About the Subgradient Method for Equilibrium Problems

Abstract

Convergence results of the subgradient algorithm for equilibrium problems were mainly obtained using a Lipschitz continuity assumption on the given bifunctions. In this paper, we first provide a complexity result for monotone equilibrium problems without assuming Lipschitz continuity. Moreover, we give a convergence result of the value of the averaged sequence of iterates beyond Lipschitz continuity. Next, we derive a rate convergence in terms of the distance to the solution set relying on a growth condition. Applications to convex minimization and min–max problems are also stated. These ideas and results deserve to be developed and further refined.

1. Introduction

The subgradient algorithm has recently enjoyed regained popularity, see for example [1,2,3,4]. However, the Lipschitz continuity assumption can be stringent even in a convex setting. It is well known that not only SVM (Support Vector Machine) but also Feed-Forward and Recurrent Neural Networks do not satisfy the Lipschitz continuity assumption, see [5]. Indeed, most $l_2$-regularized convex learning problems lack this property, while $l_2$ regularization and weight decay are ubiquitous in the learning field. The  proposed equilibrium approach will allow us not only to generalize some very recent results in convex minimization [1] and present them for min–max problems in a unified way but also to yield new insights into equilibrium problems. In order to go to the essential to share, we took the same paper outline as in [1] and we assume the reader has some basic knowledge of variational and convex analysis as can be found, for example, in [6,7].

The subgradient algorithm is a powerful tool for constructing algorithms to approximate solutions of optimization and equilibrium problems, the latter being the problem of finding $\overline{x}\in C$ such that

$$\left(\mathcal{P}\right)\mathrm{find}\overline{x}\in C,F(\overline{x},y)\ge 0\forall y\in C,$$

where C is a given nonempty closed convex subset of ${\mathit{ℝ}}^d$ and $F:C\times C\to \mathit{ℝ}$ is a bifunction. The solution set of $\left(\mathcal{P}\right)$ will be denoted by $\Gamma$. Such a problem is also known as the equilibrium problem in the sense of Blum and Oettli [8]. It is worth mentioning that variational inequalities, Nash equilibrium, the saddle-point problem, optimization, and many problems arising in applied nonlinear analysis are special cases of equilibrium problems. In this paper, we will be concerned with the convergence analysis of a subgradient method for solving problem $\left(\mathcal{P}\right)$ in which F is not assumed to be Lipschitz continuous. For more simplicity and clarity, we suppose that F verifies the usual conditions:

$\left(\mathrm{A}1\right)F(x,x)=0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in C$;

$\left(\mathrm{A}2\right)F\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}F(x,y)+F(y,x)\le 0$ for all $x,y\in C$;

$\left(\mathrm{A}3\right){lim\; sup}_{t\downarrow 0}F(tz+(1-t)x,y)\le F(x,y)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}x,y,z\in C$;

$\left(\mathrm{A}4\right)$ for each $x\in C,y\mapsto F(x,y)$ is convex and lower semicontinuous.

2. The Main Results

Let us state the subgradient method which generates a sequence ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}$ by Algorithm 1:

Algorithm 1 (SGM)

Step 0: Choose $x_0\in C$ and set $k=0$.

Step 1: Let $x_k\in C$ and compute ${\eta }_k\in {\partial }_{2}F(x_k,x_k)$, where ${\partial }_{2}F$ stands for the convex subdifferential of F with respect to the second variable.

If ${\eta }_k=0$, then stop.

Step 2: Update $x_{k+1}=P_C\left(x_k-{\alpha }_k{\eta }_k\right)$, where ${\alpha }_k=\frac{{\beta }_k}{\parallel {\eta }_k\parallel }$, for all $k\ge 0$ with ${\beta }_k>0$.

Now we are in a position to provide a complexity result.

Theorem 1.

Suppose F verifies conditions $A_1$ to $A_4$ and note by T the total number of iterations. Set ${\overline{x}}_T=\frac{{\sum }_{i=0}^T{x}_i}{T+1}$, ${\beta }_k=\frac{c}{\sqrt{T+1}}$, and $c>0$ and let $\overline{x}$ be a solution of $\left(\mathcal{P}\right)$. Then, $x_k\in \mathcal{B}(\overline{x},r)$ for $k=0,1,···,T$ with $r=\sqrt{\parallel x_0-\overline{x}{\parallel }^2+c^2}$. Moreover, we have

$$F(\overline{x},{\overline{x}}_T)\le \frac{r^2{L}_{\mathcal{B}}}{2r\sqrt{T+1}},$$

(1)

where $L_{\mathcal{B}}$ is the local Lipschitz continuity constant of $F(\overline{x},·)$ on the smallest open convex set $\mathcal{O}$ such that the closed ball $\mathcal{B}(\overline{x},r)\subset \mathcal{O}$.

