Convex Sets Research Articles

A robust multi-objective optimization model for grid-scale design of sustainable cropping patterns: A case study

In the face of growing population, water scarcity, and increasing food demand, there is a pressing need to shift towards optimized, resource-efficient, climate-resilient, and sustainable agricultural practices. In light of that, designing a sustainable cropping pattern that considers the procedure of resource allocation based on land capabilities and crop features is vital for ensuring long-term food production and safeguarding the delicate balance of ecosystems. Motivated by this imperative, this study proposes a comprehensive framework that integrates Geographic Information System (GIS), System Dynamics (SD), and optimization to address the sustainable design of cropping patterns. The framework assesses grid-scale land suitability, models dynamic water resource interactions, and optimizes resource allocation based on crop calendar considerations. For the first time, a dynamic crop inventory is integrated into the cropping pattern optimization process, addressing food security concerns in a comprehensive manner. In order to evaluate the effect of uncertainties on the designed system, a robust optimization model is developed based on convex sets. The results demonstrate the advantages of the robust model in situations with uncertainty. Despite a 5% reduction in profit compared to the deterministic solution, the robust design achieves a 25% decrease in irrigation, highlighting the cost of ensuring sustainability. The deterministic approach prioritizes crops based on their economic value, whereas the robust solution considers the volume of irrigation required for a sustainable design. The managerial implications emphasize the importance of prioritizing water-efficient and climate-resilient agricultural practices to guarantee long-term food security.

Strict power convexity of solutions to fully nonlinear elliptic partial differential equations in two dimensional convex domains

Strict power convexity of solutions to fully nonlinear elliptic partial differential equations in two dimensional convex domains

Multiple critical points in closed sets via minimax theorems

In this paper, we apply our minimax theory ([Ricceri B. On a minimax theorem: an improvement, a new proof and an overview of its applications. Minimax Theory Appl. 2017;2:99–152; Ricceri B. A more complete version of a minimax theorem. Appl Anal Optim. 2021;5:251–261; Ricceri B. Addendum to “A more complete version of a minimax theorem”. Appl Anal Optim. 2022;6:195–197]) with the one developed by A. Moameni in [Moameni A. Critical point theory on convex subsets with applications in differential equations and analysis. J Math Pures Appl. 2020;141:266–315.] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical elliptic problem: THEOREM. – Let Ω ⊂ R n ( n ≥ 3 ) be a smooth bounded domain and let 1 < q < 2 ≤ p < 2 n n − 2 . Then, for every r , ν > 0 , there exists λ ∗ > 0 with the following property: for every λ ∈ ] 0 , λ ∗ [ , μ ∈ ] − λ ∗ , λ ∗ [ , and for every convex dense set S ⊂ H − 1 ( Ω ) , there exists φ ~ ∈ S , with ‖ φ ~ ‖ H − 1 ( Ω ) < r , such that the problem { − Δu = λ ( | u | 4 n − 2 u + ν | u | q − 2 u + μ | u | p − 2 u + φ ~ ) in Ω u = 0 on ∂Ω has at least two solutions whose norms in H 0 1 ( Ω ) are less than or equal to r.

A Solution Method for Arbitrary Polyhedral Convex Set Optimization Problems

A Solution Method for Arbitrary Polyhedral Convex Set Optimization Problems

On some relations between the perimeter, the area and the visual angle of a convex set

Abstract We establish some relations between the perimeter, the area and the visual angle of a planar compact convex set. Our first result states that Crofton’s formula is the unique universal formula relating the visual angle, the length and the area. After that we give a characterization of convex sets of constant width by means of the behavior of their isotopic sets at infinity. Also for this class of convex sets we prove that the existence of an isotopic circle is enough to ensure that the considered set is a disc.

Length-Minimizing LED Trees

In this paper, we introduce a previously not studied type of Euclidean tree called LED (Leaves of Equal Depth) tree. LED trees can be used, for example, in computational phylogeny, since they are a natural representative of the time evolution of a set of species in a feature space. This work is focused on LED trees that are length minimizers for a given set of leaves and a given isomorphism type. The underlying minimization problem can be seen as a variant of the classical Euclidean Steiner tree problem. Even though it has a convex objective function, it is rather non-trivial, since it has a non-convex feasible set. The main contribution of this paper is that we prove the uniqueness of a stationary point of the length function on the feasible set. Moreover, we prove several geometrical characteristics of the length minimizers that are analogous to the properties of Steiner minimal trees. We also explore some geometrical and topological properties of the feasible set. At the end, to demonstrate the applicability of our theoretical results, we show an example of an application in historical linguistics.

