delta x)\approx \Theta (x)+{\Theta }^{\prime }(x)\Delta x=0.$$

$$\Delta x=-({\Theta }^{\prime }(x){)}^{-1}\Theta (x),$$

$${\Theta }^{\prime }(x)=\left(\begin{array}{cccc}{f}_{1,x_1,x_1}^{\prime \prime }& f_{1,x_1,x_2}^{\prime \prime }& \cdots & f_{1,x_1,x_m}^{\prime \prime }\\ f_{2,x_2,x_1}^{\prime \prime }& f_{2,x_2,x_2}^{\prime \prime }& \cdots & f_{2,x_2,x_m}^{\prime \prime }\\ ⋮& ⋮& \ddots & ⋮\\ f_{m,x_m,x_1}^{\prime \prime }& f_{m,x_m,x_2}^{\prime \prime }& \cdots & f_{m,x_m,x_m}^{\prime \prime }\end{array}\right)$$

$$\Delta x_k=-({\Theta }^{\prime }(x_k){)}^{-1}\Theta (x_k).$$

$$L_1\Vert \Delta x\Vert \le \Vert {\Theta }^{\prime }(x)\Delta x\Vert \le L_2\Vert \Delta x\Vert$$

$$\Vert \Theta (x_{k+1})\Vert =\Vert \Theta (x_k+\Delta x_k)-\Theta (x^{\ast })\Vert =\Vert {\Theta }^{\prime }(\xi )\Delta x_k\Vert ,$$

$$\Vert {\Theta }^{\prime }(\xi )\Delta x_k\Vert =\Vert {\Theta }^{\prime }(\xi )({\Theta }^{\prime }(x_k){)}^{-1}\Theta (x_k)\Vert .$$

$$L_1\Vert \Delta x_k\Vert \le \Vert {\Theta }^{\prime }(\xi )\Delta x_k\Vert =\Vert {\Theta }^{\prime }(\xi )({\Theta }^{\prime }(x_k){)}^{-1}\Theta (x_k)\Vert \le c_k\frac{\Vert \Delta x_{k-1}\Vert }{N_{k-1}}.$$

$$\Vert \Theta (x_k)\Vert =\Vert o(\Delta x_{k-1})\Vert \le \frac{\Vert \Delta x_{k-1}\Vert }{N_{k-1}}$$

$$\Vert o(\Delta x)\Vert \le \epsilon (\Delta x)\Vert \Delta x\Vert ,$$

$$N_k=N(\Delta x_k)=\frac{1}{\epsilon (\Delta x_k)}.$$

$$\Vert \Delta x_k\Vert \le c_k\frac{\Vert \Delta x_{k-1}\Vert }{L_1{N}_{k-1}}=q_k\Vert \Delta x_{k-1}\Vert .$$

$$q_k=\frac{c_k}{L_1{N}_{k-1}}{\to }_{k}0.$$

$$\Vert f_i(u)-f_i(v)\Vert \le L_i\Vert u-v\Vert$$

$${\phi }_i(x)=\frac{1}{\mu (D)}{\int }_D{f}_i(z+x)dz.$$

$${\phi }_{i,x_i}^{\prime }(x)=\frac{\partial {\phi }_i(x)}{\partial x_i}=\frac{1}{\mu (D)}{\int }_D\frac{\partial f_i(x+y)}{\partial x_i}dy.$$

$${\Phi }_i(x)=\frac{1}{\mu (D)}{\int }_D{\phi }_i(x+y)dy,$$

$${\Phi }_i^{\prime }(x)=\frac{1}{\mu (D)}{\int }_D{\phi }_i^{\prime }(z+x)dz.$$

$$L_i^{\prime }=\frac{2L_i}{d^2(D)}.$$

$$\frac{{\lambda }_k}{d^2(D_k)}<{\epsilon }_k,$$

$$\Theta (x)=\left(\begin{array}{c}{\Phi }_{1,x_1}^{\prime }(x)\\ {\Phi }_{2,x_2}^{\prime }(x)\\ \dots \\ {\Phi }_{m,x_m}^{\prime }(x),\end{array}\right)$$

