Non-convex Research Articles

Systematic Literature Review on Robust Optimization in Solving Sustainable Development Goals (SDGs) Problems during the COVID-19 Pandemic

Handling uncertainty is important in decision making, especially for SDGs problems. Robust Optimization (RO) is an applied optimization method that can be employed to handle optimization under uncertain data. With SDGs problems, many uncertain data have been considered in decision making. With RO, the data uncertainties are assumed to lay within a compact, convex continuous set. There are three special sets that can be used to represent the data, i.e., box, ellipsoidal, or polyhedral uncertainty sets. These special sets lead the SDGs problems to a computationally tractable optimization model, such that the global optimal solution is attained. However, literature reviews on the application of RO in SDGs decision-making is sparse, especially during the COVID-19 pandemic period. This paper examines the following topics: (1) the purposes of studies of RO and SDGs during the COVID-19 pandemic, (2) the state-of-the-art in RO-SDGs to determine the research objectives, and (3) the SDGs type of problems that have been modeled using RO. A systematic literature review is conducted in this paper, wherein discussion is based on a PRISMA (Preferred Reporting Items for Systematic Review and Meta-Analyses) flowchart. To this end, the database reference searching conducted on the Scopus, Science Direct, and SAGE databases, is completed using the help RStudio software. The analysis was carried out on two datasets, assisted by the output visualization using RStudio software with the “bibliometrix” package, and using the ‘biblioshiny()’ command to create a link to the “shiny web interface”. In this paper, the research gap on application of RO to SDGs problems is analyzed in order to identify the research objectives, methods, and specific RO-SDGs problems. As a result, the application of RO to SDGs problems is rare; this finding provides a motivation to conduct a further study of RO and SDGs during the COVID-19 pandemic. An expansion is presented using the key phrase “Operations Research and Optimization Modeling”, or “OROM”. SDGs in Indonesia may be referenced as an example of the capacity building available through RO/OROM.

Knowledge vector representation of three-dimensional convex polyhedrons and reconstruction of medical images using knowledge vector

Three-dimensional image construction and reconstruction plays an important role in various applications of the real world in the field of computer vision. In the last three decades, researchers are continually working in this area because construction and reconstruction is an important approach in medical imaging. Reconstruction of the 3D image allows us to find the lesion information of the patients which could offer a new and accurate approach for the diagnosis of the disease and it adds a clinical value. Considering this, a novel approach is proposed for the construction and reconstruction of the image. First, a syntactic pattern recognition-based algorithm is implemented to extract the features from the 2D image. The proposed algorithm takes an input image and extracts the features from the image and these features (Knowledge vector) consist of direction code and length. In addition, a unique and novel algorithm is developed that can rebuild an image using a knowledge vector. Reconstruction allows us to investigate the interior details of 3D images, such as the object’s size, form, and structure. The proposed algorithm performance is assessed on a medical imaging dataset, and the findings are outperformed. Performances of the proposed algorithms are evaluated on Kaggle brain MRI dataset and Medical MRI datasets which is collected from Pentagram research institute, Hyderabad. As per the experimental analysis, the proposed method gives 93.89% of accuracy on Kaggle brain MRI dataset and 97.24% on Medical MRI dataset which is better than exiting state of art methods.

Recognition of Affine-Equivalent Polyhedra by Their Natural Developments

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two convex polyhedra are isometric or not by only using their developments. We study a similar problem of whether it is possible to understand that two convex polyhedra in Euclidean 3-space are affine-equivalent by only using their developments.

Dual Function of 4-Aminothiophene in Surface-Enhanced Raman Scattering Application as an Internal Standard and Adsorbent for Controlling Au Nanocrystal Morphology.