Optimizing Entanglement in Two‐Qubit Systems

ABSTRACTWe investigate entanglement in two‐qubit systems using a geometric representation based on the minimum of essential parameters. The latter is achieved by requiring subsystems with the same entropy, regardless of whether the state of the entire system is pure or mixed. The geometric framework is provided by a convex set that forms a right‐triangle, whose points are linked to the absolute value of just two of the coherences of the system under study. As a result, we find that optimized states of two qubits host pairs of identical populations while reducing the number of coherences involved, so they are X‐shaped. A geometric ‐measure of entanglement is introduced as the distance between the points in that represent entangled states and the closest point that defines separable states. It is shown that reproduces the results of the Hill‐Wootters concurrence , so that can be interpreted as a distance‐like entanglement measure. However, unlike , the measure also distinguishes the rank of states. The universality of the two‐qubit X‐states ensures the utility of our geometric model for studying entanglement of two‐qubit states in any configuration. To show the applicability of our approach far beyond time‐independent cases, we construct a time‐dependent two‐qubit state, traced out over the complementary components of a pure tetra‐partite system, and find that its one‐qubit states share the same entropy. The entanglement measure results bounded from above by the envelope of the minima of such entropy.

Asymptotic one-dimensional symmetry for the Fisher–KPP equation

Let u be a solution of the Fisher–KPP equation \partial_{t} u=\Delta u+f(u) , t&gt;0 , x\in\mathbb{R}^{N} , with an initial datum u_{0} . We address the following question: does u become locally planar as t\to+\infty ? Namely, does u(t_{n},x_{n}+\cdot) converge locally uniformly, up to subsequences, towards a one-dimensional function, for any sequence ((t_{n},x_{n}))_{n\in\mathbb{N}} in (0,+\infty)\times\mathbb{R}^{N} such that t_{n}\to+\infty as n\to+\infty ? This question is in the spirit of the celebrated De Giorgi’s conjecture concerning stationary solutions of the Allen–Cahn equation. Some affirmative answers to the above question are known in the literature: when the support of the initial datum u_{0} is bounded or when it lies between two parallel half-spaces. Instead, the answer is negative when the support of u_{0} is “V-shaped”. We prove here that u is asymptotically locally planar when the support of u_{0} is a convex set (satisfying in addition a uniform interior ball condition), or, more generally, when it is at finite Hausdorff distance from a convex set. We actually derive the result under an even more general geometric hypothesis on the support of u_{0} . We recover in particular the aforementioned results known in the literature. We further characterize the set of directions in which u is asymptotically locally planar, and we show that the asymptotic profiles are monotone. Our results apply in particular when the support of u_{0} is the subgraph of a function with vanishing global mean.

Approximations of unbounded convex projections and unbounded convex sets

Abstract We consider the problem of projecting a convex set onto a subspace or, equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex sets by polyhedrons. The existing literature on convex projections provides methods for bounded convex sets only, in this paper we propose a method that can handle both bounded and unbounded problems. The algorithms we propose build on the ideas of inner and outer approximation. In particular, we adapt the recently proposed methods for solving unbounded convex vector optimization problems to handle also the class of projection 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.

A modified spectral projected gradient method for tensor approximations over closed convex sets

A modified spectral projected gradient method for tensor approximations over closed convex sets

An accelerated mirror descent algorithm for constrained nonconvex problems

This paper considers the composite optimization problem where the sum of a smooth nonconvex function and a proper lower semicontinuous function has to be minimized over a closed convex set. The mirror descent algorithm and its acceleration variants can be considered as a series of popular methods for solving this problem. Meanwhile, some recent contributions try to circumvent the global Lipschitz condition of smooth part in objective, and establish their global convergence by replacing it with a local one. But, to the best of our knowledge, they still require some extra conditions, including but not limited to the assumption that the abstract set is the whole space. The aim of this paper is to design an accelerated mirror descent algorithm, which uses a convex combination of essentially the whole iteration trajectory, and to show that the local Lipschitz assumption of associated Legendre function together with the Kurdyka-Łojasiewicz property is sufficient to recover its convergence results. In addition to the theoretical improvement in the convergence analysis, it also showcases possible computational advantages which provides an interesting option for practical problems.

