## Conjugate Duality and OptimizationProvides a relatively brief introduction to conjugate duality in both finite- and infinite-dimensional problems. An emphasis is placed on the fundamental importance of the concepts of Lagrangian function, saddle-point, and saddle-value. General examples are drawn from nonlinear programming, approximation, stochastic programming, the calculus of variations, and optimal control. |

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applications assume assumptions Banach space Borel structure closed concave closed convex function compatible topology concave function constraints continuous convex optimization convex set core dom q corresponding define denote dual problem element epigraph equicontinuous equivalent Example exists F is closed F is convex finite finite-dimensional fo(x formula functions f hence implies inf f(x inf(P infimum inſ integral functionals k(Ax Kuhn–Tucker condition Lagrangian Lagrangian function level sets linear functions linear space lower-semicontinuity Mackey topology maximize measurable function minimax problem minimize f(x minimizing f monotone operator neighborhood nonconvex nonempty and equicontinuous Nonlinear norm topology optimal value function optimization problems pairing parameters Proof proper convex function properties ſ h(x saddle-point saddle-value satisfies sigma-finite solves Stochastic programming subgradients subset subspace summable sup K(x sup(D supremum Theorem 18 Theorem 20 tion vector weak topology weakly compact