$${\Theta }^{\prime }(x)=\left(\begin{array}{cccc}{\Phi }_{1,x_1,x_1}^{\prime \prime }& {\Phi }_{1,x_1,x_2}^{\prime \prime }& \cdots & {\Phi }_{1,x_1,x_m}^{\prime \prime }\\ {\Phi }_{2,x_2,x_1}^{\prime \prime }& {\Phi }_{2,x_2,x_2}^{\prime \prime }& \cdots & {\Phi }_{2,x_2,x_m}^{\prime \prime }\\ ⋮& ⋮& \ddots & ⋮\\ {\Phi }_{m,x_m,x_1}^{\prime \prime }& {\Phi }_{m,x_m,x_2}^{\prime \prime }& \cdots & {\Phi }_{m,x_m,x_m}^{\prime \prime }\end{array}\right)$$

$${\phi }_{i,s}(x)=\frac{1}{\mu (D_s)}{\int }_{D_s}{f}_i(x+y)dy$$

$${\Phi }_{i,s}(x)=\frac{1}{\mu (D_s)}{\int }_{D_s}{\phi }_{i,s}(x+y)dy.$$

$${\tilde{\Theta }}_s(y,x)={\Theta }_s(y)+2L_s(y-x).$$

$$L_s\Vert z{\Vert }^2\le ({\tilde{\Theta }}_s^{\prime }(x,x)z,z)\le 3L_s\Vert z{\Vert }^2\forall z\in {\mathbb{R}}^n.$$

Full text

For the more complex case, with nonsmooth functions describing the state of each element of the system as damage, loss, or gain, the Steklov average integrals are used that turn nonsmooth functions into smooth ones.

Numerical methods for finding equilibrium points in the more general non-smooth case are constructed. In the process of optimization, the diameters of the sets, over which the averaging takes place, are decreased in accordance with the optimization steps.

All limit points are proved to be equilibrium points. Under some conditions, the convergence rate can be estimated using the Kantorovich theorem. The necessity to develop new methods for finding Nash equilibrium points in the nonsmooth case is concluded.

††journal: COMMUNICATIONS IN OPTIMIZATION THEORY

1 Introduction

Let the physical or economic state of a system be described by $m$ loss functions $f_{1}(x_{1},x_{2},\dots,x_{m}):\mathbb{R}^{m}\rightarrow\mathbb{R}$, $f_{2}(x_{1},x_{2},\dots,x_{m}):\mathbb{R}^{m}\rightarrow\mathbb{R}$, $\dots$, $f_{m}(x_{1},x_{2},\dots,x_{m}):\mathbb{R}^{m}\rightarrow\mathbb{R}$ depending on $m$ variables $x_{1},x_{2},\dots,x_{m},$ where $\mathbb{R}^{m}$ is $m$-dimensional Euclidean space. Then an equilibrium point is a state $x^{*}_{1},x^{*}_{2},\dots,x^{*}_{m}$ for which changing any $x^{*}_{j}$ leads to an increase in the corresponding function $f_{j}(\cdot)$, i.e.

[TABLE]

Equilibrium states were introduced into economics by J. Nash. In 1950-1953, his articles proving the existence of equilibrium points were published Nash1 -Nash4 .

The problem of equilibrium point finding in biology or economics is closely related to game theory and is of practical importance. Equilibrium points arise from interspecific competition in biology and intercompany competition in economics. Equilibrium points in medicine are homeostasis points gomeostasis . These are points of balance between various states of the human body, e.g. blood pressure, temperature, blood cholesterol level, pulse rate. Some balance is achieved between different pills when we take medication.

Here we consider non-cooperative games of $m$ players, none of whom can influence other players’ behavior (strategies). A player $i$ chooses independently a pure strategy $x_{i}$ from a compact convex set $S_{i}$, such that he minimizes his loss function $f_{i}(\cdot)$.

