Convex Sets Research Articles

State estimation of DC microgrids using manifold optimization and semidefinite programming

DC microgrids are becoming more common in modern systems, so computation methodologies such as the power flow, the optimal power flow, and the state estimation require being adapted to this new reality. This paper deals with the latter problem, which consists of reconstructing the state variables given voltage and power measurements. Although the model of DC grids is undoubtedly less complicated than its counterpart AC, it is still a nonlinear/non-convex optimization problem. Our approach is based on the idea of solving the problem in a matrix space. Although it may be counter-intuitive to transform from Rn to Rn×n, a matrix space exhibits better geometric properties that allow an elegant formulation and, in some cases, an efficient form to solve the optimization problem. We compare two methodologies: semidefinite programming and manifold optimization. The former relaxes the problem to a convex set, whereas the latter maintains the geometry of the original problem. A specialized gradient method is proposed to solve the problem in the matrix manifold. Extensive numerical experiments are conducted to showcase the key characteristics of both methodologies. Our study aims to shed light on the potential benefits of employing matrix space techniques in addressing operation problems in DC microgrids and power system computations in general.

Stripe Embedding: Efficient Maps with Exact Numeric Computation

We consider the fundamental problem of injectively mapping a surface mesh with disk topology onto a boundary constrained convex domain. We start from the basic observation that mapping a strip of triangles onto a rectangular shape always yields a valid embedding, if the vertices that bound the strip are sorted coherently along the sides of the rectangle. Based on this intuition, we propose a straightforward algorithm, called Stripe Embedding, that operates by decomposing the input mesh into a set of triangle strips and then embeds each strip into the target domain by means of linear interpolation between two previously embedded vertices. Thanks to its simplicity, Stripe Embedding is extremely efficient and permits to switch to an exact implementation without almost increasing its running times. Stripe Embedding is up to three orders of magnitude faster than the Tutte embedding for same numerical model and, even when implemented with costly rational numbers, it is faster than any floating point implementation of prior methods at any scale.

Hypercube Pooling for Visual Semantic Embedding

Visual Semantic Embedding ( VSE ) is a primary model for cross-modal retrieval, wherein the global feature aggregator is a crucial component of the VSE model. In recent research, the General Pooling Operator ( GPO ) aggregator, which weighs the features reconstructed from the local feature set to aggregate, facilitates the related models to achieve good retrieval performance. However, the reason for the effectiveness remains to be explored. To enhance the rationality of aggregator designs, we analyze the reason from the perspective of feature space. Indeed, for each data, the local feature set forms a hypercube containing abundant data information, and the feature learned by GPO measures the hypercube, thereby representing the data. The geometric structure of the hypercube implies that the set containing all points within the hypercube is a convex set, so the feature learned by weighted aggregation is an interior point of the hypercube. However, using the interior point to measure the hypercube leads to some problems in feature representation and model optimization, as well as the reduction of retrieval efficiency caused by weight computation. For example, the related pair’s features may be far, while the unrelated ones may be close. To measure the hypercube more clearly and alleviate the problems mentioned above, we propose Hypercube Pooling ( HCP ) aggregator. Specifically, HCP concatenates the Max and Min Pooling features as the global features. This aggregation method has multiple advantages, e.g., the learned global feature represents all hyperplanes of the hypercube that contain critical information and hypercube geometric structure. Moreover, HCP adds normalization-before-concatenation and reduces the usual setting of margin in the loss function by half to avoid gradient loss caused by the difference in the feature value and dimensionality doubling. The experimental results on the Flickr30K and MSCOCO datasets show that the HCP model has excellent performance with high efficiency, confirming the correctness of the spatial analysis and the effectiveness of the HCP aggregator.

