Wolfe duality

In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle.^[1]

Mathematical formulation

For a minimization problem with inequality constraints,

$\begin{align} &\underset{x}{\operatorname{minimize}}& & f(x) \\ &\operatorname{subject\;to} & &g_i(x) \leq 0, \quad i = 1,\dots,m \end{align}$

the Lagrangian dual problem is

$\begin{align} &\underset{u}{\operatorname{maximize}}& & \inf_x \left(f(x) + \sum_{j=1}^m u_j g_j(x)\right) \\ &\operatorname{subject\;to} & &u_i \geq 0, \quad i = 1,\dots,m \end{align}$

where the objective function is the Lagrange dual function. Provided that the functions $f$ and $g_1, \ldots, g_m$ are continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem

$\begin{align} &\underset{x, u}{\operatorname{maximize}}& & f(x) + \sum_{j=1}^m u_j g_j(x) \\ &\operatorname{subject\;to} & & \nabla f(x) + \sum_{j=1}^m u_j \nabla g_j(x) = 0 \\ &&&u_i \geq 0, \quad i = 1,\dots,m \end{align}$

is called the Wolfe dual problem.^[2] This problem employs the KKT conditions as a constraint. This problem may be difficult to deal with computationally, because the objective function is not concave in the joint variables $(u,x)$. Also, the equality constraint $\nabla f(x) + \sum_{j=1}^m u_j \nabla g_j(x)$ is nonlinear in general, so the Wolfe dual problem is typically a nonconvex optimization problem. In any case, weak duality holds.^[3]

See also

• Lagrangian duality
• Fenchel duality

References

1. Philip Wolfe (1961). "A duality theorem for non-linear programming". Quarterly of Applied Mathematics 19: 239–244.
2. "Chapter 3. Duality in convex optimization" (pdf). October 30, 2011. Retrieved May 20, 2012.
3. Geoffrion, Arthur M. (1971). "Duality in Nonlinear Programming: A Simplified Applications-Oriented Development". SIAM Review 13 (1): 1–37. doi:10.1137/1013001. JSTOR 2028848.