Consider a vector $x=(x_{1},x_{2},\dots,x_{m})\in\mathbb{R}^{m},$ called a multistrategy and comprised of the pure strategies $x_{i}\in S_{i}$. We assume that the vector $x$ belongs to the compact convex set $S=S_{1}\times S_{2}\times\dots S_{m}\in\mathbb{R}^{m},$ which is the Cartesian product of the compact sets $S_{i},i\in 1:m$, and $\mbox{int }S\neq\emptyset$.

Definition 1.1

A multistrategy $x^{*}=(x^{*}_{1},x^{*}_{2},\dots,x^{*}_{m})\in S$ of a non-cooperative game is called a non-cooperative equilibrium if the inequality (1) is true for every $j\in 1:m$ and $x_{j}\in S_{j}$.

The equilibrium point definitions in game theory, medicine, and economics are similar. In 1950, J. Nash proved the following theorem.

Theorem 1.1

Nash1 . Let $S_{i}$ be a compact convex set for any $i\in 1:m$ and $f_{i}(\cdot)$ be convex with respect to $x_{i}\in S_{i}$. Then there is a non-cooperative equilibrium in a non-cooperative game with $m$ players.

The aim of the paper is to develop numerical methods for finding equilibrium points in the nonsmooth case. While, to our knowledge, papers describing numerical methods for finding equilibrium points in special cases minarchenko have been published, no papers describing numerical methods for finding equilibrium points in the general case are found.

2 Discussion of the problem

We describe a method of searching for an equilibrium state, provided that the functions $f_{i}(x_{-i},x_{i}):S\rightarrow R$, where $x_{-i}=(x_{1},x_{2},\dots,x_{i-1},x_{i+1},\dots,x_{m})$, are convex with respect to $x_{i}$. We assume that the inclusion $\{x\in\mathbb{R}^{m}\mid f_{i}(x)<f_{i}(x_{0})\}\subset\mbox{int }S$ is true for any $i\in 1:m$, in which $x_{0}$ is a starting point. Here $\mbox{int}S$ means the interior of the set $S$.

Let us denote the coordinate vectors by $e_{1}=(1,0,0,\dots,0),e_{2}=(0,1,0,\dots,0),\\ \dots,e_{m}=(0,0,0,\dots,1)$. It is known that coordinate descent method admits no convergence for nonsmooth functions demminmax . Therefore, we will use the ideas from proudintegapp1 .

Equilibrium points finding algorithms for the smooth case

One way is to use gradient and second-order methods for the smooth case, i.e., when $f_{i}(\cdot),i\in 1:m,$ are differentiable functions with respect to the variables $x_{j},j\in 1:m.$ Denote the partial derivative of the function $f_{i}(\cdot)$ with respect to the variable $x_{i}$ by $f^{\prime}_{i,x_{i}}(\cdot)$.

Consider the vector function $\Theta(\cdot):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}:$

[TABLE]

We search a vector $x^{*}=(x^{*}_{1},x^{*}_{2},\dots,x^{*}_{m}),$ for which $\Theta(x^{*})=0.$ It is clear that the vector $x^{*}$ is an equilibrium point.

From expansion accurate to the higher order terms

[TABLE]

in which

[TABLE]

we obtain the value for the step $\Delta x$. Suppose

[TABLE]

From here we obtain

[TABLE]

where

[TABLE]

Algorithm 1 (The Newton’s method for twice continuously differentiable functions $fi(\cdot)$)

At each step $k$ we find

[TABLE]

We set $x_{k+1}=x_{k}+2^{-l}\,\Delta x_{k},$ in which $l$ is the smallest number from the set $M=\{0,1,2,\cdots\}$ for which the inequality $\|\Theta(x_{k+1})\|<\|\Theta(x_{k})\|$ is correct. It is easy to prove that there exists a number $l$ for which the inequality $\|\Theta(x_{k+1})\|<\|\Theta(x_{k})\|$ is correct. Repeat the process as long as $\|\Theta(x_{k})\|\leq\varepsilon,$ in which $\varepsilon$ is a positive small number.

