Branch-and-bound Research Articles

A branch-and-bound algorithm for parametric mixed-binary nonlinear programs

AbstractAs rapid response to changes becomes more imperative, optimization under uncertainty has continued to grow in both the continuous and mixed-integer fields. We design a branch-and-bound (BB) algorithm for mixed-binary nonlinear optimization problems with parameters in general locations. At every node of the BB tree we apply a state-of-the-art algorithm we have recently developed to approximately optimize parametric programs containing objectives and constraints biconvex in the variables and parameters. Numerical results are included.

Optimal charging scheduling for Indoor Autonomous Vehicles in manufacturing operations

Indoor Autonomous Vehicles (IAVs) have become instrumental in modern logistics, particularly in dynamic operational environments. Their consistent availability is crucial for both timely task execution and energy efficiency in smart factories. Therefore, an inefficient charging schedule can waste energy, compromising production and warehousing efficiency. In addition, formulating an effective charging schedule is challenging due to the nonlinearity of its design and the uncertainties inherent in smart factory settings. In contrast to previous studies that either simplify the problem using linearization-based methods or approach it with computationally demanding algorithms, this paper efficiently addresses the nonlinear challenge, ensuring a closer alignment with real-world conditions. Hence, a simulation environment is first designed to replicate a smart factory, including validated models of IAVs, Charging Stations (CSs), and a variety of unpredictable static and dynamic obstacles. A comprehensive dataset is then provided from this simulation setup. Utilizing this dataset, a model tailored to ascertain the travel time of IAVs, accounting for inherent uncertainties, is trained using Deep Neural Networks (DNNs). Subsequently, a Mixed-Integer Nonlinear Programming (MINLP) problem is formulated to design the optimization task. Finally, integration of the DNNs’ model with the Branch-and-Bound (BnB) approach, forming the BnBD method, streamlines the determination of the optimal charging schedule. Experimental results highlight significant improvements in charging scheduling, establishing this approach as a viable and promising solution for manufacturing operations.

Nonconvexity and computational effort in the problem of Hydro-Power spillage policy assessment

This paper aims to assess the spillage policies in the Day-Ahead Hydro-Power Plant operation. For this purpose, a combination technique is proposed to prevent erroneous spillage decisions. Given that the developed mathematical modeling lacks the convexity property, we examine the multiple solutions obtained from the Hydro-Power production optimization subject to different power output formulations. In addition, a nonlinear approach to consider friction losses in the modeling is introduced. The modeling is described by Mixed-Integer Linear (MILP) and Integer Nonlinear (MINLP) Programming models. The optimal solutions are provided by Outer Approximation (OA), Branch-and-Bound (BB), and Interior Point algorithms. The simulations show that the water is managed distinctly depending on the optimizer and solution algorithms. Finally, the findings of the spillage policy obtained by the Mixed-Integer Nonlinear model under the cubic function for power generation encourage and indicate its adequacy for a more realistic operation and management of the Mexican Hydro-Power Plant water resources.

Combinatorial lower bounds for the Generalized Traveling Salesman Problem with Neighborhoods

In this paper, we study the Generalized Traveling Salesman Problem with Neighborhoods (GTSPN), a variant of the Traveling Salesman Problem (TSP), where the goal is to find the shortest path visiting each of the given neighborhood sets represented as a set of convex regions. The GTSPN is motivated by the sequencing problem of robotic manipulators, where an operation can be achieved from multiple locations, such as the visual inspection that can be performed from several possible views. The GTSPN formulation allows for exploiting continuous optimization to find the most suitable locations for the inspection, yielding possible solution cost reduction. Moreover, instances with overlapping high-dimensional convex regions further allow modeling neighborhood sets with complex shapes. We propose a novel approach to determine the first lower bounds to the studied GTSPN by employing the Branch-and-Bound (BB) method and the Mixed-Integer Second-Order Cone Programming (MISOCP) model for particular BB subproblems. In addition, the proposed method allows for solving the GTSPN to optima. The developed lower bound determination is further exploited in the empirical evaluation of existing heuristic approaches to the GTSPN and assesses the solution quality using the relative optimality gap. Regarding the presented results, the proposed BB-based approach provides tight lower bounds and solutions with up to 20% optimality gap for the GTSPN instances with less than 15 neighborhood sets for the given limited computational time. Furthermore, the presented results support that the proposed approach is suitable for solving high-dimensional instances of the GTSPN that can be found in inspection tasks with robotic manipulators.

