$L_p$-norm Regularization Algorithms for Optimization Over Permutation Matrices

Abstract

Abstract. Optimization problems over permutation matrices appear widely in facility layout, chip design, scheduling, pattern recognition, computer vision, graph matching, etc. Since this problem is NP-hard due to the combinatorial nature of permutation matrices, we relax the variable to be the more tractable doubly stochastic matrices and add an Lp-norm (0 < p < 1) regularization term to the objective function. The optimal solutions of the Lp-regularized problem are the same as the original problem if the regularization parameter is sufficiently large. A lower bound estimation of the nonzero entries of the stationary points and some connections between the local minimizers and the permutation matrices are further established. Then we propose an Lp regularization algorithm with local refinements. The algorithm approximately solves a sequence of Lp regularization subproblems by the projected gradient method using a nonmontone line search with the Barzilai-Borwein step sizes. Its performance can be further improved if it is combined with certain local search methods, the cutting plane techniques as well as a new negative proximal point scheme. Extensive numerical results on QAPLIB and the bandwidth minimization problem show that our proposed algorithms can often find reasonably high quality solutions within a competitive amount of time.

Max-Min Fairness User Scheduling and Power Allocation in Full-Duplex OFDMA Systems

“…For the converse part, assuming that (X,α,p d ,p u ) is a feasible solution to problem (27), we claim that the answer to the 3-dimensional matching problem is yes. Define…”

“…ρ 1 ≥ 0, ρ 2 ≥ 0, q ∈ (0, 1), 1 > 0 and 2 > 0 are some parameters, and 1 is the all-one vector. It has been shown in [27], [37] that for sufficiently large ρ 1 and ρ 2 , (45) can asymptotically have the same optimal solution as (40). Analogously, in the second stage of Algorithm 1, problem (44) is replaced by the following regularized problem…”

“…The complexity analysis in Section III also suggests that the full load case is the most difficult case to handle. To overcome this issue, we propose to employ the q -norm regularization and the iterative reweighted minimization (IRM) method in [27], [37] to tighten the relaxation in (37).…”

“…Specifically, according to [27], [37], the OFDMA constraints (7) and (8) can be well approximated by the simple relaxation (37) plus a proper q −norm regularization on X. For the TDA variable α, as α i +(1−α i ) = 1, for all i ∈ M always hold true, one can also approximate the binary constraint (11) by (37) and an q −norm regularization of α.…”

A Brief Introduction to Manifold Optimization

“…It is proved in [49] that it is equivalent to an L p -regularized optimization problem over the doubly stochastic matrices, which is much simpler than the original problem. An estimation of the lower bound of the non-zero elements at the stationary points are presented.…”