Discrete Optimization Research Articles

Information theoretic clustering for coarse-grained modeling of non-equilibrium gas dynamics

We present a new framework towards the objective of learning coarse-grained models based on the maximum entropy principle. We show that existing methods for assigning clusters using the maximum entropy approach are heuristic or sub-optimal. We propose a machine learning framework informed by rate-distortion theory to learn optimal cluster assignments, aiming to improve the effectiveness of the coarse-graining process. Our approach involves transforming the discrete optimization problem into a probabilistic and continuous form. Our inverse modeling approach involves the development of a fully differentiable, adjoint-driven dynamics solver for use with gradient-based optimization techniques. The entire framework is end-to-end differentiable, facilitating backward pass gradient computation to flow through the ODE solve and probabilistic coarse-graining to train a classifier. We demonstrate application in the evolution of particle quantum states in non-equilibrium conditions. The high dimensionality of these equations poses a significant challenge for efficient and accurate computations, even for seemingly simple chemical systems. Training is performed using a loss function informed by rate-distortion theory to cluster the rovibrational states of some molecule, effectively forming a reduced order model of the master equations. We also introduce variable transformations along with several ways of improving the tractability of the training process. These allow the framework to process the extremely high-dimensional discrete optimization problem successfully. Our method is general, and to demonstrate the effectiveness in a controlled setting, we apply it to a one-dimensional Gaussian source quantization problem for which an analytical solution is known, followed by the problem of isothermal relaxation of a ▪ system with O(104) degrees of freedom.

Discrete Element Model Building and Optimization of Tomato Stalks at Harvest

The mechanical properties of tomato stalk, relevant to the harvesting and crushing of tomato vines, significantly impact its harvesting quality and efficiency. Establishing a simulation model, which accurately mirrors these properties, is foundational for designing related mechanical components. The discrete element method models tomato stalk harvesting and is optimized through mechanical tests and simulations. A blend of Plackett–Burman, steepest ascent, and central composite design modeling identified three contact model parameters influencing the maximum stalk shear force. The optimal values of these three parameters were a normal stiffness of 1.04 × 1010 N m−3, tangential stiffness of 7.59 × 109 N m−3, and bond radius of 1.06 mm. The relative error in the simulated versus measured shear force was <1%, affirming the model’s accuracy in characterizing cutting properties. These findings lay the theoretical groundwork for numerical simulations of tomato-stalk-related equipment.

An Optimized Recurrent Neural Network for re-modernize food dining bowls and estimating food capacity from images

Maintaining health and preventing chronic diseases requires awareness of the energy and nutrient content of one's diet. Due to the rapid development of computing and Artificial intelligence (AI) technology, nutritional evaluation can be performed using smartphones or wearable food images.Thechallenges of this technology include estimating the visual estimation of the amount of food in a bowl. We describe an innovative approach for determining a bowl's dimensions by sticking a paper ruler throughout the bottom and edges yet photographing the outcome. We used a variety of colored liquids, bowlsand crystallized foods to evaluate the precision of our technique for measuring food volume employing spherical bowls as boxes. Different food-related datasets can have other numbers of images, types of food dining bowls and estimated food capacity from the images represented. Image preprocessing using a wiener filter enhances the estimating of food dining bowl capacity from images. Following preprocessing, a threshold-based approach to image segmentation is used for the resulting data; finally, we used the Gradient Lion Swarm Optimized Discrete Recurrent Neural Network (GLSO-DRNN) model to present an AI-based multi-dish food identification approach. An excellent mAP = 0.96 was found between the results. The suggested model shows great promise as a means of improving dish reporting because of its high classification performance.The GLSO-DRNN performed better toproving its flexibility and efficient discrete parameter optimization. In addition to offering accurate food quantity estimates, it demonstrated significant potential for other uses in the field of smart dining technology. The results highlight aneffective ofthis advanced computational method that is improving food dining bowlsand interactions with components associated with food. In the future, we plan to improve the techniques' performance and incorporate our technology into a working Smartphone based on the cloud environment.

Manipulation order optimization in industrial pick-and-place operations: application to textile and leather industry

This work addresses the problem of the development of a robotic system for the picking of parts cut by a CNC machine and the optimization of the sequencing of this picking process. An automated parts collection system is optimized to reduce the time required to perform the task of both picking and the subsequent classification by the type of part. The automated picking system, which is located at the end of a cutting machine, uses a robot equipped with an additional axis to expand its working space. Therefore, in this proposal, the industrial equipment necessary to automate this process is designed and the process to be optimized is computationally modeled. In particular, three discrete optimization algorithms are analyzed, with different evolution strategies and operators, but all of them are free of specific configuration parameters. The whole process is shown in this research, from the design of the procedure to the design of the tool, the algorithm selection, and elements validation. Finally, the first steps towards its industrial implementation are presented, and the hypothesis behind this project is validated.