The sensitivity and quantitative accuracy of surface-enhanced Raman scattering (SERS) are the main factors that restrict its application. Here, novel Au nanoscale convex polyhedrons (Au NCPs) were designed and fabricated to solve these problems via an embedded standard, including eight pods and six small protrusions. Spherical Au seeds regrew into different sizes of Au NCPs with a face-centered cubic structure. This morphology is due to the dual mechanism of the 4-aminothiophene (4-ATP) molecule that serves as an internal standard and a surface ligand regulator combined with the regulatory role of hexadecyl trimethyl ammonium chloride. The results show that Au NCPs were enclosed by high-index {12 9 1} facets, which greatly improved the local plasma resonance and reduced the lowest SERS detectable concentration of pyrene in standard seawater to 0.5 nM. An effective reference was produced by embedding 4-ATP with a relative standard deviation value less than 2.97% (in the same batch) and 3.92% (between different batches). Our research offers a new strategy for morphological regulation of metal nanocrystals, which is useful for the preparation of highly sensitive SERS substrates and trace analysis.

An algorithmic approach to Rupert’s problem

A polyhedron P ⊂ R 3 \mathbf {P} \subset \mathbb {R}^3 has Rupert’s property if a hole can be cut into it, such that a copy of P \mathbf {P} can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert’s property. A commonly believed conjecture states that every convex polyhedron is Rupert. We prove that Rupert’s problem is algorithmically decidable for polyhedra with algebraic coordinates. We also design a probabilistic algorithm which can efficiently prove that a given polyhedron is Rupert. Using this algorithm we not only confirm this property for the known Platonic and Archimedean solids, but also prove it for one of the remaining Archimedean polyhedra and many others. Moreover, we significantly improve on almost all known Nieuwland numbers and finally conjecture, based on statistical evidence, that the Rhombicosidodecahedron is in fact not Rupert.

Evaluation of Steel Slag Optimal Replacement in Asphalt Mixture under Microwave Heating Based on 3D Polyhedral Aggregate Electromagnetic-Thermal Meso-Model

Replacing conventional aggregate with steel slag waste can boost the microwave absorption properties of asphalt mixtures and reduce pollution to protect the environment. In order to achieve the best healing in steel slag asphalt mixture, the optimum particle size and content of steel slag are essential. For this purpose, a high-efficiency algorithm for the random growth and placement of convex polyhedron aggregate is proposed in this paper. The limestone aggregate is replaced with an equal volume of steel slag, and a three-dimensional mesoscale random model of steel slag asphalt mixture is developed. The process of microwave heating is simulated by FEM. The numerical simulations are compared with the reported experimental data, which proves that the model is reliable (R2 = 99.40%). Both the volume average temperature and the uniformity of temperature distribution indicate that the steel slag replacement rate of 60% at 4.75–9.5 mm and 60% at 9.5–13.2 mm is optimal, among which the heat transfer of 4.75–9.5 mm steel slag is more uniform, and the temperature gradient is lower. Steel slag can dramatically increase the heating rate of an asphalt mixture, and the peak of the temperature gradient is around the boundary of steel slag. The reflection properties of steel slag may be related to the dielectric constant, permeability, and particle size. Excess steel slag will cause overheating in most zones of the specimen and will also depress the absorption efficiency of microwaves. The coefficient of variance for spherical (0.36) and polyhedral (0.32) aggregate specimen temperatures indicates that the aggregate’s shape has a negligible effect on the heat transfer of asphalt mixtures.

Three-dimensional femtosecond snapshots of isolated faceted nanostructures.

The structure and dynamics of isolated nanosamples in free flight can be directly visualized via single-shot coherent diffractive imaging using the intense and short pulses of x-ray free-electron lasers. Wide-angle scattering images encode three-dimensional (3D) morphological information of the samples, but its retrieval remains a challenge. Up to now, effective 3D morphology reconstructions from single shots were only achieved via fitting with highly constrained models, requiring a priori knowledge about possible geometries. Here, we present a much more generic imaging approach. Relying on a model that allows for any sample morphology described by a convex polyhedron, we reconstruct wide-angle diffraction patterns from individual silver nanoparticles. In addition to known structural motives with high symmetries, we retrieve imperfect shapes and agglomerates that were not previously accessible. Our results open unexplored routes toward true 3D structure determination of single nanoparticles and, ultimately, 3D movies of ultrafast nanoscale dynamics.

