The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity
Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathem...
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MDPI AG
2023-10-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/20/4240 |
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| author | Hong Yang Angang Cui |
| author_facet | Hong Yang Angang Cui |
| author_sort | Hong Yang |
| collection | DOAJ |
| description | Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the <i>K</i>−directional derivative, that is, uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>invex, strictly uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>invex, uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>pseudoinvex, strictly uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>pseudoinvex, uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>quasiinvex and weakly uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions. |
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| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-10T21:05:03Z |
| publishDate | 2023-10-01 |
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| spelling | doaj.art-c22a8899b67848439863140021e255bf2023-11-19T17:13:09ZengMDPI AGMathematics2227-73902023-10-011120424010.3390/math11204240The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−InvexityHong Yang0Angang Cui1School of Mathematics and Statistics, Yulin University, Yulin 719000, ChinaSchool of Mathematics and Statistics, Yulin University, Yulin 719000, ChinaMinimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the <i>K</i>−directional derivative, that is, uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>invex, strictly uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>invex, uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>pseudoinvex, strictly uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>pseudoinvex, uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>quasiinvex and weakly uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mi>K</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></semantics></math></inline-formula>quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions.https://www.mdpi.com/2227-7390/11/20/4240non-smooth programmingfractional semi-infinite programming<i>K</i>−directional derivativeuniform (<i>B<sub>K</sub></i>,<i>ρ</i>)−invexityoptimality conditions |
| spellingShingle | Hong Yang Angang Cui The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity Mathematics non-smooth programming fractional semi-infinite programming <i>K</i>−directional derivative uniform (<i>B<sub>K</sub></i>,<i>ρ</i>)−invexity optimality conditions |
| title | The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity |
| title_full | The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity |
| title_fullStr | The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity |
| title_full_unstemmed | The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity |
| title_short | The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B<sub>K</sub>,ρ</i>)−Invexity |
| title_sort | sufficiency of solutions for non smooth minimax fractional semi infinite programming with i b sub k sub ρ i invexity |
| topic | non-smooth programming fractional semi-infinite programming <i>K</i>−directional derivative uniform (<i>B<sub>K</sub></i>,<i>ρ</i>)−invexity optimality conditions |
| url | https://www.mdpi.com/2227-7390/11/20/4240 |
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