Non-convex Research Articles

Devising a numerical method for estimating the positioning accuracy of aircraft by an information- communication network of optoelectronic stations

The object of this study is the accuracy of aircraft positioning for open and covert video surveillance by an infocommunication network of optical-electronic stations along the trajectories of their movement. The task addressed is numerical assessment of the accuracy of aircraft positioning in airspace. It is proposed to use a convex polyhedron as a universal assessment of the accuracy of aircraft positioning, in which, with a given probability, the aircraft is located. It is shown that the lower estimate of this probability depends on the a priori information on the statistical properties of the errors in the estimates of the coordinates of the aircraft location, and the scattering ellipsoid, which is currently the main form of assessing the accuracy of aircraft positioning in airspace, is a special case and is always located inside a convex polyhedron. The results reported here include the following: – simulation models of open and covert video surveillance by an infocommunication network of optoelectronic stations along the trajectories of aircraft movement; – a numerical method for estimating the uncertainty region in the form of a convex polyhedron, in which, with a given probability, the aircraft is located; – dependence of change in the shapes and boundaries of the convex polyhedron on the errors of video surveillance and the mutual spatial location of the aircraft and the network of optoelectronic stations; – software implementation of methods for constructing and visualizing the shapes and boundaries of uncertainty regions in the form of convex polyhedrons and scattering ellipsoids. It is shown that the aircraft is inside the convex polyhedron with the probability P ≥ 0.8889 for any distribution, P ≥ 0.9506 for a symmetric one and P ≥ 0.9973 for a normal distribution

A Dual-Uncertainty Multi-Scenario Multi-Period Facility Location Model for Post-Disaster Humanitarian Logistics

The frequent occurrence of natural disasters creates a symmetry-breaking scenario between pre-disaster planning and post-disaster rescue operations, such as post-disaster supply–demand mismatches for materials and the risk of potential facility failures. Thus, we propose a dual-uncertainty multi-scenario multi-period facility location allocation model for humanitarian rescue. The model employs two polyhedral uncertainty sets to represent facility failure risks and demand uncertainty at disaster points. Moreover, by constructing diverse disaster scenarios, it simulates material distribution schemes across different relief periods, enhancing its realism. Given that the model integrates three subproblems—facility location, supply–demand matching analysis, and emergency material allocation—we design a hybrid algorithm (DCSA-MA) that combines the discrete crow search algorithm (DCSA) and the material allocation (MA) method for its solution. Experimental results demonstrate that the model maintains a relatively high material satisfaction rate even under significant demand fluctuations. The number of facility failures has a direct bearing on emergency rescue effectiveness. The DCSA-MA method achieves a superior material satisfaction rate compared to other algorithms across various disaster scenarios and multiple rescue periods. Furthermore, DCSA-MA outperforms other algorithms in terms of solution quality, convergence, computational time, and stability. These findings indicate that DCSA-MA is an effective and highly stable approach.

Affinely adjustable robust optimal scheduling of a power system considering the flexibility of supply and demand

Abstract As the integration of renewable energy sources, such as wind power and photovoltaics, continues, the issue of system uncertainty has become more pronounced. This paper proposes a stochastic power system dispatch method based on affinely adjustable robust optimization (AARO) with a generalized linear polyhedron (GLP) uncertainty set that can accurately quantify the flexibility of the power system supply and demand as well as enhance the optimality of dispatch strategies. First, a GLP uncertainty set was established to characterize both the temporal stochasticity and spatial correlation of multiple renewable energy outputs. A correlation envelope was employed to reflect renewable energy outputs from historical data, and a polyhedral set was proposed to accurately describe the uncertainty for model formulation, which can effectively reduce model conservatism by minimizing empty regions. Furthermore, the range of net load variations was analysed to build a demand flexibility quantification model for the power system. Next, based on the expected operational value, a robust optimization dispatch model that considers the flexible supply and demand balance is developed within the affine strategy framework. Finally, simulations of a modified 6‐bus system and modified IEEE 57‐bus system validate the effectiveness of the proposed GLP‐AARO method for power system flexibility quantification and dispatch strategy optimization.

