Abstract Nonconvex optimization is becoming the fashion to solve constrained optimization problems. Classical Lagrangian does not necessarily represent a nonconvex optimization problem. In this paper, we give conditions under which the Classical Lagrangian serves as an exact penalization of a nonconvex programming. This has a simple interpretation and is easy to solve. We use this Classical Lagrangian to provide su¢ cient conditions under which value functions are (i) Clarke di¤erentiable with di¤erential bounds, (ii) directionally di¤erentiable, and (iii) once-continuously di¤erentiable (C 1 ). Relative to the literature, we provide the most general su¢ cient conditions for the existence of directional di¤erentiable and once continuously di¤erentiable envelopes. Our theory has numerous potential examples in lattice programming, nonclassical growth theory and macroeconomics, Negishi methods, nonstationary dynamic lattice programming, and duopoly problems

Department of Economics, WP Carey School of Business, Arizona State University

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