Analyzing the Relationship between Free Locally Convex Spaces and Nuclear Space Concepts

Analyzing the Relationship between Free Locally Convex Spaces and Nuclear Space Concepts

Compositional Imprecise Probability: A Solution from Graded Monads and Markov Categories

Imprecise probability is concerned with uncertainty about which probability distributions to use. It has applications in robust statistics and machine learning. We look at programming language models for imprecise probability. Our desiderata are that we would like our model to support all kinds of composition, categorical and monoidal; in other words, guided by dataflow diagrams. Another equivalent perspective is that we would like a model of synthetic probability in the sense of Markov categories. Imprecise probability can be modelled in various ways, with the leading monad-based approach using convex sets of probability distributions. This model is not fully compositional because the monad involved is not commutative, meaning it does not have a proper monoidal structure. In this work, we provide a new fully compositional account. The key idea is to name the non-deterministic choices. To manage the renamings and disjointness of names, we use graded monads. We show that the resulting compositional model is maximal and relate it with the earlier monadic approach, proving that we obtain tighter bounds on the uncertainty.

Cooperative navigation using constrained convex generators

In this article, we suggest a technique of set-based cooperative navigation for a fleet of vehicles under communication constraints. In the suggested approach, only the distances and bearings to other vehicles or known locations can be measured. Each vehicle in the fleet must determine its location and the positions of the other vehicles based on the measurements as well as information shared among the vehicles. Since the measurement set may be nonconvex, it must be approximated by a convex set. To address this issue, we employ constrained convex generators, which is a generalization of the definition for constrained zonotopes that allows ℓ2 unit norm balls and convex cones. To keep the amount of exchanged information low the vehicles transmit an ellipse containing the state estimate to the other vehicles. The obtained estimates are worst-case bounds for the true position, which is important in certain applications such as collision avoidance. The computed sets are then applied to vehicle control algorithms, taking into account the positions of all the agents. Through numerical simulations, we illustrate the application of this method to the problem of cooperative navigation of autonomous underwater vehicles and the problem of fire detection with multiple UAVs.

Collision Detection Between Convex Objects Using Pseudodistance and Unconstrained Optimization

Collision Detection Between Convex Objects Using Pseudodistance and Unconstrained Optimization

Stabilizing and Accelerating Federated Learning on Heterogeneous Data With Partial Client Participation.

Federated learning (FL) commonly encourages the clients to perform multiple local updates before the global aggregation, thus avoiding frequent model exchanges and relieving the communication bottleneck between the server and clients. Though empirically effective, the negative impact of multiple local updates on the stability of FL is not thoroughly studied, which may result in a globally unstable and slow convergence. Based on sensitivity analysis, we define in this paper a local-update stability index for the general FL, as measured by the maximum inter-client model discrepancy after the multiple local updates that mainly stems from the data heterogeneity. It enables to determine how much the variation of client's models with multiple local updates may influence the global model, and can also be linked with the convergence and generalization. We theoretically derive the proposed local-update stability for current state-of-the-art FL methods, providing possible insight to understanding their motivation and limitation from a new perspective of stability. For example, naively executing the parallel acceleration locally at clients would harm the local-update stability. Motivated by this, we then propose a novel accelerated yet stabilized FL algorithm (named FedANAG) based on the server- and client-level Nesterov accelerated gradient (NAG). In FedANAG, the global and local momenta are elaborately designed and alternatively updated, while the stability of local update is enhanced with help of the global momentum. We prove the convergence of FedANAG for strongly convex, general convex and non-convex settings. We then conduct evaluations on both the synthetic and real-world datasets to first validate our proposed local-update stability. The results further show that across various data heterogeneity and client participation ratios, FedANAG not only accelerates the global convergence by reducing the required number of communication rounds to a target accuracy, but converges to an eventually higher accuracy.

