Convex Sets Research Articles

Stability of the F∗ Algorithm on Strong Pseudocontractive Mapping and Its Application

This paper investigates the stability of the F∗ iterative algorithm applied to strongly pseudocontractive mappings within the context of uniformly convex Banach spaces. The study leverages both analytic and numerical methods to demonstrate the convergence and stability of the algorithm. In comparison to previous works, where weak-contraction mappings were utilized, the strongly pseudocontractive mappings used in this study preserve the convergence property, exhibit greater stability, and have broader applicability in optimization and fixed point theory. Additionally, this work shows that the type of mapping employed converges faster than those in earlier studies. The results are applied to a mixed-type Volterra–Fredholm nonlinear integral equation, and numerical examples are provided to validate the theoretical findings. Key contributions of this work include the following: (i) the use of strongly pseudocontractive mappings, which offer a more stable and efficient convergence rate compared to weak-contraction mappings; (ii) the application of the F∗ algorithm to a wider range of problems; and (iii) the proposal of future directions for improving convergence rates and exploring the algorithm’s behavior in Hilbert and reflexive Banach spaces.

Symmetric Separable Convex Resource Allocation Problems with Structured Disjoint Interval Bound Constraints

Motivated by the problem of scheduling electric vehicle (EV) charging with a minimum charging threshold in smart distribution grids, we introduce the resource allocation problem (RAP) with a symmetric separable convex objective function and disjoint interval bound constraints. In this RAP, the aim is to allocate an amount of resource over a set of n activities, in which each individual allocation is restricted to a disjoint collection of m intervals. This is a generalization of classic RAPs studied in the literature in which, in contrast, each allocation is only restricted by simple lower and upper bounds, that is, m = 1. We propose an exact algorithm that, for four special cases of the problem, returns an optimal solution in [Formula: see text] time, where the term nF represents the number of flops required for one evaluation of the separable objective function. In particular, the algorithm runs in polynomial time when the number of intervals m is fixed. Moreover, we show how this algorithm can be adapted also to output an optimal solution to the problem with integer variables without increasing its time complexity. Computational experiments demonstrate the practical efficiency of the algorithm for small values of m and, in particular, for solving EV charging problems. History: Accepted by Antonia Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. 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.2023.0263 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0263 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

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.

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.

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.

Approximation of Fixed Points of Nearly Non-expansive Mapping

The aim of this paper is to prove some weak and strong convergence results of a novel modified Picard-S hybrid iteration to converge to fixed point involving nearly non-expansive mapping in the setting of uniformly convex Banach space. Our results generalize and improve the known results of the existing literature.

Inversion of the exponential X-ray transform of symmetric 2-tensors

A unique inversion of the exponential X-ray transform of some class of symmetric 2-tensor fields supported in a two-dimensional strictly convex set is presented. The approach to inversion is based on the Cauchy problem for a Beltrami-like equation associated with A-analytic maps.

Convex set reliability-based optimal attitude control for space solar power station with bounded and correlated uncertainties

Considering the bounded and correlated uncertainties in the inherently nonlinear attitude dynamics, this paper proposes an uncertain optimal attitude control for space solar power stations (SSPS), subjected to the convex set-based reliability constraint. To accurately and cost-effectively quantify the uncertainty in the SSPS attitude control system, the uncertain attitude dynamics are proposed based on nominal SSPS with convex set-based uncertainties. Given the known bounds and correlations of the random variables, the convex set-based attitude dynamics of SSPS can be conveniently constructed and transformed into corresponding uncertain state-space equations. Uncertain state and output can be solved using the extended-order matrix and uncertainty propagation method. A convex set-based Riccati equation is: developed using the convex set theory, traditional algebraic Riccati equation (ARE), and Lyapunov equation, which can obtain all the uncertain components of feedback gain, input torque, and cost function, respectively. Convex set-based time-dependent reliability (CSTR) is applied to assess the safety of the attitude control system by setting critical values and considering the time-dependent effect, which is regarded as an important constraint in the optimal attitude control issue with uncertainty. A numerical example of SSPS is provided to verify the proposed uncertain controller within the reliability-constrained optimization framework, demonstrating its effectiveness in managing nonlinear system dynamics under uncertainty.

Acceleration by stepsize hedging: Silver Stepsize Schedule for smooth convex optimization

Abstract We provide a concise, self-contained proof that the Silver Stepsize Schedule proposed in our companion paper directly applies to smooth (non-strongly) convex optimization. Specifically, we show that with these stepsizes, gradient descent computes an $$\varepsilon $$ ε -minimizer in $$O(\varepsilon ^{-\log _{\rho } 2}) = O(\varepsilon ^{-0.7864})$$ O ( ε - log ρ 2 ) = O ( ε - 0.7864 ) iterations, where $$\rho = 1+\sqrt{2}$$ ρ = 1 + 2 is the silver ratio. This is intermediate between the textbook unaccelerated rate $$O(\varepsilon ^{-1})$$ O ( ε - 1 ) and the accelerated rate $$O(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 ) due to Nesterov in 1983. The Silver Stepsize Schedule is a simple explicit fractal: the i-th stepsize is $$1 + \rho ^{\nu (i)-1}$$ 1 + ρ ν ( i ) - 1 where $$\nu (i)$$ ν ( i ) is the 2-adic valuation of i. The design and analysis are conceptually identical to the strongly convex setting in our companion paper, but simplify remarkably in this specific setting.

Linear topological invariants for kernels of differential operators by shifted fundamental solutions

AbstractWe characterize the condition $$(\Omega )$$ ( Ω ) for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $$(P\Omega )$$ ( P Ω ) and $$(P\overline{\overline{\Omega }})$$ ( P Ω ¯ ¯ ) for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space $$\{ f \in {\mathscr {E}}(X) \, | \, P(D)f = 0\}$$ { f ∈ E ( X ) | P ( D ) f = 0 } satisfies $$(\Omega )$$ ( Ω ) for any differential operator P(D) and any open convex set $$X \subseteq {\mathbb {R}}^d$$ X ⊆ R d .

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.

Hermite interpolation with retractions on manifolds

Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. A novel procedure relying on the general concept of retractions is proposed to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. The well-posedness of the method is analyzed by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. A classical result on the asymptotic interpolation error of Hermite interpolation is extended to the manifold setting. Finally numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns illustrate these results and the effectiveness of the method.

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/ .