Orientation of Convex Sets

We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a so-called interiority condition: $\circlearrowleft(ABD)=\circlearrowleft(BCD)=\circlearrowleft(CAD)=1$ imply $\circlearrowleft(ABC)=1$ for any $A, B, C, D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O but not every P3O is the order type of some planar point set; a P3O that is realizable by points is called a p-P3O. If the family is non-degenerate with respect to the orientation, i.e., always $\circlearrowleft(ABC)\ne 0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-P3O, is the order type of a point set in general position. Despite these similarities to order types, P3O's and T3O's that can arise from the orientation of pairwise intersecting convex sets, denoted by C-P3O and C-T3O, turn out to be quite different from order types: there is no containment relation among the family of all C-P3O's and the family of all p-P3O's, or among the families of C-T3O's and p-T3O's.Finally, we study properties of these orientations if we also require that the family of the underlying convex sets satisfies the (4,3) property, as a first step towards obtaining better $(p,q)$-theorems.

Alternating Projection Method for Intersection of Convex Sets, Multi-Agent Consensus Algorithms and Averaging Inequalities

The history of the alternating projection method for finding a common point of several convex sets in Euclidean space goes back to the well-known Kaczmarz algorithm for solving systems of linear equations, which was devised in the 1930s and later found wide applications in image processing and computed tomography. An important role in the study of this method was played by I.I. Eremin’s, L.M. Bregman’s, and B.T. Polyak’s works, which appeared nearly simultaneously and contained general results concerning the convergence of alternating projections to a point in the intersection of sets, assuming that this intersection is nonempty. In this paper, we consider a modification of the convex set intersection problem that is related to the theory of multi-agent systems and is called the constrained consensus problem. Each convex set in this problem is associated with a certain agent and, generally speaking, is inaccessible to the other agents. A group of agents is interested in finding a common point of these sets, that is, a point satisfying all the constraints. Distributed analogues of the alternating projection method proposed for solving this problem lead to a rather complicated nonlinear system of equations, the convergence of which is usually proved using special Lyapunov functions. A brief survey of these methods is given, and their relation to the theorem ensuring consensus in a system of averaging inequalities recently proved by the first author is shown (this theorem develops convergence results for the usual method of iterative averaging as applied to the consensus problem).

Super-resolution imaging quality enhancement method for distributed array infrared camera

Abstract To address issues related to low resolution and high noise in infrared cameras, a distributed array infrared camera imaging system utilizing four cameras is proposed. The four cameras are arranged in an unconstrained array, and the combination algorithm of Projections onto Convex Sets (POCS) and Real-Enhanced Super-Resolution Generative Adversarial Networks (Real-ESRGAN) is applied to achieve high-quality super-resolution infrared imaging. The wavelet fusion algorithm is used to preprocess four low-resolution infrared images to reduce noise. Then, the POCS algorithm is used to reconstruct the preprocessed image. Finally, the Real-ESRGAN is employed for image reconstruction, resulting in an ultra-high-resolution infrared image. The results show that compared to single infrared camera imaging, the resolution of images reconstructed using the distributed infrared camera array is increased by 0.58 times, with the Modulation Transfer Function (MTF) increased by 1.2 times. Additionally, the entropy is increased by 18.87%, the standard deviation is increased by 13.51%, and the Naturalness Image Quality Evaluator (NIQE) is reduced by 18.87%. This demonstrates a significant enhancement in the super-resolution imaging quality of the distributed infrared camera array.

Gelfand–Phillips Type Properties of Locally Convex Spaces

We let 1≤p≤q≤∞. Being motivated by the classical notions of the Gelfand–Phillips property and the (coarse) Gelfand–Phillips property of order p of Banach spaces, we introduce and study different types of the Gelfand–Phillips property of order (p,q) (the GP(p,q) property) and the coarse Gelfand–Phillips property of order p in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the GP(p,q) property. Characterizations of locally convex spaces with the introduced Gelfand–Phillips type properties are given.

