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What (who) is dual problem - definition
 = \varphi(x_1 + x_2) = \varphi(x)</math>'' for any ''<math>\varphi</math>'' in the dual space.](https://commons.wikimedia.org/wiki/Special:FilePath/Double dual nature.svg?width=200 "''x''<sub>1</sub> + ''x''<sub>2</sub>}}.
The addition +′ induced by the transformation can be defined as ''<math>[\Psi(x_1) +' \Psi(x_2)](\varphi) = \varphi(x_1 + x_2) = \varphi(x)</math>'' for any ''<math>\varphi</math>'' in the dual space.")
![multiple constrained problem]] could consider both the weight and volume of the boxes. <br />(Solution: if any number of each box is available, then three yellow boxes and three grey boxes; if only the shown boxes are available, then all except for the green box.)](https://commons.wikimedia.org/wiki/Special:FilePath/Knapsack.svg?width=200 "multiple constrained problem]] could consider both the weight and volume of the boxes. <br />(Solution: if any number of each box is available, then three yellow boxes and three grey boxes; if only the shown boxes are available, then all except for the green box.)")
Wikipedia
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality.
In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality.