Modular Construction and Optimization of the UZP Sparse Format for SpMV on CPUs

Sparse data structures are ubiquitous in modern computing, and numerous formats have been designed to represent them. These formats may exploit specific sparsity patterns, aiming to achieve higher performance for key numerical computations than more general-purpose formats such as CSR and COO. In this work we present UZP, a new sparse format based on polyhedral sets of integer points. UZP is a flexible format that subsumes CSR, COO, DIA, BCSR, etc., by raising them to a common mathematical abstraction: a union of integer polyhedra, each intersected with an affine lattice. We present a modular approach to building and optimizing UZP: it captures equivalence classes for the sparse structure, enabling the tuning of the representation for target-specific and application-specific performance considerations. UZP is built from any input sparse structure using integer coordinates, and is interoperable with existing software using CSR and COO data layout. We provide detailed performance evaluation of UZP on 200+ matrices from SuiteSparse, demonstrating how simple and mostly unoptimized generic executors for UZP can already achieve solid performance by exploiting Z-polyhedra structures.

Recovering the polyhedral geometry of fragments.

Recovering the polyhedral geometry of fragments.

Viewpoint Selection for 3D Scenes in Map Narratives

Narrative mapping, an advanced geographic information visualization technology, presents spatial information episodically, enhancing readers’ spatial understanding and event cognition. However, during 3D scene construction, viewpoint selection is heavily reliant on the cartographer’s subjective interpretation of the event. Even with fixed-angle settings, the task of ensuring that selected viewpoints align with the narrative theme remains challenging. To address this, an automated viewpoint selection method constrained by narrative relevance and visual information is proposed. Narrative relevance is determined by calculating spatial distances between each element and the thematic element within the scene. Visual information is quantified by assessing the visual salience of elements as the ratio of their projected area on the view window to their total area. Pearson’s correlation coefficient is used to evaluate the relationship between visual salience and narrative relevance, serving as a constraint to construct a viewpoint fitness function that integrates the visual salience of the convex polyhedron enclosing the scene. The chaotic particle swarm optimization (CPSO) algorithm is utilized to locate the viewpoint position while maximizing the fitness function, identifying a viewpoint meeting narrative and visual salience requirements. Experimental results indicate that, compared to the maximum projected area method and fixed-value method, a higher viewpoint fitness is achieved by this approach. The narrative views generated by this method were positively recognized by approximately two-thirds of invited professionals. This process aligns effectively with narrative visualization needs, enhances 3D narrative map creation efficiency, and offers a robust strategy for viewpoint selection in 3D scene-based narrative mapping.

Adaptive data driven multi period power supply recovery method for distribution networks

In the process of distribution network fault recovery, in order to better address the issues caused by the uncertainty of new energy power output, this paper proposes a multi period power supply recovery method for distribution networks based on adaptive data driven approach. Firstly, this method uses the historical data of new energy power output to construct an ellipsoidal uncertainty set, and forms a data driven convex hull polyhedral set by connecting the vertices of the high-dimensional ellipsoid. Then, aiming at the problem of large conservatism during the reduction process of the convex hull polyhedral set, based on the range of the box set, cutting planes are made starting from the vertices, and a data driven hyperplane polyhedral set model is constructed. Furthermore, considering the constraints of cyber-physical integration, an adaptive data driven power supply recovery model for distribution networks is established, and the column and constraint generation (C&CG) algorithm is adopted to solve the robust scheduling model. Finally, simulations on the improved IEEE-33 bus system and actual example systems verify that the adaptive data driven power supply recovery model for distribution networks can reduce conservatism and improve the robustness of the optimization results. This power supply recovery model can reduce conservatism and enhance the robustness of the optimization results. In the actual distribution network fault recovery scenarios, it can make more efficient use of new energy and ensure the stability of power supply, which strongly demonstrates the effectiveness and application value of the proposed method.

Efficient Simulation of Polyhedral Expectations with Applications to Finance

We consider the problem of estimating the expectation over a convex polyhedron specified by a set of linear inequalities. This problem encompasses a multitude of financial applications, including systemic risk quantification, exotic option pricing, and portfolio management. We particularly focus on the case where the target event is rare, which corresponds to extreme systemic failures, deep out-of-the-money options, and high target returns in the aforementioned applications, respectively. This rare-event setting renders the naive Monte Carlo method inefficient and requires the use of variance reduction techniques. To address this issue, we develop a novel and strongly efficient method for the computation of the said expectation in a general rare-event setting by exploiting the geometry of the target polyhedron and concentrating the sampling density almost within the polyhedron. The proposed method significantly outperforms the existing approaches in various numerical experiments in terms of accuracy and computational costs. Funding: This research was supported by the Early Career Scheme from the Research Grants Council of Hong Kong, University Grants Committee [Grant CUHK 24210420] and the Chinese University of Hong Kong (CUHK) Direct Grant for Research [Grant 4055206].

A Lagrange function approach to study second-order optimality conditions for infinite-dimensional optimization problems

In this paper, we focus on the second-order optimality conditions for infinite-dimensional optimization problems constrained by generalized polyhedral convex sets. Our aim is to further explore the role of the generalized polyhedral convex property, which is inspired by the findings of other authors. To this end, we employ the concept of Fréchet second-order subdifferential, a tool in variational analysis, to establish optimality conditions. Furthermore, by applying this concept to the Lagrangian function associated with the problem, we are able to derive refined optimality conditions that surpass existing results. The unique properties of generalized polyhedral convex sets play a crucial role in enabling these improvements.

On a Nash curve selection lemma through finitely many points

A celebrated theorem in real algebraic and analytic geometry (originally due to Bruhat–Cartan and Wallace, and stated later in its current form by Milnor) is the (Nash) curve selection lemma, which has wide applications. It states that each point in the closure of a semialgebraic set {\mathcal{S}}\subset\mathbb{R}^{n} can be reached by a Nash arc of \mathbb{R}^{n} such that at least one of its branches is contained in {\mathcal{S}} .The purpose of this work is to generalize the previous result to finitely many points. More precisely, let {\mathcal{S}}\subset\mathbb{R}^{n} be a semialgebraic set, let x_{1},\ldots,x_{r}\in{\mathcal{S}} be r points (that we call ‘control points’) and let 0=:t_{1}<\cdots<t_{r}:=1 be r values (that we call ‘control times’). A natural ‘logistic’ question concerns the existence of a smooth and semialgebraic (Nash) path \alpha\colon [0,1]\to{\mathcal{S}} that passes through the control points at the control times, that is, \alpha(t_{k})=x_{k} for k=1,\ldots,r . The necessary and sufficient condition to guarantee the existence of \alpha when the number of control points is large enough and they are in general position is that {\mathcal{S}} is connected by analytic paths. The existence of generic real algebraic sets that do not contain rational curves confirms that the analogous result involving polynomial paths (instead of Nash paths) is only possible under additional restrictions. A sufficient condition is that {\mathcal{S}}\subset\mathbb{R}^{n} has, in addition, dimension n .A related problem concerns the approximation by a Nash path of an existing continuous semialgebraic path \beta\colon[0,1]\to\mathcal{S} with control points x_{1},\ldots,x_{r} \in{\mathcal{S}} and control times 0=:t_{1}<\cdots<t_{r}:=1 . As one can expect, apart from the restrictions on {\mathcal{S}} , some restrictions on \beta are needed. A sufficient condition is that the (finite) set of values \eta(\beta) at which \beta is not smooth is contained in the set of regular points of {\mathcal{S}} and \eta(\beta) does not meet the set of control times.If {\mathcal{S}}\subset\mathbb{R}^{n} is a finite union (connected by analytic paths) of n -dimensional convex polyhedra, we can even ‘estimate’ (using Bernstein’s polynomials) the degree of the involved polynomial path. This requires: (1) a polynomial double curve selection lemma for convex polyhedra involving only degree 3 cuspidal curves; (2) to find the simplest polynomial paths that connect two convex polyhedra (whose union is connected by analytic paths), and (3) some improvements concerning well-known bounds for Bernstein’s polynomials (and their high order derivatives) to approximate continuous functions that are not differentiable on their whole domain.

Многомерное расширение классических чисел Каталана для решения непрерывных задач анализа случайных точечных изображений

Предложено обобщение классических чисел Каталана на многомерный случай, базирующееся на словарно-символьной статистике. Представлены формулировки вероятностно-комбинаторных задач, решение которых приводит сначала к двумерному, а затем и трехмерному обобщению классической последовательности Каталана. Изложенный подход возник и вместе с программами компьютерной алгебры применялся при переводе в дискретную форму ряда непрерывных вероятностных задач, связанных с анализом случайных точечных изображений. Эффективность приведенного в работе многомерного расширения чисел Каталана продемонстрирована на примере решения с их помощью классической баллотировочной задачи Бертрана. Purpose. The purpose of the research is to systematize and find an explicit analytical form of generalized Catalan numbers based on a word-symbolic representation, as well as their application for applied problems related to the processing of random point images. Methodology. An original mechanism for reducing solvable continuous problems to their discrete analogues is proposed. In the practical implementation of the chosen research scheme, there are two types of problems, namely, discrete-combinatorial problems leading to generalized Catalan numbers and problems of calculating multidimensional integrals over convex polyhedra in n-dimensional space requiring the development of high-speed computer algebra programs. The main content. A generalization of the classical Catalan numbers to the multidimensional case is proposed, based on word-symbolic statistics using a limited alphabet. Probabilistic-combinatorial problems are explicitly formulated; their solution leads to a two-dimensional and then to a three dimensional generalization of the classical Catalan sequence. The main advantages of the new extension of Catalan numbers are the ease of generalization of word-symbolic problems to the multidimensional case and the unity and universality of their formulations. Findings. Word-symbolic problems leading to multidimensional generalized Catalan numbers are formulated. An explicit form of generalized Catalan numbers in two-dimensional and three dimensional cases is obtained. A number of probabilistic problems related to the analysis of random point images have been solved. Scientific novelty and originality. The originality and scientific novelty of the article contains both the obtained results and the developed methods to solve them. The article presents new previously unknown probabilistic formulas required for solving problems related to the registration and analysis of random point images. A package of programs has been created for high-speed integrating multidimensional integral expressions over areas limited by a system of hyperplanes in n-dimensional space.

Min–max tracking model predictive control for linear parameter‐varying systems using polyhedral invariant sets

Abstract In this paper, a min–max tracking model predictive control (MPC) method for linear parameter‐varying (LPV) systems using polyhedral invariant sets is proposed. The method aims to expand the error state stabilizable domain and improve dynamic performance while handling asymmetric system constraints and guaranteeing robust stability under parameter uncertainty, with low computational burden. Firstly, the augmented dynamic formulation is constructed based on the original state‐space model and reference trajectory dynamic model to obtain the tracking error state. Secondly, a min–max optimization problem considering parameter variation and system constraints is formulated based on the tracking error state. Thirdly, a sequence of optimal control laws is offline obtained by solving the optimization problem to design the nested corresponding polyhedral invariant sets. These sets have a larger error state stabilizable domain than ellipsoidal invariant sets. An interpolation method is applied to improve dynamic tracking performance during online control. Finally, the simulation results and comparative analysis substantiate the effectiveness of the proposed min–max tracking MPC method.