On Domatic Number of Some Rotationally Symmetric Graphs

Domination is a well-known graph theoretic concept due to its significant real-world applications in several domains, such as design and communication network analysis, coding theory, and optimization. For a connected graph Γ = V , E , a subset U of V Γ is called a dominating set if every member present in V − U is adjacent to at least one member in U . The domatic partition is the partition of the vertices V Γ into the disjoint dominating set. The domatic number of the graph Γ is the maximum cardinality of the disjoint dominating sets. In this paper, we improved the results for the middle and central graphs of a cycle, respectively. Furthermore, we discuss the domatic number for some other cycle-related graphs and graphs of convex polytopes.

Digital twin-driven robust bi-level optimisation model for COVID-19 medical waste location-transport under circular economy

COVID-19 medical waste collection-transport system, including the location of the related facilities, transport, and disposal, is a critical component in the circular economy. To promote the circularity of the waste management system and mitigate the spread of novel coronavirus, how to optimise COVID-19 medical waste location-transport strategies remains an open but urgent issue. In this paper, a novel digital twin-driven conceptual framework is proposed to improve the strategic decision on the location of temporary disposal centres and, subsequently, the operational decision on the transport of COVID-19 medical waste in the presence of hierarchical relationships amongst stakeholders, circular economy, environmental regulations, service level, and uncertainty in infection probability. The polyhedral uncertainty set is introduced to characterise stochastic infection probability. Digital twin technology is further used to estimate the upper and lower bound of the uncertainty set. Such a problem is formulated as a digital twin-driven robust bi-level mixed-integer programming model to minimise total infection risks on the upper level and total costs on the lower level. A hybrid solution strategy is designed to combine dual theory, Karush-Kuhn-Tucker (KKT) conditions, and a branch-and-bound approach. Finally, a real case study from Maharashtra in India is presented to evaluate the proposed model. Results demonstrate that the solution strategy performs well for such a complex problem because the CPU time required to conduct all experiments is less than one hour. Under a given uncertainty level of 36 and perturbation ratio of 20%, a regional transport strategy is preferred from generation points to transfer points, while a cross-regional one is usually implemented from transfer points to disposal centres. It is of significance to determine the bound of available temporary disposal centres. Using digital technology (e.g., digital twin) to accurately estimate the amount of COVID-19 medical waste is beneficial for controlling the pandemic. Reducing infection risks relative to cost is the prioritised goal in cleaning up COVID-19 medical waste within a relatively long period.

Properties of 3-Triangulations for p-Toroid

In this paper, a method for constructing a toroid and its decomposition into convex pieces is considered. A graph of connection for 3-triangulable toroid is introduced in such a way that these pieces are represented by graph nodes. It is shown that connected, nonorientable graph can serve as a graph of connection for some of the toroids. The relationship between graphs that can be realized on surfaces of different genus and corresponding toroids is considered. Keywords: 3-triangulation of polyhedra, toroids, piecewise convex polyhedra, graph of connection

On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra

AbstractWe characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class . In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter continuity statement is based on Alexandrov geometry methods that we discuss, respectively, establish first.

Quantum states of H2+ and H2 in an icosahedral potential well

We investigate the potential energy surfaces (PESs) of the hydrogen-based cation H2+ and the neutral molecule H2 confined inside an infinite potential well in the shape of a regular icosahedron. The numerical computations are performed using the diffusion Monte Carlo method and are based on an analytical technique for obtaining simple equations of the surfaces of convex polyhedra proposed by S. Onaka. Different states and different orientations of the molecules inside the confining potential well, as well as various sizes of the latter, are studied. We provide a detailed symmetry analysis and consistent labeling of the H2+ states considered. The results show that the icosahedral confinement is closely isotropic in its inner region, leading to PESs that develop pronounced minima, as in the case of simpler confinement geometries. Shape-specific effects can be evidenced when the nuclei are in contact with the confining wall.

