Design of General Projection Neural Networks for Solving Monotone Linear Variational Inequalities and Linear and Quadratic Optimization Problems
Most existing neural networks for solving linear variational inequalities (LVIs) with the mapping Mx + p require positive definiteness (or positive semidefiniteness) of M. In this correspondence, it is revealed that this condition is sufficient but not necessary for an LVI being strictly monotone (o...
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description | Most existing neural networks for solving linear variational inequalities (LVIs) with the mapping Mx + p require positive definiteness (or positive semidefiniteness) of M. In this correspondence, it is revealed that this condition is sufficient but not necessary for an LVI being strictly monotone (or monotone) on its constrained set where equality constraints are present. Then, it is proposed to reformulate monotone LVIs with equality constraints into LVIs with inequality constraints only, which are then possible to be solved by using some existing neural networks. General projection neural networks are designed in this correspondence for solving the transformed LVIs. Compared with existing neural networks, the designed neural networks feature lower model complexity. Moreover, the neural networks are guaranteed to be globally convergent to solutions of the LVI under the condition that the linear mapping Mx + p is monotone on the constrained set. Because quadratic and linear programming problems are special cases of LVI in terms of solutions, the designed neural networks can solve them efficiently as well. In addition, it is discovered that the designed neural network in a specific case turns out to be the primal-dual network for solving quadratic or linear programming problems. The effectiveness of the neural networks is illustrated by several numerical examples. |
doi_str_mv | 10.1109/TSMCB.2007.903706 |
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In this correspondence, it is revealed that this condition is sufficient but not necessary for an LVI being strictly monotone (or monotone) on its constrained set where equality constraints are present. Then, it is proposed to reformulate monotone LVIs with equality constraints into LVIs with inequality constraints only, which are then possible to be solved by using some existing neural networks. General projection neural networks are designed in this correspondence for solving the transformed LVIs. Compared with existing neural networks, the designed neural networks feature lower model complexity. Moreover, the neural networks are guaranteed to be globally convergent to solutions of the LVI under the condition that the linear mapping Mx + p is monotone on the constrained set. Because quadratic and linear programming problems are special cases of LVI in terms of solutions, the designed neural networks can solve them efficiently as well. In addition, it is discovered that the designed neural network in a specific case turns out to be the primal-dual network for solving quadratic or linear programming problems. 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(IEEE) 2007</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c378t-35b033bad40b70f888017a03cb5da0189372c4f924325f62a21c6348191dfaa23</citedby><cites>FETCH-LOGICAL-c378t-35b033bad40b70f888017a03cb5da0189372c4f924325f62a21c6348191dfaa23</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/4305278$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>314,780,784,796,27924,27925,54758</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/4305278$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc><backlink>$$Uhttps://www.ncbi.nlm.nih.gov/pubmed/17926722$$D View this record in MEDLINE/PubMed$$Hfree_for_read</backlink></links><search><creatorcontrib>Hu, Xiaolin</creatorcontrib><creatorcontrib>Wang, Jun</creatorcontrib><title>Design of General Projection Neural Networks for Solving Monotone Linear Variational Inequalities and Linear and Quadratic Optimization Problems</title><title>IEEE transactions on cybernetics</title><addtitle>TSMCB</addtitle><addtitle>IEEE Trans Syst Man Cybern B Cybern</addtitle><description>Most existing neural networks for solving linear variational inequalities (LVIs) with the mapping Mx + p require positive definiteness (or positive semidefiniteness) of M. In this correspondence, it is revealed that this condition is sufficient but not necessary for an LVI being strictly monotone (or monotone) on its constrained set where equality constraints are present. Then, it is proposed to reformulate monotone LVIs with equality constraints into LVIs with inequality constraints only, which are then possible to be solved by using some existing neural networks. General projection neural networks are