Proof. 

For all $x\in C$, we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}{-x\parallel }^2& \le & \parallel x_k-x-{\alpha }_k{\eta }_k{\parallel }^2\hfill \\ & \le & \parallel x_k{-x\parallel }^2-2{\alpha }_k\langle {\eta }_k,x_k-x\rangle +{\beta }_k^2.\hfill \end{array}$$

On the other hand, ${\eta }_k\in {\partial }_{2}F(x_k,x_k)$ implies that $F(x_k,x)\ge \langle {\eta }_k,x-x_k\rangle$, which combined with the monotonicity of F ensures that $\langle {\eta }_k,x_k-x\rangle \ge F(x,x_k)$. Consequently,

$$\parallel x_{k+1}{-x\parallel }^2\le {\parallel x_k-x\parallel }^2-2\frac{{\beta }_k}{{\eta }_k}F(x,x_k)+{\beta }_k^2.$$

(2)

Setting $x=\overline{x}$, we have

$$\parallel x_{k+1}-\overline{x}{\parallel }^2\le {\parallel x_k-\overline{x}\parallel }^2+{\beta }_k^2,$$

(3)

because ${\beta }_k>0$ together with $F(\overline{x},x_k)\ge 0$. In other words, the sequence ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}$ is quasi-Fejér monotone to the solution set $\Gamma$.

From the latter inequality, we infer that

$$\parallel x_{k+1}-\overline{x}{\parallel }^2 \le \parallel x_0-\overline{x}{\parallel }^2+\sum _{i=0}^k{\beta }_i^2\le {\parallel x_0-\overline{x}\parallel }^2+c^2,\mathrm{for}0\le k\le T.$$

(4)

Relation (4) assures that $x_k\in \mathcal{B}(\overline{x},r)\mathrm{for}0\le k\le T$.

In light of the consequence of [6]—Proposition 9.13, we have that $F(\overline{x},·)$ is locally Lipschitz continuous with constant $L_{\mathcal{B}}$ on $\mathcal{O}$ which, for $0\le k\le T$, in turn gives that $\parallel {\eta }_k\parallel \le L_{\mathcal{B}}$. This combined with (2) leads to

$$\parallel x_{k+1}-\overline{x}{\parallel }^2\le {\parallel x_k-\overline{x}\parallel }^2-2\frac{{\beta }_k}{L_{\mathcal{B}}}F(\overline{x},x_k)+{\beta }_k^2.$$

(5)

From which we derive

$$\sum _{i=0}^T{\beta }_{i}F(\overline{x},x_k)\le \frac{L_{\mathcal{B}}}{2}\left(\parallel x_0-\overline{x}{\parallel }^2+\sum _{i=0}^T{\beta }_i^2\right),$$

or equivalently

$$\sum _{i=0}^T\frac{c}{\sqrt{T+1}}F(\overline{x},x_k)\le \frac{L_{\mathcal{B}}}{2}\left(\parallel x_0-\overline{x}{\parallel }^2+c^2\right).$$

Using the convexity of the function $F(\overline{x},·)$, we finally obtain

$$F(\overline{x},{\overline{x}}_T)=F\left(\overline{x},\sum _{i=0}^T\frac{1}{T+1}{x}_i\right)\le \sum _{i=0}^T\frac{1}{T+1}F(\overline{x},x_i)\le \frac{r^2{L}_{\mathcal{B}}}{2r\sqrt{T+1}}.$$

This completes the proof. □

A convergence result of the values of the averaged sequence of iterates generated by (SGM) is provided in the next Theorem.

Theorem 2.

Suppose that the bifunction F verifies conditions $A_1$ to $A_4$. Set ${\overline{x}}_k=\frac{{\sum }_{i=0}^k{\beta }_i{x}_i}{{\sum }_{i=0}^k{\beta }_i}$, ${\sigma }_k={\sum }_{i=0}^k{\beta }_i$, and ${\left({\beta }_k\right)}_{k\in \mathit{\mathbb{N}}}\in l^2/l^1$ and let $\overline{x}$ be a solution of $\left(\mathcal{P}\right)$. Then, the sequence ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}\subset \mathcal{B}(\overline{x},\overline{r})$ where $\overline{r}=\sqrt{\parallel x_0-\overline{x}{\parallel }^2+\overline{c}}$ and $\overline{c}>0$ is such that ${\sum }_{k=0}^{\infty }{\beta }_k^2\le \overline{c}$. Moreover,

$$\forall k\in \mathit{\mathbb{N}},F(\overline{x},{\overline{x}}_k)\le \frac{r^2{L}_{\mathcal{B}}}{2{\sigma }_k},$$

(6)

where $L_{\mathcal{B}}$ is the local Lipschitz continuity constant of $F(\overline{x},·)$ on the smallest open convex set $\mathcal{O}$ such that $\mathcal{B}(\overline{x},r)\subset \mathcal{O}$.