Bender–Knuth Billiards in Coxeter Groups

Abstract Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$ , where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involutions $\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$ indexed by elements of I. For arbitrary W and for each $i\in I$ , we introduce an operator $\tau _i\colon W\to W$ (depending on $\mathscr {L}$ ) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on $\mathscr {L}$ that coincides with $\mathrm {BK}_i$ in type A. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$ , we consider the operator $\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$ . We say W is futuristic if for every nonempty finite convex set $\mathscr {L}$ , every Coxeter element c and every $u\in W$ , there exists an integer $K\geq 0$ such that $\mathrm {Pro}_c^K(u)\in \mathscr {L}$ . We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$ , and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of W, then $\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$ ; this allows us to determine the smallest integer $\mathrm {M}(c)$ such that $\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$ for all $\mathscr {L}$ . We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$ , $\widetilde C$ , or $\widetilde G_2$ .

Comparison of recent advances in set-membership techniques: Application to state estimation, fault detection and collision avoidance

There has been a vast body of research on set-membership techniques in recent years. These algorithms compute convex sets that contain the state of a dynamical system given bounds on disturbance and noise signals. Recently, a thorough comparison of zonotopes-based methods against interval arithmetic and ellipsoids has been presented in the literature. However, two main issues were left unexplored: (i) added conservatism in the presence of bounds of different kinds such as ℓ2-norm for the disturbance and an ℓ∞-norm bound on the noise: (ii) set-membership methods can be used in different settings apart from guaranteed state estimation, such as fault detection and isolation and collision avoidance of autonomous vehicles. In this paper, we extend this comparison by considering state estimation, fault detection and isolation, and collision avoidance for interval arithmetic, ellipsoids, zonotopes, constrained zonotopes, polytopes and constrained convex generators in the presence of various combination of bounds for the exogenous signals. The main objective is to compare accuracy, computation time and the scalability of the growth of the data structures required by each set representation. The results indicate that intervals, ellipsoids and zonotopes have a much worse accuracy. The recently introduced Constrained Convex Generators have a negligible increase in computation time in comparison with constrained zonotopes but have a better accuracy when bounds for disturbances, noise and initial conditions are heterogeneous or at least not polytopic.

Floorplanning with I/O Assignment via Feasibility-Seeking and Superiorization Methods.

The feasibility-seeking approach offers a systematic framework for managing and resolving intricate constraints in continuous problems, making it a promising avenue to explore in the context of floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be expressed as the union of convex sets. However, conventional projection-based algorithms for feasibility-seeking do not guarantee convergence in such situations, which are also heavily influenced by the initialization. We present a quantitative property about the choice of the initial point that helps good initialization and analyze the occurrence of the oscillation phenomena for bad initialization. In implementation, we introduce a resetting strategy aimed at effectively reducing the problem of algorithmic divergence in the projection-based method used for the feasibility-seeking formulation. Furthermore, we introduce the novel application of the superiorization method (SM) to floorplanning, which bridges the gap between feasibility-seeking and constrained optimization. The SM employs perturbations to steer the iterations of the feasibility-seeking algorithm towards feasible solutions with reduced (not necessarily minimal) total wirelength. Notably, the proposed algorithmic flow is adaptable to handle various constraints and variations of floorplanning problems, such as those involving I/O assignment. To evaluate the performance of Per-RMAP, we conduct comprehensive experiments on the MCNC benchmarks and GSRC benchmarks. The results demonstrate that we can obtain legal floorplanning results 166× faster than the branch-and-bound (B&B) method while incurring only a 5% wirelength increase compared to the optimal results. Furthermore, we evaluate the effectiveness of the algorithmic flow that considers the I/O assignment constraints, which achieves an 6% improvement in wirelength. Besides, considering the soft modules with a larger feasible solution space, we obtain 15% improved runtime compared with PeF, the state-of-the-art analytical method. Moreover, we compared our method with Parquet-4 and Fast-SA on GSRC benchmarks which include larger-scale instances. The results highlight the ability of our approach to maintain a balance between floorplanning quality and efficiency.