designed in this correspondence for solving the transformed LVIs. Compared with existing neural networks, the designed neural networks feature lower model complexity. Moreover, the neural networks are guaranteed to be globally convergent to solutions of the LVI under the condition that the linear mapping Mx + p is monotone on the constrained set. Because quadratic and linear programming problems are special cases of LVI in terms of solutions, the designed neural networks can solve them efficiently as well. In addition, it is discovered that the designed neural network in a specific case turns out to be the primal-dual network for solving quadratic or linear programming problems. The effectiveness of the neural networks is illustrated by several numerical examples.</description><subject>Algorithms</subject><subject>Artificial Intelligence</subject><subject>Automation</subject><subject>Computer Simulation</subject><subject>Constraint optimization</subject><subject>Constraints</subject><subject>Convergence</subject><subject>Councils</subject><subject>Cybernetics</subject><subject>Design optimization</subject><subject>Global convergence</subject><subject>Inequalities</subject><subject>Linear Models</subject><subject>Linear programming</subject><subject>linear variational inequality (LVI)</subject><subject>Mapping</subject><subject>Mathematical models</subject><subject>Neural networks</subject><subject>Neural Networks (Computer)</subject><subject>Pattern Recognition, Automated - methods</subject><subject>Projection</subject><subject>Quadratic programming</subject><subject>recurrent neural network</subject><subject>Recurrent neural networks</subject><subject>Regression analysis</subject><subject>Studies</subject><issn>1083-4419</issn><issn>2168-2267</issn><issn>1941-0492</issn><issn>2168-2275</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2007</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><sourceid>EIF</sourceid><recordid>eNp9kcFu1DAQhiMEoqXwAAgJWRzoKcuM7cTxERYolbYtqIVr5CSTykvW3toJCJ6ij4zTXUDiwGlGM9__jzR_lj1FWCCCfnV1ebZ8s-AAaqFBKCjvZYeoJeYgNb-feqhELiXqg-xRjGsA0KDVw-wAleal4vwwu31L0V475nt2Qo6CGdjH4NfUjtY7dk7TPDmn8bsPXyPrfWCXfvhm3TU7886P3hFbWUcmsC8mWDOrkuDU0c1kBjtaisy47jczt58m04UEtuxiO9qN_Xknmq82A23i4-xBb4ZIT_b1KPv8_t3V8kO-ujg5Xb5e5a1Q1ZiLogEhGtNJaBT0VVUBKgOibYrOAFZaKN7KXnMpeNGX3HBsSyEr1Nj1xnBxlB3vfLfB30wUx3pjY0vDYBz5KdbJsCx0VWIiX_6XLKv0ehQqgS_-Add-Cukfya2UiCXifBd3UBt8jIH6ehvsxoQfNUI9p1rfpVrPqda7VJPm-d54ajbU_VXsY0zAsx1giejPWgoouKrEL7K1pyA</recordid><startdate>20071001</startdate><enddate>20071001</enddate><creator>Hu, Xiaolin</creator><creator>Wang, Jun</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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In this correspondence, it is revealed that this condition is sufficient but not necessary for an LVI being strictly monotone (or monotone) on its constrained set where equality constraints are present. Then, it is proposed to reformulate monotone LVIs with equality constraints into LVIs with inequality constraints only, which are then possible to be solved by using some existing neural networks. General projection neural networks are designed in this correspondence for solving the transformed LVIs. Compared with existing neural networks, the designed neural networks feature lower model complexity. Moreover, the neural networks are guaranteed to be globally convergent to solutions of the LVI under the condition that the linear mapping Mx + p is monotone on the constrained set. Because quadratic and linear programming problems are special cases of LVI in terms of solutions, the designed neural networks can solve them efficiently as well. In addition, it is discovered that the designed neural network in a specific case turns out to be the primal-dual network for solving quadratic or linear programming problems. The effectiveness of the neural networks is illustrated by several numerical examples.</abstract><cop>United States</cop><pub>IEEE</pub><pmid>17926722</pmid><doi>10.1109/TSMCB.2007.903706</doi><tpages>8</tpages></addata></record> |
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subjects | Algorithms Artificial Intelligence Automation Computer Simulation Constraint optimization Constraints Convergence Councils Cybernetics Design optimization Global convergence Inequalities Linear Models Linear programming linear variational inequality (LVI) Mapping Mathematical models Neural networks Neural Networks (Computer) Pattern Recognition, Automated - methods Projection Quadratic programming recurrent neural network Recurrent neural networks Regression analysis Studies |
title | Design of General Projection Neural Networks for Solving Monotone Linear Variational Inequalities and Linear and Quadratic Optimization Problems |
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