Proof. 

We again have that

$$\parallel x_{k+1}-\overline{x}{\parallel }^2\le {\parallel x_k-\overline{x}\parallel }^2-2\frac{{\beta }_k}{{\eta }_k}F(\overline{x},x_k)+{\beta }_k^2.$$

(7)

Therefore,

$$\parallel x_{k+1}-\overline{x}{\parallel }^2\le \parallel x_0-\overline{x}{\parallel }^2+\sum _{i=0}^k{\beta }_i^2\le {\parallel x_0-\overline{x}\parallel }^2+\overline{c}.$$

(8)

This ensures that ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}\subset \mathcal{B}(\overline{x},\overline{r})$. Following the same lines as in the proof of Theorem 1, we obtain

$$\parallel x_{k+1}-\overline{x}{\parallel }^2 \le {\parallel x_k-\overline{x}\parallel }^2-2\frac{{\beta }_k}{L_{\mathcal{B}}}F(\overline{x},x_k)+{\beta }_k^2,\mathrm{for}k\in \mathit{\mathbb{N}}.$$

(9)

This together with the convexity of the function $F(\overline{x},·)$ yields to

$$F(\overline{x},{\overline{x}}_k)=F\left(\overline{x},\sum _{i=0}^k\frac{{\beta }_i}{{\sigma }_k}{x}_i\right)\le \sum _{i=0}^k\frac{{\beta }_i}{{\sigma }_k}F(\overline{x},x_i)\le \frac{r^2{L}_{\mathcal{B}}}{2{\sigma }_k},$$

which leads to the announced result. □

Remark 1.

We can obtain more than the convergence of the averaged sequence of the iterates. Actually, we have

$$\underset{k\to +\infty }{lim\; inf}F(\overline{x},x_k)=0,$$

and the whole sequence ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}$ converges to a solution of $\left(\mathcal{P}\right)$.

Indeed, for all $k\ge 0$, the inequality

$$\parallel x_{k+1}-\overline{x}{\parallel }^2\le {\parallel x_k-x\parallel }^2-2\frac{{\beta }_k}{L_{\mathcal{B}}}F(\overline{x},x_k)+{\beta }_k^2,$$

leads to

$$\sum _{k=0}^{\infty }{\beta }_{k}F(\overline{x},x_k)<+\infty .$$

Since ${\displaystyle \sum _{k=0}^{\infty }{\beta }_k=+\infty }$, this implies that ${\displaystyle \underset{k\to +\infty }{lim\; inf}F(\overline{x},x_k)=0}$.

Now, with the sequence ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}$ being quasi-Fejér convergent to the set Γ, namely verifying (3) with ${\left({\beta }_k\right)}_{k\in \mathit{\mathbb{N}}}\in l^2$, this implies its boundedeness. Further, in view of (9), which is still valid for all $x\in C$ together with both lower semicontinuity and upper hemi-continuity assumptions, we obtain that any cluster point $x^{*}$ of ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}$ belongs to Γ. Consequently, the whole sequence converges to $x^{*}$, see for example [9], and we retrieve the main results in [10].

A convergence result based on a growth property.

Corollary 1.

Suppose in addition to hypotheses $A_1$ to $A_4$ that F verifies the following growth property:

$$there exists\gamma >0such thatF(\overline{x},x)\ge \gamma dis{t}^2(x,\Gamma )for allx\in Cand some\overline{x}\in \Gamma .$$

Consider the sequence ${\left({\beta }_k\right)}_{k\in \mathit{\mathbb{N}}}$ given by ${\beta }_k=\frac{L_{\mathcal{B}}}{\gamma (k+1)}$ for all $k\in \mathit{\mathbb{N}}$. Then, for all $k\in \mathit{\mathbb{N}}$, ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}\subset \mathcal{B}(\overline{x},\overline{r})$, and we have

$$dis{t}^2(x_{k+1},\Gamma )\le \frac{L_{\mathcal{B}}^2}{{\gamma }^2(k+1)},$$

(10)

where $L_{\mathcal{B}}$ is the local Lipschitz continuity constant of $F(\overline{x},·)$ on the smallest open convex set $\mathcal{O}$ such that $\mathcal{B}(\overline{x},r)\subset \mathcal{O}$.