Let the inequality

[TABLE]

hold true for some $L_{1},L_{2}>0$ and any $\Delta x$. We assume that $x$ belongs to a small neighborhood of an equilibrium point $x^{*}$, in which $\Theta(x^{*})=0$ and the optimization process takes place with full step $\Delta x_{k}$ i.e. $l=0$. Then it is possible to obtain an estimation of the convergence rate of the Newton’s method. We have

[TABLE]

in which $\xi$ is a point on the line, connecting $x_{k+1}$ and $x^{*}$, and between them. Let’s substitute the expression for $\Delta x_{k}$ from (4). Therefore, we obtain

[TABLE]

Due to the continuity of the matrix $\Theta^{\prime}(\cdot)$ and the fact that at each step $k$ according to the choice of the step $\Theta(x_{k})=o(\Delta x_{k-1})$, and also considering the assumption (5) we obtain a chain of inequalities

[TABLE]

Here $c_{k}=\|\Theta^{\prime}(\xi)(\Theta^{\prime}(x_{k}))^{-1}\|$. We have

[TABLE]

in which $N_{k-1}=N(\Delta x_{k-1})\rightarrow_{k}\infty$ as $\|\Delta x_{k-1}\|\rightarrow 0$. The latter follows from the definition of the infinitesimal function $o(\cdot)$. We have

[TABLE]

in which $\varepsilon(\Delta x)\rightarrow 0$ as $\Delta x\rightarrow 0.$ Therefore, we can put

[TABLE]

From the inequalities written we get

[TABLE]

Superlinear convergence follows from here, since

[TABLE]

We finally obtain

Theorem 2.1

Let the assumption (5) hold true for the twice continuously differentiable functions $f_{i}(\cdot),i\in 1:m.$ Then the Newton’s method will converge with superlinear velocity in a small neighborhood of $x^{*}$.

This optimization process requires the existence of continuous second mixed derivatives with respect to the variables $x_{i},x_{j},i,j\in 1:m,$ of the functions $f_{i}(\cdot),i\in 1:m,$ and the existence of the inverse matrix $(\Theta^{\prime}(x_{k}))^{-1}$ at any step $k$. Unfortunately, the assumption (5) does not always hold. Moreover, the theorem 2.1 is true in a small neighborhood of the point $x^{*}$ which is to be reached.

3 Solution of the problem for the nonsmooth case

Let us use the ideas of the paper proudintegapp1 . We assume, that $f_{i}(\cdot),i\in 1:m,$ are Lipschitz functions with constants $L_{i}$ i.e.

[TABLE]

for all $u,v\in\mathbb{R}^{m}$. We construct functions

[TABLE]

in which $D$ is an arbitrary convex compact set, $0\in\mbox{int}D,\mu(D)>0$ is the Lebesgue measure of the set $D$ and the integral is the Lebesgue integral. It is not difficult to verify that the function $\varphi_{i}(\cdot)$ is convex with respect to the variable $x_{i}.$

The function $f_{i}(\cdot)$ has the partial derivative with respect to $x_{i}$ almost everywhere (a.e.) on the set $S$. In proudintegapp1 it was proven that the function $\varphi_{i}(\cdot)$ is continuously differentiable with respect to the variable $x_{i}$. The partial derivative $\varphi_{i}(\cdot)$ with respect to $x_{i}$ can be calculated by the formula proudintegapp1

[TABLE]

The functions $\varphi_{i}(\cdot),i\in 1:m,$ have an equilibrium point according to Nash’s theorem.

