Lagrangian duality

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.

Usually the term "dual problem" refers to the Lagrangian dual problem but other dual problems are used, for example, the Wolfe dual problem and the Fenchel dual problem. The Lagrangian dual problem is obtained by forming the Lagrangian, using nonnegative Lagrange multipliers to add the constraints to the objective function, and then solving for some primal variable values that minimize the Lagrangian. This solution gives the primal variables as functions of the Lagrange multipliers, which are called dual variables, so that the new problem is to maximize the objective function with respect to the dual variables under the derived constraints on the dual variables (including at least the nonnegativity).

In general given two dual pairs of separated locally convex spaces ${\displaystyle \left(X,X^{*}\right)}$ and ${\displaystyle \left(Y,Y^{*}\right)}$ and the function ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$, we can define the primal problem as finding ${\displaystyle {\hat {x}}}$ such that ${\displaystyle f({\hat {x}})=\inf _{x\in X}f(x).\,}$ In other words, ${\displaystyle f({\hat {x}})}$ is the infimum (greatest lower bound) of the function ${\displaystyle f}$.

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