OPTIMIZING CARTON PRODUCT DELIVERY BY SOLVING TRAVELLING SALESMAN PROBLEM AT PACKAGING COMPANIES

Optimization of delivery routes is one form of increasing business productivity to achieve the company's main goal of distributing each product to customers. The Traveling Salesman Problem (TSP) is a combinatorial optimization problem where a salesman goes on a distribution journey that starts from the depot, then visits all customers exactly once, and ends with returning to the depot. This study aims to optimize the distribution route of carton products for packaging companies using the TSP model. The research methodology includes observation, interviews, and document studies to understand the distribution process of carton products at packaging companies. To complete the TSP model, Branch and Bound (BB) and Nearest Neighbor (NN) methods are applied to find the best solution for determining the distribution route of carton products. The way the BB method works is by utilizing branch cuts and boundaries to reduce search space and speed up the resolution process. In the NN method, the nearest point is chosen to get the shortest route distance. Research findings show that the use of the BB method results in a mileage difference from the initial route of 125 km (53% more efficient) and a fuel cost difference of 121,231 IDR (45% cheaper). Meanwhile, the NN method results in a mileage difference from the initial route of 115 km (48.94%) and a fuel cost difference of 109,901 IDR (41.27%). So, the method that produces the best solution is the BB method. The limitations of this study lie in the scale of the model used and the assumptions underlying the analysis. Future research can broaden the scope of the model and consider other factors that may affect the distribution of carton products. The results of this study contribute to improving the efficiency of carton product distribution.

Neural network assisted branch and bound algorithm for dynamic berth allocation problems

One of the key challenges in maritime operations at container terminals is the need to improve or optimize berth operation schedules, thus allowing terminal operators to maximize the efficiency of quay usage. Given a set of vessels and a set of berths, the goal of the dynamic berth allocation problem is to determine the allocation of each vessel to a berth and the berthing time that minimizes the total service time. This problem can be solved using exact solution methods such as branch and bound (BB) algorithms or heuristic methods, however, exact methods do not scale to large-scale terminal operations. To this end, this paper proposes a BB algorithm in which branching decisions are made with a deep neural network. The proposed exact algorithm utilizes the search order of nodes based on the output of the neural network, with the goal of speeding up the search. Three types of solution representations are compared, along with machine learning models are created for each of them. Computational results confirm the effectiveness of the proposed method, which leads to computation times that are on average around half of those without the neural network.

A Branch-and-Bound Algorithm for the Molecular Ordered Covering Problem.

The Discretizable Molecular Distance Geometry Problem (DMDGP) plays a key role in the construction of three-dimensional molecular structures from interatomic distances acquired through nuclear magnetic resonance (NMR) spectroscopy, with the primary objective of validating a sequence of distance constraints related to NMR data. This article addresses the escalating complexity of the DMDGP encountered with larger and more flexible molecules by introducing a novel strategy via the Molecular Ordered Covering Problem, which optimizes the ordering of distance constraints to improve computational efficiency in DMDGP resolution. This approach utilizes a specialized Branch-and-Bound (BB) algorithm, tested on both synthetic and actual protein structures from the protein data bank. Our analysis demonstrates the efficacy of the previously proposed greedy heuristic in managing complex molecular scenarios, highlighting the BB algorithm's utility as a validation mechanism. This research contributes to ongoing efforts in molecular structure analysis, with possible implications for areas such as protein folding, drug design, and molecular modeling.