Substitute $f_{i}(\cdot),i\in 1:m,$ for $\Phi_{i}(\cdot):\mathbb{R}^{m}\rightarrow\mathbb{R},i\in 1:m,$ defined by

[TABLE]

in which the functions $\varphi_{i}(\cdot),i\in 1:m,$ and the set $D$ are defined above (see (6)). We take the integral of the integral, since in this way we obtain the twice continuously differentiable functions $\Phi_{i}(\cdot),i\in 1:m,$ and the stationary points of $\Phi_{i}(\cdot)$ are $\varepsilon(D)-$ stationary points of $f(\cdot)$ proudintegapp1 .

Since $\varphi_{i}(\cdot)$ is a Lipschitz proudintegapp1 , we will have

[TABLE]

.

We have proved that the functions $\Phi_{i}(\cdot),i\in 1:m,$ have Lipschitz second derivatives proudintegapp1 . If $D$ is a ball or a cube centered at zero with the diameter $d(D)$, then the functions $\Phi_{i}(\cdot),i\in 1:m,$ have Lipschitz second derivatives $\Phi^{\prime\prime}_{i}(\cdot)$ with constant proudintegapp1

[TABLE]

We can apply the Newton’s method to the functions $\Phi_{i}(\cdot),i\in 1:m,$ to find the equilibrium points. In the process of optimization we will consistently decrease the step $\lambda_{k}$ and the diameter $d(D_{k})$ so that the inequality

[TABLE]

was true for some sequence $\{\varepsilon_{k}\}$ in which $\varepsilon_{k}\rightarrow+0$ as $k\rightarrow\infty.$ We will prove that the inequality (8) guarantees that any limit point of a sequence obtained by the Newton’s method using the functions $\Phi^{\prime}_{i}(\cdot),\Phi^{\prime\prime}_{i}(\cdot),i\in 1:m,$ is an equilibrium point of the functions $f_{i}(\cdot),i\in 1:m$.

**The Newton’s method for finding equilibrium points for $f_{i}(\cdot)$, $i\in 1:m,$ using the functions $\Phi_{i}(\cdot)$ **

Calculate $\Theta(z_{k})$ and $\Theta^{\prime}(z_{k})$

[TABLE]

[TABLE]

accordingly to (7) for the twice differentiable functions $\Phi_{i}(\cdot),i\in 1:m,$ when $D$ is a ball or a cube.

Take a sequence of sets $\{D_{s}\},s=1,2,\dots$ with non-empty interior, the diameters $d(D_{s})$ of which tend to zero in $s\rightarrow\infty$. Let $D_{s}=B^{m}_{r_{s}}(0)=\{v\in\mathbb{R}^{n}\mid\|v\|\|\leq r_{s}\}$ for $r_{s}\rightarrow+0$ and $s\rightarrow\infty$. Let us introduce for $i\in 1:m$ the following sequence of functions

[TABLE]

and

[TABLE]

The difference between (7) and (9) is that (7) is written for a constant $D$, while (9) is written for a set $D_{s}$ depending on the parameter $s$.

Construct the functions $\Theta_{s}(\cdot)$ for the functions $\Phi_{i,s}(\cdot),i\in 1:m,$ as written above. Let the inequality $\|\Phi^{\prime\prime}_{i,s}(\cdot)\|\leq{L}_{s}$ hold true in which $\Phi^{\prime\prime}_{i,s}(\cdot)$ is the matrix of the second mixed derivatives. In proudintegapp1 it was proved that $L_{s}=\frac{L}{d(D_{s})},$ in which $L=\mbox{max}\,_{i\in 1:m}L_{i}$.

It follows from here that, depending on the selected metric of the space $\mathbb{R}^{m}$, the norm $\|\Theta^{\prime}_{s}(\cdot)\|$ is proportional to $L_{s}$. Suppose $\|\Theta^{\prime}_{s}(\cdot)\|\leq L_{s}.$

Define the vector-function $\tilde{\Theta}_{s}(\cdot):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ as a function of $y$:

[TABLE]

Then we have the inequality for the matrix $\tilde{\Theta}^{\prime}_{s}(\cdot)$

[TABLE]

Let us construct the Newton’s method for finding the roots of the equation $\Theta_{s}(x)=0$ using the function $\tilde{\Theta}_{s}(\cdot)$. We will use the rule of consistent reduction of the length $\lambda_{k}$ of $k^{th}$ step and the diameter $d(D_{k})$.