Multi-objective optimization approach for coverage path planning of mobile robot

AbstractCoverage path planning (CPP) is a subfield of path planning problems in which free areas of a given domain must be visited by a robot at least once while avoiding obstacles. In some situations, the path may be optimized for one or more criteria such as total distance traveled, number of turns, and total area covered by the robot. Accordingly, the CPP problem has been formulated as a multi-objective optimization (MOO) problem, which turns out to be a challenging discrete optimization problem, hence conventional MOO algorithms like Non-dominated Sorting Genetic Algorithm-2 (NSGA-II) do not work as it is. This study implements a modified NSGA-II to solve the MOO problem of CPP for a mobile robot. In this paper, the proposed method adopted two objective functions: (1) the total distance traveled by the robot and (2) the number of turns taken by the robot. The two objective functions are used to calculate energy consumption. The proposed method is compared to the hybrid genetic algorithm (HGA) and the traditional genetic algorithm (TGA) in a rectilinear environment containing obstacles of various complex shapes. In addition, the results of the proposed algorithm are compared to those generated by HGA, TGA, oriented rectilinear decomposition, and spatial cell diffusion and family of spanning tree coverage in existing research papers. The results of all comparisons indicate that the proposed algorithm outperformed the existing algorithms by reducing energy consumption by 5 to 60%. This paper provides the facility to operate the robot in different modes.

Accelerating Cutting-Plane Algorithms via Reinforcement Learning Surrogates

Discrete optimization belongs to the set of N P-hard problems, spanning fields such as mixed-integer programming and combinatorial optimization. A current standard approach to solving convex discrete optimization problems is the use of cutting-plane algorithms, which reach optimal solutions by iteratively adding inequalities known as cuts to refine a feasible set. Despite the existence of a number of general-purpose cut-generating algorithms, large-scale discrete optimization problems continue to suffer from intractability. In this work, we propose a method for accelerating cutting-plane algorithms via reinforcement learning. Our approach uses learned policies as surrogates for N P-hard elements of the cut generating procedure in a way that (i) accelerates convergence, and (ii) retains guarantees of optimality. We apply our method on two types of problems where cutting-plane algorithms are commonly used: stochastic optimization, and mixed-integer quadratic programming. We observe the benefits of our method when applied to Benders decomposition (stochastic optimization) and iterative loss approximation (quadratic programming), achieving up to 45% faster average convergence when compared to modern alternative algorithms.

DOGE-Train: Discrete Optimization on GPU with End-to-End Training

We present a fast, scalable, data-driven approach for solving relaxations of 0-1 integer linear programs. We use a combination of graph neural networks (GNN) and a Lagrange decomposition based algorithm. We make the latter differentiable for end-to-end training and use GNNs to predict its algorithmic parameters. This allows to retain the algorithm's theoretical properties including dual feasibility and guaranteed non-decrease in the lower bound while improving it via training. We overcome suboptimal fixed points of the basic solver by additional non-parametric GNN update steps maintaining dual feasibility. For training we use an unsupervised loss. We train on smaller problems and test on larger ones showing strong generalization performance with a GNN comprising only around 10k parameters. Our solver achieves significantly faster performance and better dual objectives than its non-learned version, achieving close to optimal objective values of LP relaxations of very large structured prediction problems and on selected combinatorial ones. In particular, we achieve better objective values than specialized approximate solvers for specific problem classes while retaining their efficiency. Our solver has better any-time performance over a large time period compared to a commercial solver.

Using Clustering to Strengthen Decision Diagram Bounds for Discrete Optimization

Offering a generic approach to obtaining both upper and lower bounds, decision diagrams (DDs) are becoming an increasingly important tool for solving discrete optimization problems. In particular, they provide a powerful and often complementary alternative to other well-known generic bounding mechanisms such as the LP relaxation. A standard approach to employ DDs for discrete optimization is to formulate the problem as a Dynamic Program and use that formulation to compile a DD top-down in a layer-by-layer fashion. To limit the size of the resulting DD and to obtain bounds, one typically imposes a maximum width for each layer which is then enforced by either merging nodes (resulting in a so-called relaxed DD that provides a dual bound) or by dropping nodes (resulting in a so-called restricted DD that provides a primal bound). The quality of the DD bounds obtained from this top-down compilation process heavily depends on the heuristics used for the selection of the nodes to merge or drop. While it is sometimes possible to engineer problem-specific heuristics for this selection problem, the most generic approach relies on sorting the layer’s nodes based on objective function information. In this paper, we propose a generic and problem-agnostic approach that relies on clustering nodes based on the state information associated with each node. In a set of computational experiments with different knapsack and scheduling problems, we show that our approach generally outperforms the classical generic approach, and often achieves drastically better bounds both with respect to the size of the DD and the time used for compiling the DD.