Proof. 

The beginning of Theorem 2 ensures that ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}$ lies in $\mathcal{B}(\overline{x},\overline{r})$ and assures again that $\parallel {\eta }_k\parallel \le L_{\mathcal{B}}$. Remember that (7) reads as

$$\parallel x_{k+1}-\overline{x}{\parallel }^2\le {\parallel x_k-\overline{x}\parallel }^2-2\frac{{\beta }_k}{{\eta }_k}F(\overline{x},x_k)+{\beta }_k^2.$$

(11)

By taking $\overline{x}=pro{j}_{\Gamma }\left(x_k\right)$, we obtain

$$dis{t}^2(x_{k+1},\Gamma )\le {\parallel x_{k+1}-\overline{x}\parallel }^2\le dis{t}^2(x_k,\Gamma )-2\frac{{\beta }_k}{L_{\mathcal{B}}}F(\overline{x},x_k)+{\beta }_k^2.$$

This implies

$$dis{t}^2(x_{k+1},\Gamma )\le dis{t}^2(x_k,\Gamma )-\frac{2}{k+1}dis{t}^2(x_k,\Gamma )+\frac{L_{\mathcal{B}}^2}{{\gamma }^2{(k+1)}^2},$$

or equivalently

$${(k+1)}^{2}dis{t}^2(x_{k+1},\Gamma )\le \left({(k+1)}^2-2(k+1)\right)dis{t}^2(x_k,\Gamma )+\frac{L_{\mathcal{B}}^2}{{\gamma }^2}.$$

Following the same lines as in the proof of [1]—Corollary 1, for all $k\in \mathit{\mathbb{N}}$, we further have

$${(k+1)}^{2}dis{t}^2(x_{k+1},\Gamma )-k^{2}dis{t}^2(x_k,\Gamma )\le \frac{L_{\mathcal{B}}^2}{{\gamma }^2}.$$

Summing the last inequality for $i=1$ to k leads to

$${(k+1)}^{2}dis{t}^2(x_{k+1},\Gamma )=\sum _{i=0}^k\left({(i+1)}^{2}dis{t}^2(x_{i+1},\Gamma )-i^{2}dis{t}^2(x_i,\Gamma )\right)\le \frac{L_{\mathcal{B}}^2}{{\gamma }^2}(k+1).$$

From which we derive

$$dis{t}^2(x_{k+1},\Gamma )\le \frac{L_{\mathcal{B}}^2}{{\gamma }^2(k+1)}.$$

□

3. Applications

In the convex minimization case, (SGM) coincides with the classical subgradient method, and we recover the results obtained in the convex setting in [1]. Indeed, just take $F(x,y)=f\left(y\right)-f\left(x\right)$, f being a proper convex lower semicontinuous convex function, clearly ${\partial }_{2}F(x_k,x_k)=\partial f\left(x_k\right)$, and we retrieve

$$\left(SGM\right)\left\{\begin{array}{c}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}{\eta }_k\in \partial f\left(x_k\right)\hfill \\ \mathrm{U}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}{x}_{k+1}=P_C(x_k-{\alpha }_k{\eta }_k).\hfill \end{array}\right.$$

Theorem 1 reduces to

$$f\left({\overline{x}}_k\right)-f\left(\overline{x}\right)\le \frac{r^2{L}_{\mathcal{B}}}{2r\sqrt{T+1}},$$

(12)

where $\overline{x}$ is a minimizer of f and $L_{\mathcal{B}}$ is the local Lipschitz continuity constant of f on the smallest open convex set $\mathcal{O}$ such that $\mathcal{B}(\overline{x},r)\subset \mathcal{O}$. Theorem 2, in turn, leads to the fact that the sequence ${\left(x_k\right)}_{k\in \mathit{\mathbb{N}}}\subset \mathcal{B}(\overline{x},\overline{r})$, where $\overline{r}=\sqrt{\parallel x_0-\overline{x}{\parallel }^2+\overline{c}}$ and $\overline{c}>0$, is such that ${\sum }_{k=0}^{\infty }{\beta }_k^2\le \overline{c}$. Moreover, for all $k\in \mathit{\mathbb{N}}$, we obtain the following convergence result in terms of the suboptimality error:

$$f\left({\overline{x}}_k\right)-f\left(\overline{x}\right)\le \frac{r^2{L}_{\mathcal{B}}}{2{\sigma }_k},$$

(13)

${\overline{x}}_k,{\sigma }_k,\overline{r},\overline{c}$ were defined in Theorem 2, and ${\left({\beta }_k\right)}_{k\in \mathit{\mathbb{N}}}\in l^2/l^1$ and $\overline{x}$ are minimizers of f.