Multi-objective branch-and-bound for the design-for-control of water distribution networks with global bounds

This study investigates a design-for-control (DfC) problem formulation for the joint minimization of pressure-induced leakage, maximization of resilience, and minimization of cost in water distribution networks (WDN). The DfC problem, which consists in simultaneously installing new valves and/or pipes and optimizing valve control settings, results in a challenging optimization problem belonging to the class of non-convex multi-objective mixed-integer non-linear programs (MOMINLP). Due to their complex mathematical structure, multi-objective WDN design-for-control problems have previously been solved using general-purpose evolutionary algorithms or local deterministic methods, which do not provide guarantees on the quality of the returned solutions. While branch-and-bound (BB) frameworks have been proposed to approximate the Pareto fronts of MOMINLPs with global bounds, they rely on the availability of attainable solutions which, for WDN design-for-control problems, can be hard to identify. Moreover, in the absence of general-purpose solvers, the performance of multi-objective BB implementations depends, for a given application, on the choice of adequate branching and lower bounding strategies. In this study, we investigate a multi-objective BB algorithm based on tailored branching, lower bounding and additional upper bounding strategies to efficiently approximate the Pareto front of WDN design-for-control problems with bounds of ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon$$\\end{document}-non-dominance. The proposed algorithm is applied to the optimal DfC of two case study networks and is shown to outperform alternative solution methods.

A branch-and-bound based globally optimal solution to 2D multi-robot relative pose estimation problems

The relative pose estimation problem is important for multi-robot systems when they execute a variety of tasks cooperatively, such as cooperative sensing, multi-robot search and rescue, formation control. The finding of reliable solutions to this problem is essential, however, algorithms that guarantee to provide globally optimal solutions are still in demand. In this paper, an alternative dimensionality reduced objective function is proposed for the planar ground multi-robot relative pose estimation problem, and a new branch-and-bound (BnB) based method is developed, through which initialization is not required and the globally optimal solution is guaranteed under arbitrary noise levels. We demonstrate that the globally optimal solutions obtained via the algorithm proposed in this paper are superior than that obtained by existing classic methods such as SE-Sync algorithm when noise levels are high. Moreover, the method proposed is simple and easy to implement.

KD-Club: An Efficient Exact Algorithm with New Coloring-Based Upper Bound for the Maximum k-Defective Clique Problem

The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.

Efficient k-Clique Listing: An Edge-Oriented Branching Strategy

k-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(k · m · (δ/2)k-2 ), where m is the number of edges in the graph and δ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge ). We explore various edge orderings for EBBkC such that it achieves a time complexity of O( m · δ + k · m · (τ/2)k-2 ), where τ is an integer related to the maximum truss number of the graph and we have τ < δ. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m · δ) and O(k · m · (τ/2)k-2 ) are bounded by O(k · m · (δ/2)k-2 ). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC based) algorithms.

Sub‐optimal Internet of Thing devices deployment using branch and bound method

AbstractThe Internet of Thing (IoT) network deployments are widely investigated in 4G and 5G systems and will still be key technical systems to drive massive connectivity of 6G systems. In 6G, IoT systems will operate with 6G new technologies such as Integrated Sensing and Communications. The IoT systems of 6G will be a platform to collect information in real world and create new use cases and business models. As the IoT devices and cellular networks are getting smarter, the IoT ecosystem allows us to bridge between human life and digital life and accelerate the transition towards a hyper‐connected world. Optimal and scalable IoT network design has been investigated in many research groups but key challenges in this topic still remain. An IoT devices deployment problem is investigated to minimise the transmission and computation cost among network nodes. The IoT devices deployment problem is formulated as Mixed‐Integer Nonlinear Programming problem. After relaxing the constraints and transforming the problem to a mixed integer linear programming (MILP) problem, the authors propose a new branch and bound (BB) method with a machine learning function and solve the MILP problem as a sub‐optimal solution. In the numerical analysis, the authors evaluate both conventional BB method and the proposed BB method with weighting factors and compare the objective function values, the number of explored nodes, and computational time. The performances of the proposed BB method are significantly improved under the given simulation configuration. The author finds the optimal mapping of IoT devices to fusion nodes.

