1 Introduction
2 Preliminaries
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monotone on C if$$ f(x,y)+f(y,x)\leq 0, \quad \forall x, y\in C; $$
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pseudomonotone on C if$$ f(x,y) \geq 0\quad \Longrightarrow\quad f(y,x)\leq 0,\quad \forall x, y\in C; $$
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Lipschitz-type continuous on C if there exist two positive constants \(c_{1}\) and \(c_{2}\) such that$$ f(x,y)+ f(y,z)\geq f(x,z)-c_{1} \Vert x-y \Vert ^{2} -c_{2} \Vert y-z \Vert ^{2},\quad \forall x, y,z\in C. $$
3 Main results
4 Application to variational inequality problems
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monotone on C if$$ \langle Ax-Ay,x-y\rangle \geq 0,\quad \forall x, y\in C; $$
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pseudomonotone on C if$$ \langle Ax,y-x\rangle \geq 0\quad \Longrightarrow\quad \langle Ay,x-y\rangle \leq 0,\quad \forall x, y\in C; $$
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L-Lipschitz continuous on C if there exists a positive constant L such that$$ \Vert Ax-Ay \Vert \leq L \Vert x-y \Vert ,\quad \forall x, y\in C. $$
5 Numerical experiments
Size | Average times (sec) | Average iterations | |||||
---|---|---|---|---|---|---|---|
k
|
m
| Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 |
5 | 10 | 1.399695 | 1.957304 | 6.356185 | 37 | 54 | 171 |
10 | 5 | 2.168317 | 2.916557 | 6.551182 | 56 | 75 | 179 |
20 | 50 | 2.834138 | 3.785376 | 8.711813 | 58 | 80 | 186 |
50 | 20 | 5.292192 | 6.570650 | 10.418191 | 111 | 138 | 220 |