In optimization, there is a notion of the duality gap, $\Delta$ which is always positive $\Delta\geq 0$. The condition $\Delta=0$, called strong duality, is sometimes quite convenient in the analysis of optimization problems. There are a lot of sufficient conditions (e.g., Slater's condition) which guarantee $\Delta = 0$ (see also here), and there are also many examples (in optimization books) where the aforementioned sufficient conditions fail but $\Delta=0$ still holds.
However, I am less familiar with examples or classes of problems which are known to possess a strictly positive duality gap $\Delta>0$. So, with this question I am hoping we can compile a list of interesting examples which demonstrate this behavior (with particular attention to properties of the functions, such as convexity and/or smoothness).
An example is interior point methods with logarithmic barrier. For reference, cf. Boyd & Vandenberghe "Convex Optimization" chap. $11$ "Interior-point methods" with its $70$ pages on the subject.
The goal is to find $x \in \mathbb R ^n$ that minimizes $f_0(x)$,
subject to constraints $f_i(x) \le 0, i=1\dots m$, and $Ax=b$,
where $f_0$ and $f_i: \mathbb R^n \to \mathbb R$ are convex and twice continuously differentiable,
$A$ is a $p \times n$ matrix, $p \lt n$, with rank $p$,
and $b \in \mathbb R ^p$.
(And we assume the problem is strictly feasible: $\exists x$ satisfying $\forall i, f_i(x) \le 0$, and $Ax=b$).
The method adds a logarithmic barrier term $\Phi(x) = - \sum_{i=1}^m \log(-f_i(x))$ to the objective function $tf_0(x)$.
($t$ is a parameter which $\to \infty$. Usually people define the new objective function as $f_0(x) + \mu \Phi(x)$, where $\mu \to 0$. This is equivalent, but Boyd & Vandenberghe chose this because it makes some proofs easier, when differentiating on $t$).
The method principle is to solve the new optimization problem ($tf_0(x) + \Phi(x)$), for some value of $t$, with Newton's method, then grow $t$ and repeat, using the previous optimum as initial point for Newton's method.
A very nice property of logarithmic barriers (and only this class of functions) is that the duality gap associated to optima $x^*$ and $\lambda^*$ for a specific value of $t$, is simply $m/t$. It depends only upon the number of constraints and the value of parameter $t$, not upon the problem other characteristics.
One consequence is that, contrary to simplex method (for linear problems) where the optimum is reached after a finite number of steps, here the number of steps is in theory infinite. But in practice, a sufficient precision is reached faster than with simplex method.
Note that in the original problem, Slater's condition holds, so the duality gap is $0$. It is only on problems with added logarithmic barrier that the duality gap is $m/t \gt 0$, with this quantity $\to 0$ when $t \to \infty$.