# Wolfe duality

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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,

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

where the objective function is the Lagrange dual function. Provided that the functions ${\displaystyle f}$ and ${\displaystyle g_{1},\ldots ,g_{m}}$ are convex and continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem

{\displaystyle {\begin{aligned}&{\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{aligned}}}

is called the Wolfe dual problem.[2] This problem employs the KKT conditions as a constraint. Also, the equality constraint ${\displaystyle \nabla f(x)+\sum _{j=1}^{m}u_{j}\nabla g_{j}(x)}$ is nonlinear in general, so the Wolfe dual problem may be a nonconvex optimization problem. In any case, weak duality holds.[3]

## 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.