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# An improved projection algorithm for variational inequality problem with multivalued mapping

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• In this paper, we propose a new projection method (extragradient method) for solving the generalized variational inequality problem. Our method is well-defined and is proven to be globally convergent to a solution of the problem under the assumptions that the multivalued mapping is continuous and satisfies a condition that is strictly weaker than the pseudomonotonicity. Finally, to support our results and to illustrate the numerical behavior of the proposed algorithm, some fundamental experiments are provided and also compared with recent works.

Mathematics Subject Classification: Primary: 65K05, 65K15, 90C33.

 Citation: • • Table 1.  Example 1

 Alg1 Alg1' $n$ $x^{0}$ $Iter$ $cpu$ $\beta$ $\lambda$ $Iter$ $cpu$ $\beta$ $\lambda$ $(1/n, ..., 1/n)$ $05$ $0.0933$ $2.3$ $30$ $05$ $0.1013$ $4.6$ $30$ $10$ $(1, ..., 1)$ $05$ $0.07693$ $2.3$ $30$ $05$ $0.1084$ $4.6$ $30$ $(-2, ..., -2)$ $05$ $0.1752$ $2.3$ $30$ $05$ $0.1502$ $4.6$ $30$ $(1/n, ..., 1/n)$ $08$ $0.4548$ $4.6$ $30$ $05$ $0.3672$ $1.3$ $600$ $50$ $(1, ..., 1)$ $09$ $0.5292$ $2.3$ $25$ $05$ $0.3637$ $1.3$ $600$ $(-2, ..., -2)$ $25$ $2.9980$ $2.3$ $30$ $25$ $2.7703$ $03$ $04$ $(1/n, ..., 1/n)$ $19$ $4.6537$ $3.3$ $30$ $18$ $4.7851$ $1.3$ $30$ $100$ $(1, ..., 1)$ $08$ $1.2500$ $5.3$ $30$ $18$ $4.9539$ $1.3$ $30$ $(-2, ..., -2)$ $50$ $17.3849$ $1.5$ $30$ $50$ $18.6507$ $1.3$ $30$ $(1/n, ..., 1/n)$ $84$ $122.7375$ $1.5$ $30$ $07$ $4.7840$ $4.6$ $30$ $200$ $(1, ..., 1)$ $88$ $133.7441$ $3.3$ $30$ $07$ $4.8564$ $4.6$ $30$ $(-2, ..., -2)$ $100$ $167.8360$ $3.3$ $30$ $100$ $165.3262$ $03$ $30$

Table 2.  Example 1

 Alg2 Alg2' $n$ $x^{0}$ $Iter$ $cpu$ $\beta$ $\theta$ $Iter$ $cpu$ $\beta$ $\theta$ $(1/n, ..., 1/n)$ $10$ $0.1044$ $09$ $0.01$ $10$ $0.2166$ $7.2$ $0.01$ $10$ $(1, ..., 1)$ $10$ $0.2146$ $05$ $0.01$ $10$ $0.2255$ $2.8$ $0.01$ $(-2, ..., -2)$ $13$ $0.2455$ $09$ $0.01$ $13$ $0.2849$ $8.1$ $0.01$ $(1/n, ..., 1/n)$ $25$ $0.7580$ $8.1$ $0.01$ $28$ $1.7506$ $4.3$ $0.01$ $50$ $(1, ..., 1)$ $27$ $1.6024$ $5.5$ $0.001$ $27$ $1.9405$ $8.3$ $0.01$ $(-2, ..., -2)$ $53$ $4.5289$ $09$ $0.01$ $53$ $5.6964$ $4.3$ $0.01$ $(1/n, ..., 1/n)$ $43$ $9.9579$ $6.1$ $0.01$ $43$ $9.2976$ $3.3$ $0.01$ $100$ $(1, ..., 1)$ $32$ $6.6368$ $6.9$ $0.01$ $45$ $13.3148$ $34$ $0.01$ $(-2, ..., -2)$ $103$ $33.9310$ $6.1$ $0.01$ $103$ $35.7461$ $34$ $0.01$ $(1/n, ..., 1/n)$ $116$ $179.6938$ $6.6$ $0.01$ $116$ $181.6362$ $6.6$ $0.01$ $200$ $(1, ..., 1)$ $113$ $159.2788$ $3.4$ $0.01$ $112$ $161.1741$ $3.4$ $0.01$ $(-2, ..., -2)$ $203$ $369.6150$ $6.6$ $0.01$ $203$ $358.1106$ $6.6$ $0.01$

Table 3.  Example 1

 Alg3 $n$ $x_{0}$ $Iter$ $cpu$ $l$ $(1/n, \ldots , 1/n)$ $35$ $1.7592$ $0.95$ $10$ $(1, \ldots , 1)$ $44$ $1.6183$ $0.95$ $(-2, \ldots , -2)$ $43$ $1.7283$ $0.95$ $(1/n, \ldots , 1/n)$ $64$ $12.3182$ $0.95$ $50$ $(1, \ldots , 1)$ $129$ $14.9710$ $0.9$ $(-2, \ldots , -2)$ $127$ $15.1543$ $0.9$ $(1/n, \ldots , 1/n)$ $90$ $37.3068$ $0.95$ $100$ $(1, \ldots , 1)$ $214$ $82.9337$ $0.95$ $(-2, \ldots , -2)$ $213$ $88.2488$ $0.95$ $(1/n, \ldots , 1/n)$ $122$ $226.3886$ $0.95$ $200$ $(1, \ldots , 1)$ $390$ $687.8674$ $0.95$ $(-2, \ldots , -2)$ $391$ $718.4568$ $0.95$