Solving Maximum Early Jobs Time and Range of Lateness Jobs Times Problem Using Exact and Heuristic Methods

In this paper, a solution to one of the Bicriteria Machine Scheduling Problems (BCMSP) is proposed. This problem focuses on the maximum early jobs time and range of lateness jobs time on a single machine (1//(E_max,R_L )). First, we derive a subproblem 1//(E_max+R_L ) from the main problem which is a special case for the suggested problem. Secondly, both exact complete enumeration and Branch and Bound (BAB) with two new lower bounds with some heuristic methods to solve the problems are proposed. The results prove the accuracy of BAB to solve the problem for n≤110 jobs in a reasonable time. In addition, the accuracy of the suggested heuristic methods is compared with the results of the exact methods.

CubeSat Mission Scheduling Method Considering Operational Reliability

Mission scheduling is an effective method to increase the value of satellite missions and can greatly improve satellite resource management and quality of service. Based on the priority-based task scheduling model, this paper proposes a CubeSat scheduling method that takes operational reliability into account, considering the impact of scheduling results on reliable operation. In this method, the available energy and the time window are used as scheduling resources, and the average state of charge of the lithium battery and the number of task start-ups are defined as two indices to measure its reliability. To meet the mission requirements and energy availability of photovoltaic (PV) solar panel and battery constraints, the scheduling model is constructed with an objective function that includes mission priority and reliability index. The branch and bound (BB) method and analytical hierarchy process (AHP) method are used to solve the scheduling problem. The example analysis compares different scheduling results and verifies the effectiveness of the proposed scheduling method. Compared with the existing methods, it comprehensively considers the mission value and operational reliability of the CubeSat, improves the energy reserve level of the CubeSat, and reduces the surge current caused by the start-up of tasks.

Using Local Search Methods for Solving Two Multi-Criteria Machine Scheduling Problems

In this paper, we have improved solutions for two of the Multi-Criteria Machine Scheduling Problems (MCMSP). These problems are to maximize early jobs time and range of lateness jobs times (1//(E_max,R_L ), and the second problem is maximum tardy jobs time and range of lateness jobs times (1//(T_max,R_L ) in a single machine with Multi-Objective Machine Scheduling Problems (MOMSP) 1//(E_max+R_L ) and 1//(T_max+R_L ) which are derived from the main problems respectively. The Local Search Methods (LSMs), Bees Algorithm (BA), and a Simulated Annealing (SA) are applied to solve all suggested problems. Finally, the experimental results of the LSMs are compared with the results of the Branch and Bound (BAB) method for a reasonable time. These results are ensuring the efficiency of LSMs.

Location Information Assisted Beamforming Design for Reconfigurable Intelligent Surface Aided Communication Systems

The large overhead arising from conventional channel estimations in reconfigurable intelligent surface (RIS) aided millimeter-wave communication systems, may offset the performance gain brought by the RIS. To tackle this issue, we propose a location information assisted beamforming design without the requirement of the channel training process. First, we establish the geometrical relationship between the channel model and the user location, and mathematically derive an approximate channel state information (CSI) error bound based on the user location error region. Then, for combating the negative impact of the location error on the communication performance, we formulate a worst-case robust beamforming optimization problem to optimize the beamformer at the base station (BS) and the phase-shift matrix at the RIS. To solve this non-convex problem, we develop a novel relaxed alternating optimization process (RAOP) by utilizing various optimization tools, such as the Lagrange multiplier, the matrix inversion lemma, the semidefinite relaxation (SDR), as well as the branch-and-bound (BnB). Additionally, we prove sufficient conditions for the SDR to yield rank-one solutions, and modify the BnB to acquire the phase-shift solution under an arbitrary constraint of possible phase-shift values. Finally, we analyse the convergence and complexity of the proposed RAOP, and carry out simulations for performance evaluations. Compared to the conventional non-robust beamforming, our method performs better and shows strong robustness against the location-error-related CSI uncertainty. Compared to the robust beamforming based on the S-procedure and penalty convex-concave procedure (CCP), our method with BnB shows the advantages of being able to converge faster and handle arbitrary phase-shift argument sets.