Description of the Newton’s method for finding for the equilibrium points using $\Phi_{i,s}(\cdot)$.

Let a point $x_{k}$ were constructed at the step $k$. Construct the point $x_{k+1}$. Take by definition $\tilde{\Theta}_{s,k}(\cdot)=\tilde{\Theta}_{s}(\cdot,x_{k})$. The dependence of $s$ on $k$ will be written as $s=s(k)$.

We calculate $\Delta x_{k}=-(\tilde{\Theta}_{s,k}^{\prime}(x_{k}))^{-1}\tilde{\Theta}_{s,k}(x_{k})$ at each step $k$. We set $x_{k+1}=x_{k}+2^{-l}\,\Delta x_{k},$ in which $l$ is the smallest number from the set $M=\{0,1,2,\cdots\}$ for which $\|\tilde{\Theta}_{s,k}(x_{k+1})\|<\|\tilde{\Theta}_{s,k}(x_{k})\|$.

It is possible to prove that $\|\tilde{\Theta}_{s,k}(x_{k+1})\|<\|\tilde{\Theta}_{s,k}(x_{k})\|$ for small $\|\Delta x_{k}\|$ and $\|\Delta x_{k}\|\rightarrow 0$ as $k\rightarrow\infty$ for fixed $s$ in a small neighborhood of the equilibrium point $x^{*}$. We assume that we reach a small surrounding of the equilibrium point $x^{*}$ for big $k$ in which the process takes place with the full step $\Delta x_{k}$.

If the inequality

[TABLE]

is fulfilled for a sequence $\{\varepsilon_{k}\},\varepsilon_{k}\rightarrow+0$ as $k\rightarrow\infty$, then we decrease the diameter $d(D_{s(k)})$ of $D_{s(k)}$ and increase $k,s=s(k)$.

The inequality (11) holds true for $\tilde{\Theta}^{\prime}_{s(k),k}(\cdot)$ and all $s,k$. Firstly, we prove that

[TABLE]

and the sequence $\{x_{k}\}$ has a limit point $x^{*}$.

We have the expansion of the function $\tilde{\Theta}_{s(k),k}(\cdot)$ in the neighborhood of the point $x_{k}$

[TABLE]

After substitution $\Delta x_{k}=-(\tilde{\Theta}_{s(k),k}^{\prime}(x_{k}))^{-1}\tilde{\Theta}_{s,k}(x_{k})$ in this expansion we obtain

[TABLE]

Let us prove that $o_{s(k),k}(\Delta x_{k})$ is an infinitesimal function with respect to $\Delta x_{k}$ as $k\rightarrow\infty$. Since $\tilde{\Theta}_{s(k),k}(\cdot)$ was obtained from ${\Theta}_{s}(\cdot)$ through adding the linear function, $o_{s(k),k}(\cdot)$ is the same infinitesimal function in the expansion of ${\Theta}_{s}(\cdot)$ in the surrounding of $x_{k}$. Now we will obtain the upper bound for $o_{s(k),k}(\cdot).$

The following expansion takes place

[TABLE]

Since the function ${\Theta}_{s}(\cdot)$ is continuously differentiable for each $s$, according to the midpoint theorem, we have

[TABLE]

in which $\xi\in[x_{k},x_{k+1}].$ Let us substitute this difference in the Taylor series and use the Lipschitzness of ${\Theta}^{\prime}_{s}(\cdot)$ with the constant $\frac{2L}{d^{2}(D_{s})}$. Therefore, since $\Theta^{\prime}_{s}(\cdot)$ is Lipschitz with the constant $\frac{2L}{d^{2}(D_{s})}$ proudintegapp1 , we obtain

[TABLE]

From here

[TABLE]

Hence, if

[TABLE]

holds true during optimization, then uniform infinitesimality of $o_{s(k),k}(\cdot)$ with respect to $s=s(k)$ and $k$ follows from here. However, we organize our process in such a way that the limit equality (16) was correct.