Constrained optimization of objective functions determined from random forests

In this paper, we examine a data‐driven optimization approach to making optimal decisions as evaluated by a trained random forest, where these decisions can be constrained by an arbitrary polyhedral set. We model this optimization problem as a mixed‐integer linear program. We show this model can be solved to optimality efficiently using pareto‐optimal Benders cuts for ensembles containing a modest number of trees. We consider a random forest approximation that consists of sampling a subset of trees and establish that this gives rise to near‐optimal solutions by proving analytical guarantees. In particular, for axis‐aligned trees, we show that the number of trees we need to sample is sublinear in the size of the forest being approximated. Motivated by this result, we propose heuristics inspired by cross‐validation that optimize over smaller forests rather than one large forest and assess their performance on synthetic datasets. We present two case studies on a property investment problem and a jury selection problem. We show this approach performs well against other benchmarks while providing insights into the sensitivity of the algorithm's performance for different parameters of the random forest.

An Automated Space-Based Graph Generation Framework for Building Energy Consumption Estimation

The 3D information in Building Information Modeling (BIM) has received significant interest for smart city applications. Recently, employing Industry Foundation Classes (IFC) for BIM in data-driven methods for Building Energy Consumption Estimation (BECE) has gained momentum because of the enriched geometric and semantic information. However, despite extensive studies on applying the IFC data in BECE analysis, employing the full potential of the BIM remains poor due to its complex data model and incompatibility with data-driven algorithms. This paper proposes a framework to extract accurate semantic, geometry, and topology information from the room-level (space) IFC schema by introducing new geo-computation algorithms to deal with these challenges. Additionally, we define a new topological weighted relationship between spaces in different stories by combining common geometry area with energy resistance value. Eventually, the proposed weighted space-based graph will be constructed to decrease the original complexity of the IFC model, and it is compatible with graph-based machine learning algorithms. The results are promising, with more than 90% accuracy in extracting the geometry information for the convex and non-convex polyhedron rooms and 100% accuracy in detecting vertical and horizontal adjacent rooms. This study confirms the proposed approach’s efficiency, accuracy, and feasibility for space-based BECE analysis.

Simple pursuit problem with phase restrictions of two coordinated evaders on time scales

In a finite-dimensional Euclidean space, the problem of a simple pursuit by a group of pursuers of two evaders is considered on a given time scale. It is assumed that all evaders use the same control and do not leave the convex polyhedral set. The pursuers use counter-strategies based on information about the initial positions and control history of the evaders. The set of admissible controls for each of the participants is a unit ball centered at zero, the target sets are the origin. The goal of the pursuers' group is the capture by two pursuers of at least one evader or the capture of two evaders. In terms of the initial positions and game parameters a sufficient capture condition has been obtained. The study uses the method of resolving functions as a basic one, which allows to obtain sufficient conditions for the solvability of the approach problem in some guaranteed time.

Model-Free Assortment Pricing with Transaction Data

We study the problem when a firm sets prices for products based on the transaction data, that is, which product past customers chose from an assortment and what were the historical prices that they observed. Our approach does not impose a model on the distribution of the customers’ valuations and only assumes, instead, that purchase choices satisfy incentive-compatible constraints. The uncertainty set of potential valuations of each past customer can then be encoded as a polyhedral set, and our approach maximizes the worst case revenue, assuming that new customers’ valuations are drawn from the empirical distribution implied by the collection of such polyhedra. We study the single-product case analytically and relate it to the traditional model-based approach. Then, we show that the optimal prices in the general case can be approximated at any arbitrary precision by solving a compact mixed-integer linear program. We further design three approximation strategies that are of low computational complexity and interpretable. In particular, the cutoff pricing heuristic has a competent provable performance guarantee. Comprehensive numerical studies based on synthetic and real data suggest that our pricing approach is uniquely beneficial when the historical data has a limited size or is susceptible to model misspecification. This paper was accepted by Omar Besbes, revenue management and market analytics. Funding: This work was supported by the Rotman School of Management (TD-MDAL Research Grant) and the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2015-06757, RGPIN-2020-04038, RGPIN-2020-06054, RGPIN-2021-04295]. Supplemental Material: The e-companion and data files are available at https://doi.org/10.1287/mnsc.2022.4651 .

