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]

## References

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9. ^ Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1993). "XII. Abstract Duality for Practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 306. Berlin: Springer-Verlag. pp. xviii+346. doi:10.1007/978-3-662-06409-2_4. ISBN 3-540-56852-2. MR 1295240.
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13. ^ Shapiro, Jeremy F. (1979). Mathematical programming: Structures and algorithms. New York: Wiley-Interscience [John Wiley & Sons]. pp. xvi+388. ISBN 0-471-77886-9. MR 0544669.