Duality gap

 ( Linear Programming ) 

In optimization problems in applied mathematics, the duality gap is the difference between the dual problem. If $d^*$ is the optimal dual value and $p^*$ is the optimal primal value then the duality gap is equal to $p^*\; -\; d^*$. 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.

(2025). 9781441920263, Springer. ISBN 9781441920263

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

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

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

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

where $F^*$ is the convex conjugate in both variables.

(2025). 9783642028854, Springer. ISBN 9783642028854

Ernö Robert Csetnek (2025). 9783832525033, Logos Verlag Berlin GmbH. ISBN 9783832525033

C. Zălinescu (2025). 9789812380678, World Scientific Publishing Co. Inc. . ISBN 9789812380678

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.

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(2025). 354035445X, Springer-Verlag. . 354035445X

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Leon S. Lasdon (2025). 9780486419992, Dover Publications, Inc.. ISBN 9780486419992

Claude Lemaréchal (2025). 9783540428770, Springer-Verlag. ISBN 9783540428770

Michel Minoux (1986). 9780471901709, A Wiley-Interscience Publication. John Wiley & Sons, Ltd.. ISBN 9780471901709

Jeremy F. Shapiro (1979). 9780471778868, Wiley-Interscience John. . ISBN 9780471778868

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