Novel Fuzzy Ostrowski Integral Inequalities for Convex Fuzzy-Valued Mappings over a Harmonic Convex Set: Extending Real-Valued Intervals Without the Sugeno Integrals

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski inequalities. Their formulations incorporate integrability concepts for fuzzy-valued mappings (FVMs), applying the Kaleva integral and a Kulisch–Miranker fuzzy order relation. The fuzzy order relation is constructed via a level-wise approach based on the Kulisch–Miranker order within the fuzzy number space. Additionally, numerical examples illustrate the effectiveness and significance of the proposed theoretical model. Various applications are explored using different means, and some complex cases are derived.

A projection-free method for solving convex bilevel optimization problems

AbstractWhen faced with multiple minima of an inner-level convex optimization problem, the convex bilevel optimization problem selects an optimal solution which also minimizes an auxiliary outer-level convex objective of interest. Bilevel optimization requires a different approach compared to single-level optimization problems since the set of minimizers for the inner-level objective is not given explicitly. In this paper, we propose a new projection-free conditional gradient method for convex bilevel optimization which requires only a linear optimization oracle over the base domain. We establish $$O(t^{-1/2})$$ O ( t - 1 / 2 ) convergence rate guarantees for our method in terms of both inner- and outer-level objectives, and demonstrate how additional assumptions such as quadratic growth and strong convexity result in accelerated rates of up to $$O(t^{-1})$$ O ( t - 1 ) and $$O(t^{-2/3})$$ O ( t - 2 / 3 ) for inner- and outer-levels respectively. Lastly, we conduct a numerical study to demonstrate the performance of our method.

Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems: even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds on an exact or approximate algorithm for the weighted sum scalarization and is therefore applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate version of the dual variant of Benson’s outer approximation algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the German Research Foundation [Project 398572517]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0220 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0220 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property}

A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian product. In this paper Siciak's result is generalized to the setting of Siciak-Zakharyuta functions with growth given by a compact convex set, along with discussing why this generalization does not work in the weighted setting.

Unsupervised 3D seismic data reconstruction using a weighted-attentive deep-learning framework

The physical and cost limitations of seismic data acquisition often result in the spatial undersampling of data, which has a detrimental impact on subsequent data processing. Seismic data reconstruction plays a crucial role in recovering missing traces arising from the acquisition process. In recent years, deep learning (DL) has emerged as an intelligent solution for this purpose. We introduce an innovative approach called the weighted-attentive DL framework for unsupervised 3D seismic data reconstruction, which combines an attentive transformer network (ATNet) with a conventional projection onto convex sets (POCS) algorithm. Our framework follows the plug-and-play concept, requiring only the original subsampled data to achieve robust performance. It accomplishes this by iteratively updating ATNet parameters, wherein ATNet serves as the reconstruction operator during the DL process. This approach allows us to simultaneously recover missing traces and filter out random noise, with a key feature being its enhanced attention to the observed signal compared with convolutional neural networks. In addition, we incorporate the POCS algorithm into our framework to introduce a linear decay weight for the aliasing space containing noise and signal during the reconstruction process. We rigorously evaluate our method against four existing approaches: adaptive POCS, fast dictionary learning sequential generalized K-means, optimally damped rank-reduction, and DenseNet methods. Our comparative experiments, conducted on synthetic and field data examples, clearly demonstrate the substantial improvements achieved by our method in terms of reconstruction and denoising when compared with the four benchmarked methods.

Fixed point theorems of semigroup of isometry mappings and α-nonexpansive mappings on weak⁎ compact convex sets

Fixed point theorems of semigroup of isometry mappings and α-nonexpansive mappings on weak⁎ compact convex sets

Distributed Online Learning Algorithm for Noncooperative Games Over Unbalanced Digraphs.

This article investigates constrained online noncooperative games (NGs) of multiagent systems over unbalanced digraphs, where the cost functions of players are time-varying and are gradually revealed to corresponding players only after decisions are made. Moreover, in the problem, the players are subject to local convex set constraints and time-varying coupling nonlinear inequality constraints. To the best of our knowledge, no result about online games with unbalanced digraphs has been reported, let alone constrained online games. To seek the variational generalized Nash equilibrium (GNE) of the game online, a distributed learning algorithm is proposed based on gradient descent, projection, and primal-dual methods. Under the algorithm, sublinear dynamic regrets and constraint violations are established. Finally, online electricity market games illustrate the algorithm.

Properties of generalized polyhedral convex multifunctions

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

Efficient Convex PCA with Applications to Wasserstein GPCA and Ranked Data

Convex PCA, which was introduced by Bigot et al. modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this article is 3-fold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.

Intrinsically Hölder Sections in Metric Spaces

We introduce the notion of intrinsically Holder graphs in metric spaces that generalized the one of ¨ intrinsically Lipschitz sections. This concept is relevant because it has many properties similar to Holder maps but ¨ is profoundly different from them. We prove some relevant results as the Ascoli-Arzela compactness Theorem, ` Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over R or C, a convex set and an equivalence relation for intrinsically Holder graphs. These last three properties are new also in the Lipschitz case.

Non-orthogonal unicast and multicast transmission with IRS-assisted Rate-Splitting Multiple Access in multiuser MISO systems

Abstract. In MISO systems, novel communication technologies such as NOMA and multicast transmission have attracted widespread attention to meet the diverse needs of users. However, traditional NOMA and multicast transmission schemes have certain limitations in dynamic interference management and resource allocation. To resolve this issue, this paper proposes a RSMA scheme assisted by phase quantized Intelligent Reflecting Surface (IRS) for downlink MISO systems with nonorthogonal transmission of unicast and multicast signals. In this scheme, through reconfiguring unicast and multicast signals and combining them into a super-common stream and partial unicast messages, RSMA achieves dynamic interference management of multi-user while effectively separating unicast and multicast streams. To maximize the weighted sum rate(WSR), we put forward an alternating optimization(AO) algorithm that utilizes Weighted Minimum Mean Square Error (WMMSE) estimation and convex set intersection representation, which jointly optimizes the equivalent beamforming matrix,rate allocation and phase shift matrix. Simulation results illustrate that the proposed single-layer RSMA exhibits performance closer to that of DPC encoding compared to traditional NOMA and MULP schemes. Furthermore, we extensively investigate the relationship between the WSR of RSMA and system parameters, providing important guidance for practical engineering applications.

Normal Approximation of Random Gaussian Neural Networks

In this paper, we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian neural network and (the law of) a random Gaussian vector. Our main results concern deep random Gaussian neural networks with a rather general activation function. The upper bounds show how the widths of the layers, the activation function, and other architecture parameters affect the Gaussian approximation of the output. Our techniques, relying on Stein’s method and integration by parts formulas for the Gaussian law, yield estimates on distances that are indeed integral probability metrics and include the convex distance. This latter metric is defined by testing against indicator functions of measurable convex sets and so allows for accurate estimates of the probability that the output is localized in some region of the space, which is an aspect of a significant interest both from a practitioner’s and a theorist’s perspective. We illustrated our results by some numerical examples. Funding: This research was supported by the European Union’s Horizon 2020 research project WARIFA under grant agreement no. 101017385, by the PRIN project 2022 “Variational Analysis of Complex Systems in Materials Science, Physics and Biology” (CUP B53D23009290006), and by the INdAM project “Modelli ed Algoritmi per dati ad elevata dimensionalità” (CUP E53C23001670001).

Probabilistic method to fundamental gap problems on the sphere

We provide a probabilistic proof of the fundamental gap estimate for Schrödinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo [Potential Anal. 44 (2016), pp. 423–442] for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei [J. Differential Geom. 112 (2019), pp. 347–389], as well as by C. He, G. Wei, and Qi S. Zhang [Amer. J. Math. 142 (2020), pp. 1161–1191]. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.