Model Counting and Sampling via Semiring Extensions

Many decision and optimization problems have natural extensions as counting problems. The best known example is the Boolean satisfiability problem (SAT), where we want to count the satisfying assignments of truth values to the variables, which is known as the #SAT problem. Likewise, for discrete optimization problems, we want to count the states on which the objective function attains the optimal value. Both SAT and discrete optimization can be formulated as selective marginalize a product function (MPF) queries. Here, we show how general selective MPF queries can be extended for model counting. MPF queries are encoded as tensor hypernetworks over suitable semirings that can be solved by generic tensor hypernetwork contraction algorithms. Our model counting extension is again an MPF query, on an extended semiring, that can be solved by the same contraction algorithms. Model counting is required for uniform model sampling. We show how the counting extension can be further extended for model sampling by constructing yet another semiring. We have implemented the model counting and sampling extensions. Experiments show that our generic approach is competitive with the state of the art in model counting and model sampling.

Symbolic Cognitive Diagnosis via Hybrid Optimization for Intelligent Education Systems

Cognitive diagnosis assessment is a fundamental and crucial task for student learning. It models the student-exercise interaction, and discovers the students' proficiency levels on each knowledge attribute. In real-world intelligent education systems, generalization and interpretability of cognitive diagnosis methods are of equal importance. However, most existing methods can hardly make the best of both worlds due to the complicated student-exercise interaction. To this end, this paper proposes a symbolic cognitive diagnosis~(SCD) framework to simultaneously enhance generalization and interpretability. The SCD framework incorporates the symbolic tree to explicably represent the complicated student-exercise interaction function, and utilizes gradient-based optimization methods to effectively learn the student and exercise parameters. Meanwhile, the accompanying challenge is that we need to tunnel the discrete symbolic representation and continuous parameter optimization. To address this challenge, we propose to hybridly optimize the representation and parameters in an alternating manner. To fulfill SCD, it alternately learns the symbolic tree by derivative-free genetic programming and learns the student and exercise parameters via gradient-based Adam. The extensive experimental results on various real-world datasets show the superiority of SCD on both generalization and interpretability. The ablation study verifies the efficacy of each ingredient in SCD, and the case study explicitly showcases how the interpretable ability of SCD works.

GSO-Net: Grid Surface Optimization via Learning Geometric Constraints

In the context of surface representations, we find a natural structural similarity between grid surface and image data. Motivated by this inspiration, we propose a novel approach: encoding grid surfaces as geometric images and using image processing methods to address surface optimization-related problems. As a result, we have created the first dataset for grid surface optimization and devised a learning-based grid surface optimization network specifically tailored to geometric images, addressing the surface optimization problem through a data-driven learning of geometric constraints paradigm. We conduct extensive experiments on developable surface optimization, surface flattening, and surface denoising tasks using the designed network and datasets. The results demonstrate that our proposed method not only addresses the surface optimization problem better than traditional numerical optimization methods, especially for complex surfaces, but also boosts the optimization speed by multiple orders of magnitude. This pioneering study successfully applies deep learning methods to the field of surface optimization and provides a new solution paradigm for similar tasks, which will provide inspiration and guidance for future developments in the field of discrete surface optimization. The code and dataset are available at https://github.com/chaoyunwang/GSO-Net.

A dual-robot cooperative arc welding path planning algorithm based on multi-objective cross-entropy optimization

In this paper a novel discrete multi-objective cross-entropy optimization (CrMOCEO) algorithm is proposed to solve the path planning problem of dual-robot cooperative arc welding. We strive to find a low-cost, fast and more efficient solution for robotic welding of large complex components. Firstly, an optimization model of dual-robot welding path planning is established by considering various variables and constraints in the actual welding process. Then, three strategies are introduced to improve the multi-objective cross-entropy optimization (MOCEO) algorithm to better solve the discrete path planning problem. Finally, in order to verify feasibility and effectiveness of the proposed algorithm, it is used to solve the 2-, 3-, 5- and 7-objective WFG2–9 problems and plan some typical welding seams of a large complex component, the MOCEO, NSGA-Ⅱ, MOPSO and MOGWO are used for comparison. The simulation demonstrates that the CrMOCEO can obtain better solutions for multiple objectives than the other four algorithms, and the path solved by the CrMOCEO is tested in the Gazebo physical model and workshop site, the results further verified the effectiveness of the CrMOCEO algorithm. Particularly, a series of experiments provide more solutions for actual production.

