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We study conjugate and Lagrange dualities for composite optimization problemswithin the framework of abstract convexity. We provide conditions for zeroduality gap in conjugate duality. For Lagrange duality, intersection propertyis applied to obtain zero duality gap. Connection between Lagrange dual andconjugate dual is also established. Examples related to convex and weaklyconvex functions are given.

Duality for Composite Optimization Problem within the Framework of Abstract Convexity