Weighted proximity search

Abstract

Proximity search is an iterative method to solve complex mathematical programming problems. At each iteration, the objective function of the problem at hand is replaced by the Hamming distance function to a given solution, and a cutoff constraint is added to impose that any new obtained solution improves the objective function value. A mixed integer programming solver is used to find a feasible solution to this modified problem, yielding an improved solution to the original problem. This paper introduces the concept of weighted Hamming distance that allows to design a new method called weighted proximity search. In this new distance function, low weights are associated with the variables whose value in the current solution is promising to change in order to find an improved solution, while high weights are assigned to variables that are expected to remain unchanged. The weights help to distinguish between alternative solutions in the neighborhood of the current solution, and provide guidance to the solver when trying to locate an improved solution. Several strategies to determine weights are presented, including both static and dynamic strategies. The proposed weighted proximity search is compared with the classic proximity search on instances from three optimization problems: the p-median problem, the set covering problem, and the stochastic lot-sizing problem. The obtained results show that a suitable choice of weights allows the weighted proximity search to obtain better solutions, for 75$$\%$$ % of the cases, than the ones obtained by using proximity search and for 96$$\%$$ % of the cases the solutions are better than the ones obtained by running a commercial solver with a time limit.

Improved sequential insertion heuristics for berth allocation problems

“…This metric was introduced in Rodrigues et al. (2021, 2022) and provides a global perspective on the number of best and worst solutions found by a given approach when compared to a reference approach (the traditional BLH in this case). To compute the value of this metric, we start by computing for each instance $$i\in \lbrace 1, \ldots , 10\rbrace$$ with a number of vessels $$N\in \lbrace 20, \ 30, \ 40, \ 50, \ 60, \ 70\rbrace$$ the value $$$$\begin{equation} \pi _{i,N} = {\begin{cases} 1, & \text{if }\ z_{i,N}^{X}<z^{BLH}_{i,N}, \\[6pt] 0, & \text{if }\ z_{i,N}^{X}=z^{BLH}_{i,N}, \\[6pt] -1, & \text{if }\ z_{i,N}^{X}>z^{BLH}_{i,N}, \end{cases}} \end{equation}$$$$where $$z^{BLH}_{i,N}$$ and $$z_{i,N}^{X}$$ are the total waiting time obtained, respectively, by the traditional BLH and by approach …”

Weighted iterated local branching for mathematical programming problems with binary variables

“…Values of this metric greater than 100, indicate a better average performance of the WILB comparing to the LBH. The second metric (Rodrigues et al 2021) provides a global perspective on the number of best solutions found by the WILB comparing to the LBH. To calculate this value, we start by computing for each instance h ∈ {1, 2, 3, 4} in class k ∈ K the value…”

“…However, this function also does not provide guidance to distinguish between the quality of equally distant solutions, nor does it incorporate available information regarding the preferred values of variables. In a recent paper (Rodrigues et al 2021) a proximity search heuristic using different weights for individual variables was successfully applied to three different combinatorial optimization problems. This provided evidence that it can be useful, prior to exploring a neighborhood of solutions, to evaluate the possible consequences of flipping binary variables and weighting the variables accordingly when performing the neighborhood exploration.…”

The nilcatenation problem and its application for detecting money laundering activities in cryptocurrency networks

“…For future works, we consider improving the parameter setting of our local branching algorithm and developing other algorithmic features such as the inclusion of valid inequalities and new branching schemes. Moreover, we consider implementing improved variants of local branching such as Fischetti and Monaci (2014) and Rodrigues et al (2021). Also, we consider using a completely different model with tighter linear relaxation bounds.…”