On Integrality in Semidefinite Programming for Discrete Optimization

On Integrality in Semidefinite Programming for Discrete Optimization

A Discrete Adaptive Lion Optimization Algorithm for QoS-Driven IoT Service Composition with Global Constraints

A Discrete Adaptive Lion Optimization Algorithm for QoS-Driven IoT Service Composition with Global Constraints

Algorithms for the Reconstruction of Genomic Structures with Proofs of Their Low Polynomial Complexity and High Exactness

The mathematical side of applied problems in multiple subject areas (biology, pattern recognition, etc.) is reduced to the problem of discrete optimization in the following mathematical method. We were provided a network and graphs in its leaves, for which we needed to find a rearrangement of graphs by non-leaf nodes, in which the given functional reached its minimum. Such a problem, even in the simplest case, is NP-hard, which means unavoidable restrictions on the network, on graphs, or on the functional. In this publication, this problem is addressed in the case of all graphs being so-called “structures”, meaning directed-loaded graphs consisting of paths and cycles, and the functional as the sum (over all edges in the network) of distances between structures at the endpoints of every edge. The distance itself is equal to the minimal length of sequence from the fixed list of operations, the composition of which transforms the structure at one endpoint of the edge into the structure at its other endpoint. The list of operations (and their costs) on such a graph is fixed. Under these conditions, the given discrete optimization problem is called the reconstruction problem. This paper presents novel algorithms for solving the reconstruction problem, along with full proofs of their low error and low polynomial complexity. For example, for the network, the problem is solved with a zero error algorithm that has a linear polynomial computational complexity; and for the tree the problem is solved using an algorithm with a multiplicative error of at most two, which has a second order polynomial computational complexity.

A Lagrangian bounding and heuristic principle for bi-objective discrete optimization

Lagrangian relaxation is a common and often successful way to approach computationally challenging single-objective discrete optimization problems with complicating side constraints. Its aim is often twofold; first, it provides bounds for the optimal value, and, second, it can be used to heuristically find near-optimal feasible solutions, the quality of which can be assessed by the bounds. We consider bi-objective discrete optimization problems with complicating side constraints and extend this Lagrangian bounding and heuristic principle to such problems. The Lagrangian heuristic here produces non-dominated candidates for points on the Pareto frontier, while the bounding forms a polyhedral outer approximation of the Pareto frontier, which can be used to assess the quality of the candidate points. As an illustration example we consider a facility location problem in which both CO2 emission and cost should be minimized. The computational results are very encouraging, both with respect to bounding and the heuristically found non-dominated solutions. In particular, the Lagrangian bounding is much stronger than the outer approximation given by the Pareto frontier of the problem’s linear programming relaxation.

SEACDSC: secure and energy-aware clustering based on discrete sand cat swarm optimization for IoT-enabled WSN applications

SEACDSC: secure and energy-aware clustering based on discrete sand cat swarm optimization for IoT-enabled WSN applications

Special Issue “Recent Advances of Discrete Optimization and Scheduling”

This Special Issue of the journal Mathematics is dedicated to new results on the topic of discrete optimization and scheduling [...]

Advanced vehicle routing in cement distribution: a discrete Salp Swarm Algorithm approach

ABSTRACT In this research, the focus is on addressing the capacitated vehicle routing problem (CVRP), a prominent logistical challenge within transportation management. The discrete salp swarm algorithm (DSSA) has been introduced as an innovative adaptation of the traditional salp swarm algorithm (SSA), specifically redesigned to cater to discrete characteristics inherent in problems like the CVRP. This algorithm integrates the core principles of SSA with mutation and crossover techniques, enhancing its applicability for discrete problems. The paper contributes in two main areas: firstly, the development of DSSA, tailored to address the unique requirements of VRP scenarios and ensuring a balanced approach between exploration and exploitation in discrete optimization. Secondly, the effectiveness of DSSA is demonstrated through practical applications, including an 8-customer routing task and a real-world case study involving cement delivery in Vietnam. In these scenarios, DSSA consistently demonstrates superior performance over other meta-heuristic strategies, marking a noteworthy advancement in the domain of optimization and providing a potential tool for complex routing challenges in logistics and distribution systems.

Quantum approximate optimization via learning-based adaptive optimization

Combinatorial optimization problems are ubiquitous and computationally hard to solve in general. Quantum approximate optimization algorithm (QAOA), one of the most representative quantum-classical hybrid algorithms, is designed to solve combinatorial optimization problems by transforming the discrete optimization problem into a classical optimization problem over continuous circuit parameters. QAOA objective landscape is notorious for pervasive local minima, and its viability significantly relies on the efficacy of the classical optimizer. In this work, we design double adaptive-region Bayesian optimization (DARBO) for QAOA. Our numerical results demonstrate that the algorithm greatly outperforms conventional optimizers in terms of speed, accuracy, and stability. We also address the issues of measurement efficiency and the suppression of quantum noise by conducting the full optimization loop on a superconducting quantum processor as a proof of concept. This work helps to unlock the full power of QAOA and paves the way toward achieving quantum advantage in practical classical tasks.