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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.