Convexity and Well-Posed Problems

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Focusing on the stability of convex functions amidst perturbations, this book presents a modern definition of convexity through the lens of epigraphs, viewing the properties of functions as those of associated convex sets. It explores extended real-valued functions, incorporating +∞ to streamline constrained optimization by integrating constraint sets into objective functions. The inclusion of -∞ facilitates operations like inf-convolution. Emphasizing lower semicontinuity, it lays a foundation for addressing minimum problems, diverging from traditional derivative-based variational calculus.