Geometry of the skeleton graph of the no-signaling polytope for the 2223 Bell scenario and beyond

The study of no-signaling polytopes is addressed under an alternative approach, focusing on the skeleton graph of the polytopes rather than on the Bell inequalities solely. The paper starts with the study of the Bell scenario where one of the parties (Bob) has three output options (2223 scenario). We found that in this polytope the Popescu-Rohrlich (PR) vertices are interconnected, making up two networks around the polytope of local correlations, which is a new feature in relation to the 2222 polytope. We also found that in this scenario each CHSH inequality is violated by five PR vertices: the PR vertex associated to that inequality and its four adjacent PR vertices. Later, we study the facet structure of the corresponding polytope of local correlations. What we obtain is a one-to-one correspondence between some two-dimensional faces of the no-signaling polytope defined by specific loops in the adjacency networks of PR vertices, the lifting Bell functions, and the nontrivial facets of the local polytope. Finally, by studying the skeleton graphs of the polytopes for the general $222{v}_{B}$ scenarios, where ${v}_{B}$ is the number of possible Bob's outcomes, a striking high degeneracy is unveiled. We show that the set of local vertices exhibits a much higher degeneracy than the PR vertices, which in turn are more degenerate than a regular vertex.

New families of cage-like structures based on Goldberg polyhedra with non-isolated pentagons

AbstractA Goldberg polyhedron is a convex polyhedron made of hexagons and pentagons that have icosahedral rotational symmetry. Goldberg polyhedra have appeared frequently in art, architecture, and engineering. Some carbon fullerenes, inorganic cages, viruses, and proteins in nature exhibit the fundamental shapes of Goldberg polyhedra. According to Euler’s polyhedron formula, an icosahedral Goldberg polyhedron always has exactly 12 pentagons. In Goldberg polyhedra, all pentagons are surrounded by hexagons only—this is known as the isolated pentagon rule (IPR). This study systematically developed new families of cage-like structures derived from the initial topology of Goldberg polyhedra but with the 12 pentagons fused in five different arrangements and different densities of hexagonal faces. These families might be of great significance in biology and chemistry, where some non-IPR fullerenes have been created recently with chemical reactivity and properties markedly different from IPR fullerenes. Furthermore, this study has conducted an optimization for multiple objectives and constraints, such as equal edge length, equal area, planarity, and spherical shape. The optimized configurations are highly desirable for architectural applications, where a structure with a small number of different edge lengths and planar faces may significantly reduce the fabrication cost and enable the construction of surfaces with flat panels.

When Less Is More: Consequence-Finding in a Weak Theory of Arithmetic

This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived of as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication symbol, which retains many of the desirable algorithmic properties of linear arithmetic. In particular, we show that the conjunctive fragment of the theory can be effectively manipulated (analogously to the usual operations on convex polyhedra, the conjunctive fragment of linear arithmetic). As a result, we can solve the following consequence-finding problem: given a ground formula F , find the strongest conjunctive formula that is entailed by F . As an application of consequence-finding, we give a loop invariant generation algorithm that is monotone with respect to the theory and (in a sense) complete. Experiments show that the invariants generated from the consequences are effective for proving safety properties of programs that require non-linear reasoning.

Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints

We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of $$f(a^\top x)$$ , where f is a univariate concave function, a is a non-negative vector, and x is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when a has two distinct values. We show how to use these inequalities to obtain valid inequalities for general a that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when a contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework.