Nonconvex Sets Research Articles

On the topological degree of planar maps avoiding normal cones

The classical Poincare-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones. We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.

On the sample-mean method for computing hyper-volumes

Abstract Estimating hyper-volumes of convex and non-convex sets are of interest in a number of areas. In this article we develop further a simple geometric Monte Carlo method, known also as the sample-mean method, which transforms the domain to an equivalent hyper-sphere with the same volume. We first examine the performance of the method to compute the volumes of star-convex unit balls and show that it gives accurate estimates of their volumes. We then examine the use of this method for computing the volumes of nonstar-shaped domains. In particular, we develop two algorithms, which couple the sample-mean method with algebraic and geometric techniques, to generate and compute the volumes of low-dimensional stability domains in parameter space.

Network Capacity Maximization Using Route Choice and Signal Control With Multiple OD Pairs

In this paper, we investigate a hybrid dynamical system which incorporates flow swap process, green-time proportion swap process, and flow divergence for a general network with multiple Origin-Destination (OD) pairs and multiple routes, where flow swap process is specified in which traffic swaps from more costly to less costly input links, green-time proportion swap process is specified in which green time at each intersection swaps from less pressurized stages to more pressurized stages, flow may diverge at each intersection from one OD pair to other OD pairs. Unlike the dynamical system model, where bottleneck delays need to be intentionally constructed to yield the equilibrium flow vector and green-time proportion vector, we propose a novel control policy to fill the gap by only adjusting the green-time proportion vector. We derive a sufficient condition for the existence of equilibrium of the dynamical system under the mild constraints that 1) the travel cost function and stage pressure function should be continuous functions and 2) the flow and green-time proportion swap processes project all flow and green-time proportion vectors on the boundary of the feasible region onto itself. We derive the condition of unique equilibrium for fixed green-time proportion vector and show that with varying green-time proportion vector, the set of equilibria is a compact, non-convex set, and with the same partial derivative of travel cost function with respect to the flow and green-time proportion vectors. Finally, we prove the stability of the proposed dynamical system by using Lyapunov stability analysis.

The Frontier of SGD and Its Variants in Machine Learning

A Numerical optimization is a classical field in operation research and computer science, which has been widely used in areas such as physics and economics. Although optimization algorithms have achieved great success for plenty of applications, handling the big data in the best fashion possible is a very inspiring and demanding challenge in the artificial intelligence era. Stochastic gradient descent (SGD) is pretty simple but surprisingly, highly effective in machine learning models, such as support vector machine (SVM) and deep neural network (DNN). Theoretically, the performance of SGD for convex optimization is well understood. But, for the non-convex setting, which is very common for the machine learning problems, to obtain the theoretical guarantee for SGD and its variants is still a standing problem. In the paper, we do a survey about the SGD and its variants such as Momentum, ADAM and SVRG, differentiate their algorithms and applications and present some recent breakthrough and open problems.

Optimal Multivariate Gaussian Fitting with Applications to PSF Modeling in Two-Photon Microscopy Imaging

Fitting Gaussian functions to empirical data is a crucial task in a variety of scientific applications, especially in image processing. However, most of the existing approaches for performing such fitting are restricted to two dimensions and they cannot be easily extended to higher dimensions. Moreover, they are usually based on alternating minimization schemes which benefit from few theoretical guarantees in the underlying nonconvex setting. In this paper, we provide a novel variational formulation of the multivariate Gaussian fitting problem, which is applicable to any dimension and accounts for possible nonzero background and noise in the input data. The block multiconvexity of our objective function leads us to propose a proximal alternating method to minimize it in order to estimate the Gaussian shape parameters. The resulting FIGARO algorithm is shown to converge to a critical point under mild assumptions. The algorithm shows a good robustness when tested on synthetic datasets. To demonstrate the versatility of FIGARO, we also illustrate its excellent performance in the fitting of the point spread functions of experimental raw data from a two-photon fluorescence microscope.

Generalised-prox-regularity in reflexive smooth Banach spaces with smooth dual norm

In this paper, we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth Banach spaces with smooth dual norm started by the first author in [9]. We define a new class of nonconvex sets in Banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm.

Set-valued anisotropic dry friction laws: formulation, experimental verification and instability phenomenon

Many technical applications, such as brakes and metal forming processes, are affected by anisotropic frictional behavior, where the magnitude and the direction of the friction force are dependent on the sliding direction. Existing dry friction laws do not sufficiently describe all relevant macroscopic aspects of anisotropic friction, and the influence on the dynamics of mechanical systems is largely unknown. Furthermore, previous experimental work on anisotropic friction is limited and the fact that the friction force is not always acting parallel to the sliding direction is often neglected. In this paper, an anisotropic dry friction law with the capability to describe the nonsmooth behavior of stick and slip and allowing for non-convex but star-shaped sets of admissible friction forces is formulated using tools from convex analysis. The formulation of the friction law as normal cone inclusion enables the direct implementation in numerical time-stepping schemes. The stability of systems with anisotropic friction is studied and an eigenvalue analysis reveals that the anisotropic friction law is in theory capable of causing anisotropic friction-induced instability. In addition, experimental setups for detailed investigations of the frictional behavior are described. The measurements reveal complex shaped force reservoirs and confirm the validity of the presented friction law. Finally, it is shown that the presented friction law leads to a more accurate prediction of the motion of nonsmooth mechanical systems.

