Duality (optimization)

The duality gap is the difference between the values of any primal solutions and any dual solutions. If ${\displaystyle d^{*}}$ is the optimal dual value and ${\displaystyle p^{*}}$ is the optimal primal value, then the duality gap is equal to ${\displaystyle p^{*}-d^{*}}$. This value is always greater than or equal to 0 (for minimization problems). The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds.[5]

In computational optimization, another "duality gap" is often reported, which is the difference in value between any dual solution and the value of a feasible but suboptimal iterate for the primal problem. This alternative "duality gap" quantifies the discrepancy between the value of a current feasible but suboptimal iterate for the primal problem and the value of the dual problem; the value of the dual problem is, under regularity conditions, equal to the value of the convex relaxation of the primal problem: The convex relaxation is the problem arising replacing a non-convex feasible set with its closed convex hull and with replacing a non-convex function with its convex closure, that is the function that has the epigraph that is the closed convex hull of the original primal objective function.[6][7][8][9][10][11][12][13][14][15][16]

In the linear case, in the primal problem, from each sub-optimal point that satisfies all the constraints, there is a direction or subspace of directions to move that increases the objective function. Moving in any such direction is said to remove slack between the candidate solution and one or more constraints. An infeasible value of the candidate solution is one that exceeds one or more of the constraints.

In the dual problem, the dual vector multiplies the constraints that determine the positions of the constraints in the primal. Varying the dual vector in the dual problem is equivalent to revising the upper bounds in the primal problem. The lowest upper bound is sought. That is, the dual vector is minimized in order to remove slack between the candidate positions of the constraints and the actual optimum. An infeasible value of the dual vector is one that is too low. It sets the candidate positions of one or more of the constraints in a position that excludes the actual optimum.

This intuition is made formal by the equations in Linear programming: Duality.

Given a nonlinear programming problem in standard form

${\displaystyle {\begin{aligned}{\text{minimize }}&f_{0}(x)\\{\text{subject to }}&f_{i}(x)\leq 0,\ i\in \left\{1,\ldots ,m\right\}\\&h_{i}(x)=0,\ i\in \left\{1,\ldots ,p\right\}\end{aligned}}}$

with the domain ${\displaystyle {\mathcal {D}}\subset \mathbb {R} ^{n}}$ having non-empty interior, the Lagrangian function ${\displaystyle {\mathcal {L}}:\mathbb {R} ^{n}\times \mathbb {R} ^{m}\times \mathbb {R} ^{p}\to \mathbb {R} }$ is defined as

${\displaystyle {\mathcal {L}}(x,\lambda ,\nu )=f_{0}(x)+\sum _{i=1}^{m}\lambda _{i}f_{i}(x)+\sum _{i=1}^{p}\nu _{i}h_{i}(x).}$

The vectors ${\displaystyle \lambda }$ and ${\displaystyle \nu }$ are called the dual variables or Lagrange multiplier vectors associated with the problem. The Lagrange dual function ${\displaystyle g:\mathbb {R} ^{m}\times \mathbb {R} ^{p}\to \mathbb {R} }$ is defined as

${\displaystyle g(\lambda ,\nu )=\inf _{x\in {\mathcal {D}}}{\mathcal {L}}(x,\lambda ,\nu )=\inf _{x\in {\mathcal {D}}}\left\{f_{0}(x)+\sum _{i=1}^{m}\lambda _{i}f_{i}(x)+\sum _{i=1}^{p}\nu _{i}h_{i}(x)\right\}.}$

The dual function g is concave, even when the initial problem is not convex, because it is a point-wise infimum of affine functions. The dual function yields lower bounds on the optimal value ${\displaystyle p^{*}}$ of the initial problem; for any ${\displaystyle \lambda \geq 0}$ and any ${\displaystyle \nu }$ we have ${\displaystyle g(\lambda ,\nu )\leq p^{*}}$.

If a constraint qualification such as Slater's condition holds and the original problem is convex, then we have strong duality, i.e. ${\displaystyle d^{*}=\max _{\lambda \geq 0,\nu }g(\lambda ,\nu )=\inf f_{0}=p^{*}}$.

For a convex minimization problem with inequality constraints,

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

the Lagrangian dual problem is

${\displaystyle {\begin{aligned}&{\underset {u}{\operatorname {maximize} }}&&\inf _{x}\left(f(x)+\sum _{j=1}^{m}u_{j}g_{j}(x)\right)\\&\operatorname {subject\;to} &&u_{i}\geq 0,\quad i=1,\ldots ,m\end{aligned}}}$

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

${\displaystyle {\begin{aligned}&{\underset {x,u}{\operatorname {maximize} }}&&f(x)+\sum _{j=1}^{m}u_{j}g_{j}(x)\\&\operatorname {subject\;to} &&\nabla f(x)+\sum _{j=1}^{m}u_{j}\,\nabla g_{j}(x)=0\\&&&u_{i}\geq 0,\quad i=1,\ldots ,m\end{aligned}}}$

is called the Wolfe dual problem. This problem may be difficult to deal with computationally, because the objective function is not concave in the joint variables ${\displaystyle (u,x)}$. 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 is typically a nonconvex optimization problem. In any case, weak duality holds.[17]

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