Table 4.  Example 2

 Alg1 Alg1' $n$ $x^{0}$ $Iter$ $cpu$ $\beta$ $\lambda$ $Iter$ $cpu$ $\beta$ $\lambda$ $(1, ..., 1)$ $05$ $0.1080$ $2.5$ $02$ $04$ $0.1173$ $01$ $90$ $10$ $(0.9, ..., 0.9)$ $01$ $0.0194$ $09$ $02$ $01$ $0.0375$ $01$ $90$ $(-2, ..., -2)$ $01$ $0.0571$ $0.2$ $02$ $01$ $0.0404$ $01$ $90$ $(1, ..., 1)$ $08$ $0.2907$ $0.2$ $90$ $07$ $0.2838$ $04$ $90$ $100$ $(0.9, ..., 0.9)$ $01$ $0.0498$ $0.2$ $90$ $01$ $0.0490$ $01$ $90$ $(-2, ..., -2)$ $01$ $0.1063$ $0.2$ $90$ $01$ $0.0698$ $01$ $90$ $(1, ..., 1)$ $08$ $0.4193$ $0.2$ $90$ $06$ $1.6353$ $05$ $90$ $500$ $(0.9, ..., 0.9)$ $01$ $0.0781$ $0.2$ $90$ $01$ $0.2545$ $01$ $90$ $(-2, ..., -2)$ $01$ $0.2038$ $0.2$ $90$ $01$ $0.3603$ $01$ $90$ $(1, ..., 1)$ $08$ $0.9780$ $0.2$ $90$ $06$ $8.0998$ $05$ $90$ $1000$ $(0.9, ..., 0.9)$ $01$ $0.1478$ $0.2$ $90$ $01$ $1.2334$ $01$ $90$ $(-2, ..., -2)$ $01$ $0.9753$ $0.2$ $90$ $01$ $2.1603$ $01$ $90$ $(1, ..., 1)$ $08$ $2.9927$ $0.2$ $90$ $06$ $51.5017$ $5.5$ $90$ $2000$ $(0.9, ..., 0.9)$ $01$ $0.3171$ $0.2$ $90$ $01$ $8.6067$ $05$ $90$ $(-2, ..., -2)$ $01$ $1.5950$ $0.2$ $90$ $01$ $11.3070$ $05$ $90$

Table 5.  Example 2

 Alg2 Alg2' $n$ $x^{0}$ $Iter$ $cpu$ $\beta$ $\theta$ $Iter$ $cpu$ $\beta$ $\theta$ $(1, ..., 1)$ $06$ $0.1537$ $15$ $0.1$ $06$ $0.1133$ $09$ $0.1$ $10$ $(0.9, ..., 0.9)$ $01$ $0.0505$ $15$ $0.1$ $01$ $0.0157$ $09$ $0.1$ $(-2, ..., -2)$ $01$ $0.0579$ $15$ $0.1$ $01$ $0.0521$ $09$ $0.1$ $(1, ..., 1)$ $42$ $3.5722$ $1.5$ $0.1$ $09$ $0.6863$ $190$ $0.3$ $100$ $(0.9, ..., 0.9)$ $01$ $0.0605$ $15$ $0.1$ $01$ $0.0745$ $09$ $0.3$ $(-2, ..., -2)$ $01$ $0.0493$ $15$ $0.1$ $01$ $0.1952$ $90$ $0.3$ $(1, ..., 1)$ $41$ $72.716$ $1.5$ $0.1$ $09$ $17.1776$ $190$ $0.3$ $500$ $(0.9, ..., 0.9)$ $01$ $0.2051$ $15$ $0.1$ $01$ $10.5841$ $90$ $0.3$ $(-2, ..., -2)$ $01$ $11.4440$ $15$ $0.1$ $01$ $10.9202$ $90$ $0.3$ $(1, ..., 1)$ $92$ $1589.711$ $0.5$ $0.1$ $07$ $98.3717$ $600$ $0.3$ $1000$ $(0.9, ..., 0.9)$ $01$ $0.8017$ $15$ $0.1$ $01$ $1.6180$ $09$ $0.3$ $(-2, ..., -2)$ $01$ $69.5720$ $15$ $0.1$ $01$ $75.3165$ $90$ $0.3$ $(1, ..., 1)$ $38$ $3255.6828$ $1.5$ $0.1$ $11$ $841.6456$ $190$ $0.9$ $2000$ $(0.9, ..., 0.9)$ $01$ $22.9217$ $15$ $0.1$ $01$ $4.6524$ $09$ $0.3$ $(-2, ..., -2)$ $01$ $621.5315$ $15$ $0.1$ $01$ $583.3619$ $90$ $0.3$

Table 6.  Example 2

 Alg 3 $n$ $x^{0}$ $Iter$ $cpu$ $(1, ..., 1)$ $15$ $0.3515$ $10$ $(0.9, ..., 0.9)$ $15$ $0.4098$ $(-2, ..., -2)$ $17$ $0.3880$ $(1, ..., 1)$ $96$ $3.7692$ $100$ $(0.9, ..., 0.9)$ $21$ $0.7153$ $(-2, ..., -2)$ $21$ $0.7609$ $(1, ..., 1)$ $101$ $26.0059$ $500$ $(0.9, ..., 0.9)$ $21$ $6.1334$ $(-2, ..., -2)$ $21$ $5.4223$ $(1, ..., 1)$ $101$ $131.7395$ $1000$ $(0.9, ..., 0.9)$ $21$ $29.3578$ $(-2, ..., -2)$ $21$ $30.2118$ $(1, ..., 1)$ $95$ $785.6897$ $2000$ $(0.9, ..., 0.9)$ $21$ $193.5405$ $(-2, ..., -2)$ $21$ $200.4297$
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