Recommend formula for eps_duality_gap_abs

Author: stephane-caron

doc/2-PROXQP_API/2-ProxQP_api.md

@@ -71,7 +71,7 @@ $$\begin{equation}\label{eq:approx_dg_sol}
 \end{aligned}
 \end{equation}$$
 
-where $[z]_+$ and $[z]_-$ stand for the projection of z onto the positive and negative orthant. ProxQP provides the ``check_duality_gap`` option to include this duality gap in the stopping criterion. Note that it is disabled by default, as other solvers don't check in general this criterion. Enable this option if you want a stronger guarantee that your solution is optimal. ProxQP will then check the same termination condition as SCS (for more details see, e.g., SCS's [optimality conditions checks](https://www.cvxgrp.org/scs/algorithm/index.html#optimality-conditions) as well as [section 7.2](https://doi.org/10.1137/20M1366307) in the corresponding paper). The absolute and relative thresholds $\epsilon^{\text{gap}}_{\text{abs}}, \epsilon^{\text{gap}}_{\text{rel}}$ for the duality gap can differ from those $\epsilon_{\text{abs}}, \epsilon_{\text{rel}}$ for residuals because, contrary to residuals which use an infinite norm, the duality gap scales with the square root of the problem dimension (thus it is numerically harder to achieve a given duality gap for larger problems). Note finally that meeting all residual and duality-gap criteria can be difficult for ill-conditioned problems.
+where $[z]_+$ and $[z]_-$ stand for the projection of z onto the positive and negative orthant. ProxQP provides the ``check_duality_gap`` option to include this duality gap in the stopping criterion. Note that it is disabled by default, as other solvers don't check in general this criterion. Enable this option if you want a stronger guarantee that your solution is optimal. ProxQP will then check the same termination condition as SCS (for more details see, e.g., SCS's [optimality conditions checks](https://www.cvxgrp.org/scs/algorithm/index.html#optimality-conditions) as well as [section 7.2](https://doi.org/10.1137/20M1366307) in the corresponding paper). The absolute and relative thresholds $\epsilon^{\text{gap}}_{\text{abs}}, \epsilon^{\text{gap}}_{\text{rel}}$ for the duality gap can differ from those $\epsilon_{\text{abs}}, \epsilon_{\text{rel}}$ for residuals because, contrary to residuals which result from an infinite norm, the duality gap scales with the square root of the problem dimension (thus it is numerically harder to achieve a given duality gap for larger problems). A recommended choice is $\epsilon^{\text{gap}}_{\text{abs}} = \epsilon_{\text{abs}} \sqrt{\max(n, n_{\text{eq}}, n_{\text{ineq}}}$. Note finally that meeting all residual and duality-gap criteria can be difficult for ill-conditioned problems.
 
 \section OverviewAPIstructure ProxQP unified API for dense and sparse backends
 

doc/3-ProxQP_solve.md

@@ -71,7 +71,7 @@ $$\begin{equation}\label{eq:approx_dg_sol}
 \end{aligned}
 \end{equation}$$
 
-where $[z]_+$ and $[z]_-$ stand for the projection of z onto the positive and negative orthant. ProxQP provides the ``check_duality_gap`` option to include this duality gap in the stopping criterion. Note that it is disabled by default, as  other solvers don't check in general this criterion. Enable this option if you want a stronger guarantee that your solution is optimal. ProxQP will then check the same termination condition as SCS (for more details see, e.g., SCS's [optimality conditions checks](https://www.cvxgrp.org/scs/algorithm/index.html#optimality-conditions) as well as [section 7.2](https://doi.org/10.1137/20M1366307) in the corresponding paper). The absolute and relative thresholds $\epsilon^{\text{gap}}_{\text{abs}}, \epsilon^{\text{gap}}_{\text{rel}}$ for the duality gap can differ from those $\epsilon_{\text{abs}}, \epsilon_{\text{rel}}$ for residuals because, contrary to residuals which use an infinite norm, the duality gap scales with the square root of the problem dimension (thus it is numerically harder to achieve a given duality gap for larger problems). Note finally that meeting all residual and duality-gap criteria can be difficult for ill-conditioned problems.
+where $[z]_+$ and $[z]_-$ stand for the projection of z onto the positive and negative orthant. ProxQP provides the ``check_duality_gap`` option to include this duality gap in the stopping criterion. Note that it is disabled by default, as other solvers don't check in general this criterion. Enable this option if you want a stronger guarantee that your solution is optimal. ProxQP will then check the same termination condition as SCS (for more details see, e.g., SCS's [optimality conditions checks](https://www.cvxgrp.org/scs/algorithm/index.html#optimality-conditions) as well as [section 7.2](https://doi.org/10.1137/20M1366307) in the corresponding paper). The absolute and relative thresholds $\epsilon^{\text{gap}}_{\text{abs}}, \epsilon^{\text{gap}}_{\text{rel}}$ for the duality gap can differ from those $\epsilon_{\text{abs}}, \epsilon_{\text{rel}}$ for residuals because, contrary to residuals which result from an infinite norm, the duality gap scales with the square root of the problem dimension (thus it is numerically harder to achieve a given duality gap for larger problems). A recommended choice is $\epsilon^{\text{gap}}_{\text{abs}} = \epsilon_{\text{abs}} \sqrt{\max(n, n_{\text{eq}}, n_{\text{ineq}}}$. Note finally that meeting all residual and duality-gap criteria can be difficult for ill-conditioned problems.
 
 \section OverviewAsingleSolveFunction A single solve function for dense and sparse backends