Time minimization and online synchronization for multi-agent systems under collaborative temporal logic tasks

Multi-agent systems can be efficient when solving a team-wide task in a concurrent manner. However, without proper synchronization, the correctness of the combined behavior is hard to guarantee, such as to follow a specific ordering of subtasks or to perform a simultaneous collaboration. This work addresses the minimum-time task planning problem for multi-agent systems under complex global tasks stated as syntactically co-safe Linear Temporal Logic (sc-LTL) formulas. These tasks include the temporal and spatial requirements on both independent local actions and direct sub-team collaborations. The proposed solution is an anytime algorithm that combines the partial-ordering analysis of the underlying task automaton for task decomposition, and the branch and bound (BnB) search method for task assignment. We analyze the soundness, completeness and optimality regarding the minimal completion time. We show that a feasible and near-optimal solution is quickly reached while the search continues within the time budget. Furthermore, to handle fluctuations in task duration and agent failures during online execution, we propose an adaptation algorithm to synchronize execution status and re-assign unfinished subtasks dynamically to maintain correctness and optimality. Both algorithms are validated over systems of more than 10 agents via numerical simulations and hardware experiments, against several established baselines.

Profit Maximization for Cache-Enabled Vehicular Mobile Edge Computing Networks

In this paper, we investigate a multiuser cache-enabled vehicular mobile edge computing (MEC) network, where one edge server (ES) has some caching and computing capabilities to assist the task computing from the vehicular users. The introduce of caching into the MEC network significantly affects the system performance such as the latency, energy consumption and profit at the ES, which imposes a critical challenge on the system design and optimization. To solve this challenge, we firstly design the vehicular MEC network in a non-competitive environment by maximizing the profit of the ES with a predetermined threshold of user QoE, and jointly exploit the caching and computing resources in the network. We then model the optimization problem into a binary integer programming problem, and adopt the cross entropy (CE) method to obtain the effective offloading and caching decision with a low complexity. Simulation results are finally presented to verify that the proposed scheme can achieve the near optimal performance of the conventional branch and bound (BnB) scheme, while sharply reduce the computational complexity compared to the BnB.

Hybrid trilinear and bilinear programming for aligning partially overlapping point sets

In many applications, we need algorithms which can align partially overlapping point sets and are invariant to the corresponding transformations. In this work, a method possessing such properties is realized by minimizing the objective of the robust point matching (RPM) algorithm. We first show that the RPM objective is a cubic polynomial. We then utilize the convex envelopes of trilinear and bilinear monomials to derive its lower bound function. The resulting lower bound problem has the merit that it can be efficiently solved via linear assignment and low dimensional convex quadratic programming. We next develop a branch-and-bound (BnB) algorithm which only branches over the transformation variables and runs efficiently. Experimental results demonstrated better robustness of the proposed method against non-rigid deformation, positional noise and outliers in case that outliers are not mixed with inliers when compared with the state-of-the-art approaches. They also showed that it has competitive efficiency and scales well with problem size.

Hybrid Learning with New Value Function for the Maximum Common Induced Subgraph Problem

Maximum Common Induced Subgraph (MCIS) is an important NP-hard problem with wide real-world applications. An efficient class of MCIS algorithms uses Branch-and-Bound (BnB), consisting in successively selecting vertices to match and pruning when it is discovered that a solution better than the best solution found so far does not exist. The method of selecting the vertices to match is essential for the performance of BnB. In this paper, we propose a new value function and a hybrid selection strategy used in reinforcement learning to define a new vertex selection method, and propose a new BnB algorithm, called McSplitDAL, for MCIS. Extensive experiments show that McSplitDAL significantly improves the current best BnB algorithms, McSplit+LL and McSplit+RL. An empirical analysis is also performed to illustrate why the new value function and the hybrid selection strategy are effective.