The limit equality

[TABLE]

[TABLE]

follows from the inequality (12) as we decrease $d(D_{s(k)})$ in the process of optimization. The equality (13) follows from (10), (14) and (17).

It follows from the upper semicontinuity of the Clarke subdifferential and from the equality (13) that the sequence $\{x_{k}\}$ converges to a limit point $x^{*}$, in which $0\in\partial_{x_{i}}f_{i}(x^{*})$ for all $i\in 1:m,$ i.e. $x^{*}$ is the equilibrium point.

All of the above stated is true if we reach a small neighborhood of the equilibrium point. In order to do this, we are to use the coordinate descent method with some modifications for the functions $\varphi_{i}(\cdot),i\in 1:m.$

Thus, the following theorem is proved.

Theorem 3.1

Any limit points of the sequence obtained by Newton’s method with starting points from small neighborhoods of the equilibrium points, are the equilibrium points if the equality (16) is satisfied in the process of optimization for the convex with respect to $x_{i}$, Lipschitzian functions $f_{i}(\cdot),i\in 1:m,$ respectively.

The given Newton’s method is also called the modified Newton’s method. It is possible to show that there is a majorant Kantorovich function for any step $k$ kantorovichakilov . The step length and the convergence rate of the optimization method are estimated under the conditions of consistency (16) and some conditions indicated in the theorem given below. We can state the convergence of the whole sequence $\{x_{k}\}$ under the below given conditions in the theorem 3.1.

This is true for the reason that $\|\Delta x_{k}\|$ is compared with the step length of the majorant function. The conditions of the Kantorovich theorem kantorovichakilov , pp. 689-690, are fulfilled if we satisfy some requirements.

We will construct a sequence $\{x_{k}\}$ converging to the solution of the equation $\Theta(x^{*})=0$ for a ball $B^{m}_{r}(x_{0})=\{y\in\mathbb{R}^{m}\mid\|y-x_{0}\|\leq r\}$. Suppose $Q_{s,0}=[\tilde{\Theta}^{\prime}_{s,0}(x_{0})]^{-1}$, $\|Q_{s,0}{\Theta}_{s}(x_{0})\|\leq A_{s}$ and $\|Q_{s,0}\tilde{\Theta}^{\prime\prime}_{s,0}(x)\|\leq B_{s}$ for any $x\in B^{m}_{r}(x_{0})$, $\triangle x_{k}=-[\tilde{\Theta}^{\prime}_{s,0}(x_{k})]^{-1}\tilde{\Theta}_{s,k}(x_{k})$. We set $x_{k+1}=x_{k}+\triangle x_{k}$. During the optimization process, we change $s=s(k)$ and decrease the diameter $d(D_{s})$ so that the requirements of Theorem 19 were satisfied.

Theorem 3.2

*We make the following assumptions:

There exists a linear operator $Q_{s,0}=[\tilde{\Theta}^{\prime}_{s,0}(x_{0})]^{-1}$ for $s=s(0)$ and $k=0$.

If*

[TABLE]

is true for any $s$ and the consistency condition

[TABLE]

is satisfied, then the equation ${\Theta}(x)=0$ has a solution $x^{*}$ to which the Newton’s method converges with the rate

[TABLE]

for a constant $C$.

The convergence rate of the modified Newton’s method (for $q<\frac{1}{2}$) is estimated by the following inequality

[TABLE]

Remark 3.1

The convergence rate proof follows with some changes from kantorovichakilov , p. 690, since the convergence rate depends on the values $q_{k}$ and $A_{k}$. The first value is limited by the value $\frac{1}{2}$. The second value tends to zero as $k\rightarrow\infty$.