If we assume, in addition, for all $x\in C$ that

$$f\left(x\right)-f\left(\overline{x}\right)\ge \gamma dis{t}^2(x,\Gamma ),$$

(14)

then

$$dis{t}^2(x_{k+1},\Gamma )\le \frac{L_{\mathcal{B}}^2}{{\gamma }^2(k+1)}.$$

(15)

Likewise, by taking $F(v,w)=L(z,p)-L(x,y)$ with $w=(z,y)$, $v=(x,p)$ and L being a proper closed convex–concave function defined on $C\times C$, $C=Q\times P$ with $Q,P$ closed convex sets of ${\mathit{ℝ}}^n$ and ${\mathit{ℝ}}^m$, respectively, then clearly

$${\partial }_{w}F(v,w)=\left({\partial }_{1}L(·,p),-{\partial }_{2}L(x,·)\right),$$

we recover the subgradient algorithm for the saddle function considered in [11] and more recently in [12] and we extend some results in [1] to the convex–concave case. More precisely, (SGM) reduces to

$$\left(SGM\right)\left\{\begin{array}{c}compute({\mu }_k,{\gamma }_k)\in ({\partial }_{1}L(x_k,y_k),{\partial }_2(-L)(x_k,y_k))\hfill \\ Update(x_{k+1},y_{k+1})=P_C\left((x_k,y_k)-{\alpha }_k({\mu }_k,{\gamma }_k)\right).\hfill \end{array}\right.$$

Theorem 1 reads, in this case, as

$$L({\overline{x}}_T,\overline{y})-L(\overline{x},{\overline{y}}_T)\le \frac{r^2{L}_{\mathcal{B}}}{2r\sqrt{T+1}},$$

(16)

where $r=\sqrt{\parallel (x_0,y_0)-(\overline{x},\overline{y}){\parallel }^2+c^2}$ and $(\overline{x},\overline{y})$ are a saddle point of L, namely $(\overline{x},\overline{y})$ verifying $L(\overline{x},y)\le L(\overline{x},\overline{y})\le L(x,\overline{y})$ for all ($x,y)\in P\times Q$ and $L_{\mathcal{B}}$ being the local Lipschitz continuity constant of $L(·,\overline{x})-L(\overline{x},·)$ on the smallest open convex set $\mathcal{O}$ such that $\mathcal{B}(\overline{x},r)\subset \mathcal{O}$.

Theorem 2, in turn, leads to the sequence ${(x_k,y_k)}_{k\in \mathit{\mathbb{N}}}\subset \mathcal{B}\left((\overline{x},\overline{y}),\overline{r}\right)$, where $\overline{r}=\sqrt{\parallel (x_0,y_0)-(\overline{x},\overline{y}){\parallel }^2+\overline{c}}$ and $\overline{c}>0$ is such that ${\sum }_{k=0}^{\infty }{\beta }_k^2\le \overline{c}$. Moreover, for all $k\in \mathit{\mathbb{N}}$, we obtain the following convergence result in terms of a merit function:

$$L({\overline{x}}_k,\overline{y})-L(\overline{x},{\overline{y}}_k)\le \frac{r^2{L}_{\mathcal{B}}}{2{\sigma }_k},$$

(17)

$({\overline{x}}_k,{\overline{y}}_k),{\sigma }_k,\overline{r},\overline{c}$ were defined in Theorem 2 and ${\beta }_k\in l^2/l^1$ and $(\overline{x},\overline{y})$ are saddle points of L.

Now, if in addition, we assume for all $(x,y)\in C$ that

$$L(x,\overline{y})-L(\overline{x},y)\ge \gamma dis{t}^2((x,y),\Gamma ),$$

(18)

then

$$dis{t}^2\left((x_{k+1},y_{k+1}),\Gamma \right)\le \frac{L_{\mathcal{B}}^2}{{\gamma }^2(k+1)}.$$

(19)

4. Conclusions

To conclude, the equilibrium approach allowed us not only to generalize some very recent results obtained by X. Li, L. Zhao, D. Zhu, and A. M-Ch.; use them in convex minimization [3]; and present them for min–max problems in a unified way but also to yield new insights into equilibrium problems. More precisely, a complexity result for monotone equilibrium problems was provided without assuming Lipschitz continuity. Moreover, a convergence result of the value of the averaged sequence of iterates beyond Lipschitz continuity was also given. Finally, a rate convergence in terms of the distance to the solution set relying on a growth condition was derived. The proposed results were declined in the convex minimization context as well as in the convex–concave min–max setting.