Statistical Inference for the Population Landscape via Moment-Adjusted Stochastic Gradients

SummaryModern statistical inference tasks often require iterative optimization methods to compute the solution. Convergence analysis from an optimization viewpoint informs us only how well the solution is approximated numerically but overlooks the sampling nature of the data. In contrast, recognizing the randomness in the data, statisticians are keen to provide uncertainty quantification, or confidence, for the solution obtained by using iterative optimization methods. The paper makes progress along this direction by introducing moment-adjusted stochastic gradient descent: a new stochastic optimization method for statistical inference. We establish non-asymptotic theory that characterizes the statistical distribution for certain iterative methods with optimization guarantees. On the statistical front, the theory allows for model misspecification, with very mild conditions on the data. For optimization, the theory is flexible for both convex and non-convex cases. Remarkably, the moment adjusting idea motivated from ‘error standardization’ in statistics achieves a similar effect to acceleration in first-order optimization methods that are used to fit generalized linear models. We also demonstrate this acceleration effect in the non-convex setting through numerical experiments.

A second welfare theorem in a non-convex economy: The case of antichain-convexity

We introduce the notion of an antichain-convex set to extend Debreu (1954)’s version of the second welfare theorem to economies where either the aggregate production set or preference relations are not convex. We show that – possibly after some redistribution of individuals’ wealth – the Pareto optima of some economies which are marked by certain types of non-convexities can be spontaneously obtained as valuation quasiequilibria and equilibria: both equilibrium notions are to be understood in Debreu (1954)’s sense. From a purely structural point of view, the mathematical contribution of this work is the study of the conditions that guarantee the convexity of the Minkowski sum of finitely many possibly non-convex sets. Such a study allows us to obtain a version of the Minkowski\\Hahn–Banach separation theorem which dispenses with the convexity of the sets to be separated and which can be naturally applied in standard proofs of the second welfare theorem; in addition – and equally importantly – the study allows to get a deeper understanding of the conditions on the single production sets of an economy that guarantee the convexity of their aggregate.

Voltage Analytics for Power Distribution Network Topology Verification

Distribution grids constitute complex networks of lines oftentimes reconfigured to minimize losses, balance loads, alleviate faults, or for maintenance purposes. Topology monitoring becomes a critical task for optimal grid scheduling. While synchrophasor installations are limited in low-voltage grids, utilities have an abundance of smart meter data at their disposal. In this context, a statistical learning framework is put forth for verifying single-phase grid structures using non-synchronized voltage data. The related maximum likelihood task boils down to minimizing a non-convex function over a non-convex set. The function involves the sample voltage covariance matrix and the feasible set is relaxed to its convex hull. Asymptotically in the number of data, the actual topology yields the global minimizer of the original and the relaxed problems. Under a simplified data model, the function turns out to be convex, thus offering optimality guarantees. Prior information on line statuses is also incorporated via a maximum a-posteriori approach. The formulated tasks are tackled using solvers having complementary strengths. Numerical tests using real data on benchmark feeders demonstrate that reliable topology estimates can be acquired even with a few smart meter data, while the non-convex schemes exhibit superior line verification performance at the expense of additional computational time.

Exact Formula for the Second-Order Tangent Set of the Second-Order Cone Complementarity Set

The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the second-order cone, and calculating the second-order directional derivative of the projection operator over the second-order cone. As an application, we derive second-order necessary optimality conditions for the mathematical program with second-order cone complementarity constraints.

Nonconvex Constrained Consensus of Discrete-Time Heterogeneous Multi-Agent Systems With Arbitrarily Switching Topologies

This paper mainly focuses on the velocity-constrained consensus problem of discrete-time heterogeneous multi-agent systems with nonconvex constraints and arbitrarily switching topologies, where each agent has first-order or second-order dynamics. To solve this problem, a distributed algorithm is proposed based on a contraction operator. By employing the properties of the stochastic matrix, it is shown that all agents' position states could converge to a common point and second-order agents' velocity states could remain in corresponding nonconvex constraint sets and converge to zero as long as the joint communication topology has one directed spanning tree. Finally, the numerical simulation results are provided to verify the effectiveness of the proposed algorithms.

Distributed Finite-Time Rotating Encirclement Control of Multiagent Systems With Nonconvex Input Constraints

This paper is devoted to the distributed finite-time rotating encirclement control problem of multiagent systems with nonconvex input constraints. Three distributed finite-time estimators are designed to estimate the targets' geometric center, the reference polar radius, and the reference polar angular. Then, the estimated values are employed to get the reference trajectory for each agent, which guarantees that the references can uniformly distribute on a circle and have the same angular velocity. The distributed finite-time control laws are proposed for all agents to drive their positions tracking reference trajectories by introducing a constraint operator to cope with the nonconvex input constraints, which guarantees that the control inputs can invariably lie in the corresponding nonconvex constraint sets. A numerical simulation is performed to demonstrate the correctness and effectiveness of the proposed control scheme.

Limiting proper minimal points of nonconvex sets in finite-dimensional spaces

In this paper, limiting proper minimal points of nonconvex sets in Euclidean finite-dimensional spaces are investigated. The relationships between these minimal points and Borwein, Benson, and Henig proper minimal points, under appropriate assumptions, are established. Furthermore, a density property is derived and a linear characterization of limiting proper minimal points is provided.

On reserve and double covering problems for the sets with non-Euclidean metrics

The article is devoted to Circle covering problem for a bounded set in a two-dimensional metric space with a given amount of circles. Here we focus on a more complex problem of constructing reserve and multiply coverings. Besides that, we consider the case where covering set is a multiply-connected domain. The numerical algorithms based on fundamental physical principles, established by Fermat and Huygens, are suggested and implemented. This allows us to solve the problems for the cases of non-convex sets and non-Euclidean metrics. Preliminary results of numerical experiments are presented and discussed. Calculations show the applicability of the proposed approach.

Minimax theorems in a fully non-convex setting

In this paper, we establish two minimax theorems for functions $f:X\times I\to {\bf R}$, where $I$ is a real interval, without assuming that $f(x,\cdot)$ is quasi-concave. Also, some related applications are presented.

Gradient Projection Method for Optimization Problems with a Constraint in the Form of the Intersection of a Smooth Surface and a Convex Closed Set

The gradient projection method is generalized to the case of nonconvex sets of constraints representing the set-theoretic intersection of a smooth surface with a convex closed set. Necessary optimality conditions are studied, and the convergence of the method is analyzed.

Sparse Ultrasound Imaging via Manifold Low-Rank Approximation and Non-Convex Greedy Pursuit.

Model-based image reconstruction has improved contrast and spatial resolution in imaging applications such as magnetic resonance imaging and emission computed tomography. However, these methods have not succeeded in pulse-echo applications like ultrasound imaging due to the typical assumption of a finite grid of possible scatterer locations in a medium–an assumption that does not reflect the continuous nature of real world objects and creates a problem known as off-grid deviation. To cope with this problem, we present a method of dictionary expansion and constrained reconstruction that approximates the continuous manifold of all possible scatterer locations within a region of interest. The expanded dictionary is created using a highly coherent sampling of the region of interest, followed by a rank reduction procedure. We develop a greedy algorithm, based on the Orthogonal Matching Pursuit, that uses a correlation-based non-convex constraint set that allows for the division of the region of interest into cells of any size. To evaluate the performance of the method, we present results of two-dimensional ultrasound imaging with simulated data in a nondestructive testing application. Our method succeeds in the reconstructions of sparse images from noisy measurements, providing higher accuracy than previous approaches based on regular discrete models.

A Non-Euclidean Gradient Descent Framework for Non-Convex Matrix Factorization

We study convex optimization problems that feature low-rank matrix solutions. In such scenarios, non-convex methods offer significant advantages over convex methods due to their lower space complexity, as well as practical faster convergence. Under mild assumptions, these methods feature global convergence guarantees. In this paper, we extend the results on this matter by following a different path. We derive a non-Euclidean optimization framework in the non-convex setting that takes nonlinear gradient steps on the factors. Our framework enables the possibility to further exploit the underlying problem structures, such as sparsity or low-rankness on the factorized domain, or better dimensional dependence of the smoothness parameters of the objectives. We prove that the non-Euclidean methods enjoy the same rigorous guarantees as their Euclidean counterparts under appropriate assumptions. Numerical evidence with Fourier ptychography and FastText applications, using real data, shows that our approach can enhance solution quality, as well as convergence speed over the standard non-convex approaches.

Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_{k}(x):= \left\{\begin{array}{llll} \alpha_{k} \delta^{ij}(x) &x_{n}>0 \beta_{k} \delta^{ij}(x) &x_{n}<0. \end{array}\right. $$ In particular, we will show that the mean value sets associated to such an operator need not be convex as αk and βk converge to 1. This example then leads to an example of nonconvex mean value sets for smooth aij(x).