Duality gap

In optimization problems in applied mathematics, the duality gap is the difference between the primal and 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.[1]

In general given two dual pairs separated locally convex spaces ${\displaystyle \left(X,X^{*}\right)}$ and ${\displaystyle \left(Y,Y^{*}\right)}$. Then given the function ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$, we can define the primal problem by

${\displaystyle \inf _{x\in X}f(x).\,}$

If there are constraint conditions, these can be built into the function ${\displaystyle f}$ by letting ${\displaystyle f=f+I_{\text{constraints}}}$ where ${\displaystyle I}$ is the indicator function. Then let ${\displaystyle F:X\times Y\to \mathbb {R} \cup \{+\infty \}}$ be a perturbation function such that ${\displaystyle F(x,0)=f(x)}$. The duality gap is the difference given by

${\displaystyle \inf _{x\in X}[F(x,0)]-\sup _{y^{*}\in Y^{*}}[-F^{*}(0,y^{*})]}$

where ${\displaystyle F^{*}}$ is the convex conjugate in both variables.[2][3][4]

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.[5][6][7][8][9][10][11][12][13]