Proof. It is easy to satisfy to the conditions of the theorem, since $A_{s(k)}\rightarrow_{k}0$ and we can decrease the diameters of the sets $D_{s}$ when $q_{s}=A_{s}B_{s}<\frac{1}{2}$ and the point $x_{k}$ can be considered as a new starting point.

At each step $k$ there is a majorant function $\psi_{s,k}(\cdot)$

[TABLE]

Since

[TABLE]

the step length $\triangle_{k}=\|x_{k+1}-x_{k}\|$ does not exceed the step length $t_{k+1}-t_{k}$ of the Newton’s method for the equation $\psi_{s}(t)=0$ the solution of which we denote by $t_{s}$ thus the following can be written:

[TABLE]

For the existence of the majorant equation $\psi_{s}(t)=0,t\in\mathbb{R},$ for the operator equation $\Theta(x)=0$, as it follows from the Taylor formula XVII.2.5 kantorovichakilov , it is sufficient that the following integral inequality is correct

[TABLE]

for big enough $k$, which is correct if

[TABLE]

Let us denote by

[TABLE]

[TABLE]

Let us note that

[TABLE]

According to the Taylor expansion for a second degree polynomial we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

However,

[TABLE]

[TABLE]

Therefore,

[TABLE]

By analogy, from (22) we obtain

[TABLE]

From here

[TABLE]

From (23) and (24), taking into account $q_{s,k}\leq\frac{1}{2}$, we obtain the following estimations

[TABLE]

Consequently, $q_{s,k}\leq\frac{1}{2}[2q_{s,0}]^{2^{k}}=\frac{1}{2}[2q_{s}]^{2^{k}}$

[TABLE]

in which $q_{s,0}=q_{s}$ and $A_{s,0}=A_{s}$.

From here and (20), (21) we obtain

[TABLE]

[TABLE]

since

[TABLE]

and $B_{s}$ is an upper bound for the norm of the second derivatives and can only increase as $s\rightarrow\infty$. Passing to the limit on $k=k(s)\rightarrow\infty$ in (26), we obtain the inequality (18).

We will use the modified Newton method $\Delta x_{k}=-(\tilde{\Theta}_{s,0}^{\prime}(x_{0}))^{-1}\tilde{\Theta}_{s,k}(x_{k})$ for solving the equality $\Theta_{s}(x)=0$. We denote the obtained sequence as $\{x^{\prime}_{k}\}$. Suppose

[TABLE]

where $c_{0}=-\frac{1}{\psi^{\prime}_{s}(t_{0})}=\frac{1}{2}$ and $t_{0}=0$. Let us replace the modified Newton method for the equation $\psi_{s}(t)=0$ with the equation

[TABLE]

and we will solve it by using the successive approximations method. Suppose

[TABLE]

We can write

[TABLE]

However,

[TABLE]

so that

[TABLE]

Therefore,

[TABLE]

We can obtain the similar inequality for $t^{*}_{s}-t^{\prime}_{k-1}$. Consequently,

[TABLE]

The inequality

[TABLE]

similar to the inequality (20), is correct for the modified Newton’s method. Using this inequality, we get

[TABLE]

Passing to the limit in $s$ and considering

[TABLE]

in which $C$ is a constant for all $s$, we obtain the inequality (19).

The theorem is proved.

4 Conclusion

We propose a method for finding equilibrium points as the limit points of a sequence obtained by applying the numerical method described above. The coordinate descent method slowly converges to an equilibrium point in the general case, but by changing the initial points, one can obtain all equilibrium points with minimal intermediate calculations.

A method for finding J. Nash equilibrium points using the matrices of second mixed derivatives (generalized matrices of second mixed derivatives) of the original functions is suggested. Such methods, under certain conditions, converge much faster than the coordinate descent method, but require more calculations at each step.

To speed up the convergence of the method, it is proposed to decrease consistently the diameter of the set $D_{m}$ on which the integration is performed, and the step length of the optimization process. We give the rules for successive decrease of the diameter of the set $D_{m}$ and the step length. The Kantorovich theorem is used to estimate the convergence rate.