A study on the siting of emergency medical facilities under uncertain demand-A case study of Wuhan country parks.

At the beginning of 2020, the novel coronavirus broke out as a sudden public health emergency worldwide, with the number of confirmed patients constantly rising, which brought huge pressure to the medical system. Many countries and regions have noticed the positive role of emergency medical facilities in combating the COVID-19 pandemic. Therefore, the analysis of the location and construction of emergency medical facilities for public health emergencies has practical significance. This paper mainly discusses the use of urban suburban parks as the construction sites for emergency medical facilities and builds a maximum service quality level model for emergency medical facilities in response to public health emergencies. Considering the suddenness and unpredictability of public health emergencies, this study introduces polyhedral uncertainty sets to describe the uncertainty of the number of confirmed patients and transforms the model into an easily solvable mixed-integer programming model through the Bertsimas and Sim robust optimization method. The GAMS software is used for programming and the CPLEX solver is called to solve the model. Taking 13 urban suburban parks in Wuhan as an example, the optimal location plan and patient allocation of emergency medical facilities are determined, verifying the feasibility and effectiveness of the model. The results show that the model effectively promotes the determination of location plans and patient transfer routes. It is expected that in the event of a sudden public health emergency in a city, it can provide reference and basis for decision-makers to deal with public health emergencies.

A problem of simple group pursuit with possible dynamical disturbance in dynamics and phase constraints

In finite-dimensional Euclidian space, we treat the problem of pursuit of one evader by a group of pursuers, which is described by a system of the form z˙i = ai(t)ui − v, ui ∈ Ui, v ∈ V, where the functions i(t) are equal to 1 for all t, except for a certain interval of a given length, on which they are equal to zero (to each pursuer there corresponds its own interval). This fact can be interpreted in such a way that each of the pursuers has a possible failure of the control device at any previously unknown moment in time, and the length of the time interval needed to fix the failure is given, while in the process of fixing the failure the pursuers have no possibility to carry out a capture. The target sets are convex compact sets, and the evader does not leave the bounds of the convex polyhedral set. We obtain sufficient conditions for solvability of the pursuit problem.

The negative energy N-body problem has finite diameter

The Jacobi-Maupertuis metric lets us reformulate the classical N-body problem at fixed energy E as a geodesic flow problem on a space whose metric depends parametrically on E. We only consider the case E<0 in which case there is a non-empty Hill boundary along which the metric degenerates. Our main result is the resulting metric space has finite diameter. As a corollary the space admits no metric rays, answering a question of Burgos (Proc Amer Math Soc 150: 1729–1733, 2022). This main result is an immediate corollary of a theorem asserting that all points of the space are a fixed bounded distance from the Hill boundary. Our proof of this last theorem relies ultimately on a game of escape from the boundary of a polyhedral convex cone in a Euclidean space into the interior of said cone. Motivation for our work comes from that of Maderna (Ann Math 192: 499–550, 2020) and from the desire to right a wrong promulgated in Montgomery (Regul Chaot Dyn 28: 374–394, 2023).

Supply–Demand Dynamics Quantification and Distributionally Robust Scheduling for Renewable-Integrated Power Systems with Flexibility Constraints

