Multi-criteria decision-making often requires finding a small representative set from the database. A recently proposed method is the regret minimization set (RMS) query. RMS returns a size $r$ subset $S$ of dataset $D$ that minimizes the regret-ratio (the difference between the score of top-1 in $S$ and the score of top-1 in $D$, for any possible utility function). RMS is not shift invariant, causing inconsistency in results. Further, existing work showed that the regret-ratio is often a made-up number and users may mistake its absolute value. Instead, users do understand the notion of rank. Thus it considered the problem of finding the minimal set $S$ with a rank-regret (the rank of top-1 tuple of $S$ in the sorted list of $D$) at most $k$, called the rank-regret representative (RRR) problem. Corresponding to RMS, we focus on the min-error version of RRR, called the rank-regret minimization (RRM) problem, which finds a size $r$ set to minimize the maximum rank-regret for all utility functions. Further, we generalize RRM and propose the restricted RRM (i.e., RRRM) problem to optimize the rank-regret for functions restricted in a given space. Previous studies on both RMS and RRR did not consider the restricted function space. The solution for RRRM usually has a lower regret level and can better serve the specific preferences of some users. Note that RRM and RRRM are shift invariant. In 2D space, we design a dynamic programming algorithm 2DRRM to return the optimal solution for RRM. In HD space, we propose an algorithm HDRRM that introduces a double approximation guarantee on rank-regret. Both 2DRRM and HDRRM are applicable for RRRM. Extensive experiments on the synthetic and real datasets verify the efficiency and effectiveness of our algorithms. In particular, HDRRM always has the best output quality in experiments.