a function is the conjugate of the conjugate, typically written as The biconjugate is useful for showing when strong or weak duality hold (via the perturbation function). For any the inequality follows from the . For proper functions, if and only if is convex and lower semi-continuous by Fenchel–Moreau theorem. Convex minimization A () is one of the form find when given a convex function and a convex subset Dual problem In optimization theory, the states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. In general given two dual pairs separated locally convex spaces and Then given the function we can define the primal problem as finding such that If there are constraint conditions, these can be built into the function by letting where is the indicator function. Then let be a perturbation function such that The with respect to the chosen perturbation function is given by where is the convex conjugate in both variables of The duality gap is the difference of the right and left hand sides of the inequality This principle is the same as weak duality. If the two sides are equal to each other, then the problem is said to satisfy strong duality. There are many conditions for strong duality to hold such as: where is the perturbation function relating the primal and dual problems and is the biconjugate of ; the primal problem is a linear optimization problem; Slater's condition for a convex optimization problem. Lagrange duality For a convex minimization problem with inequality constraints, subject to for the Lagrangian dual problem is subject to for where the objective function is the Lagrange dual function defined as follows: See also Notes References External links Analysis