The growing penetration of renewable energy sources (RES) has exacerbated operational flexibility deficiencies in modern power systems under time-varying conditions. To address the limitations of existing flexibility management approaches, which often exhibit excessive conservatism or risk exposure in managing supply–demand uncertainties, this study introduces a data-driven distributionally robust optimization (DRO) framework for power system scheduling. The methodology comprises three key phases: First, a meteorologically aware uncertainty characterization model is developed using Copula theory, explicitly capturing spatiotemporal correlations in wind and PV power outputs. System flexibility requirements are quantified through integrated scenario-interval analysis, augmented by flexibility adjustment factors (FAFs) that mathematically describe heterogeneous resource participation in multi-scale flexibility provision. These innovations facilitate the formulation of physics-informed flexibility equilibrium constraints. Second, a two-stage DRO model is established, incorporating demand-side resources such as electric vehicle fleets as flexibility providers. The optimization objective aims to minimize total operational costs, encompassing resource activation expenses and flexibility deficit penalties. To strike a balance between robustness and reduced conservatism, polyhedral ambiguity sets bounded by generalized moment constraints are employed, leveraging Wasserstein metric-based probability density regularization to diminish the probabilities of extreme scenarios. Third, the bilevel optimization structure is transformed into a solvable mixed-integer programming problem using a zero-sum game equivalence. This problem is subsequently solved using an enhanced column-and-constraint generation (C&amp;CG) algorithm with adaptive cut generation. Finally, simulation results demonstrate that the proposed model positively impacts the flexibility margin and economy of the power system, compared to traditional uncertainty models.

Robust Positively Invariant Conditions for Perturbed Linear Discrete-Time Systems Using Dual Optimization

This paper presents both sufficient and necessary conditions for polyhedral sets and symmetric polyhedral sets to be robust positively invariant sets within perturbed linear discrete-time systems. These conditions are derived through the application of optimization and dual optimization theory. By leveraging the definition of a robust positively invariant set and employing the Pontryagin difference, we have obtained robust positively invariant conditions in optimized forms. Through the use of dual optimization theory, various equivalent forms are introduced, offering additional tools for verifying that polyhedral sets are indeed robust positively invariant sets for perturbed linear discrete-time dynamic systems. The efficacy of these conclusions is further evidenced by numerical examples.

Volume-Increasing Inextensional Deformations of Platonic Polyhedra

It is known that the volume of a convex polyhedron can be increased by suitable isometric deformation of its surface resulting in a non-convex shape. Deformation patterns and the associated enclosed volumes of the Platonic polyhedra were theoretically and numerically investigated by a few authors in the past. In this paper, a generic diamond-shaped folding pattern for all Platonic polyhedra is presented, optimised to achieve the maximum enclosed volumes. The numerically obtained volume increases (44.70%, 25.12%, 13.86%, 10.61%, and 4.36% for the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively) improve the existing results (44.00%, 24.62%, 13.58%, 9.72%, and 4.27%, respectively). Quasi-rigid inflatable membrane representations of such deformed polyhedra imply a significant change of structural shape due to initial inflation and subsequent compressive stresses transverse to the crease lines.

A Solution Method for Arbitrary Polyhedral Convex Set Optimization Problems

A Solution Method for Arbitrary Polyhedral Convex Set Optimization Problems

The metric projection over a polyhedral set through the relative interiors of its faces

Abstract We first characterize the region of the n-dimensional Euclidean space for which two optimization problems with the square distance function as common objective function, but different constraints, are equivalent. The affine hull of a certain face of a closed convex set $$C\subseteq {\mathbb {R}}^n$$ C ⊆ R n is the constraint associated to one problem and the whole closed convex set C is the constraint associated to the other problem. Such optimization problems are best approximation problems which can be reformulated in terms of the metric projection. Using the language of the metric projection, we characterize the region of $${\mathbb {R}}^n$$ R n which is metrically projected over a face of C in the same way that it is projected over the affine hull of the face itself. The metric projection over such a face is the one associated to the entire closed convex set, and the metric projection over the affine hull of such a face is the one associated to the affine hull. It turns out that this region is the closure of the inverse image, through the metric projection over the entire closed convex set, of the relative interior of the face. We also characterize analytically the closure of the regions of $${\mathbb {R}}^n$$ R n that are projected over the relative interiors of the faces of a polyhedral set, through the metric projection of the polyhedral set itself. We show that these regions are polyhedral convex sets by explicitly characterizing them through systems of linear inequalities.