Nonconvex Feasibility Problem Research Articles

An Inertial Parametric Douglas–Rachford Splitting Method for Nonconvex Problems

In this paper, we propose an inertial parametric Douglas–Rachford splitting method for minimizing the sum of two nonconvex functions, which has a wide range of applications. The proposed algorithm combines the inertial technique, the parametric technique, and the Douglas–Rachford method. Subsequently, in theoretical analysis, we construct a new merit function and establish the convergence of the sequence generated by the inertial parametric Douglas–Rachford splitting method. Finally, we present some numerical results on nonconvex feasibility problems to illustrate the efficiency of the proposed method.

The construction of multidimensional membership functions and its application to feasibility problems

We present a novel idea of the membership function using logical expressions over inequalities. This is achieved by introducing the multidimensional membership function (distending function) using inequalities by including more variables instead of just one. In this paper, we concentrate on the application of this new approach to different kinds of feasibility problems. This new concept serves as a good tool for describing various kinds of feasibility regions. Another result here is that we present algorithms that can handle linear, nonlinear, convex and non-convex feasibility problems. Usually, in practice only the conjunction operator is used in the case of feasibility problems. A novelty of our result is that we drop this restriction, and based on our construction we will describe regions where other types of operators can also be used. We shall introduce the fundamentals of this new concept and we will show how it can be used in practical applications. Later, this approach may be applied to neural networks as well.

Event-Triggered Fault Detection Filtering of Fuzzy-Model-Based Systems With Prescribed Performance

This article deals with the fault detection filtering problem for a nonlinear dynamic system in the Takagi–Sugeno fuzzy framework. An event-triggered scheme considering network bandwidth utilization rate and fault occurrence probability is utilized, which can conserve communication resources and reduce computational burden. The fault detection fuzzy filter is used as a residual generator, and the proposed event-based fault detection scheme is formulated as a fuzzy filtering problem. Moreover, the generated residual is robust against exogenous disturbances while being sensitive to system faults, and the resulting fault detection system has asymptotic stability with a specified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> error property. The corresponding solvability conditions of the fault detection filter are constructed by converting the nonconvex feasibility problem into a series of optimization problems. Finally, a numerical simulation is conducted to demonstrate the feasibility and validity of the proposed scheme.

Solving Nonconvex Feasibility Problem on a Sphere and a Closed Ball by Douglas–Rachford Algorithm

In recent times, a variety of nonconvex feasibility problems have been solved empirically by employing Douglas–Rachford (DR) splitting methods. However, this theory is not adequate in explaining the observed success and is more concerned with the local convergence. In this paper, we study the convergence of the DR splitting method for finding a point of intersection of a closed ball and a sphere in the [Formula: see text]-dimensional Euclidean spaces. Also, we provide the region for the global convergence of the DR splitting method.

The fun is finite: Douglas-Rachford and Sudoku puzzle – Finite termination and local linear convergence

In recent years, the Douglas-Rachford splitting method has been shown to be effective at solving many non-convex optimization problems. In this paper we present a local convergence analysis for non-convex feasibility problems and show that both finite termination and local linear convergence are obtained. For a generalization of the Sudoku puzzle, we prove that the local linear rate of convergence of Douglas-Rachford is exactly $\frac{\sqrt{5}}{5}$ and independent of puzzle size. For the $s$-queens problem we prove that Douglas-Rachford converges after a finite number of iterations. Numerical results on solving Sudoku puzzles and $s$-queens puzzles are provided to support our theoretical findings.

Event-Triggered Fuzzy Control of Repeated Scalar Nonlinear Systems and its Application to Chua’s Circuit System

This paper addresses the problem of event-triggered $\mathcal {H}_{\infty }$ control for continuous Takagi-Sugeno fuzzy systems with repeated scalar nonlinearities. A feasible stability solution is first proposed based on the fuzzy-rule-dependent Lyapunov functional methods and positive definite diagonally dominant matrix techniques, which not only reduces the conservativeness of the resulting closed-loop dynamic system, but also ensures the concerned fuzzy system is asymptotically stable with a specified $\mathcal {H}_{\infty }$ disturbance attenuation performance. Then, sufficient conditions are presented for the existence of admissible state-feedback controller, and the cone complementarity linearization approach is employed to convert the non-convex feasibility problem into a sequential minimization one subject to linear matrix inequalities, which can be validly solved by employing standard numerical software. In the end, a numerical example and a Chua’s chaotic circuit system are provided to show the applicability of the proposed theories.

A factorization method for completely positive matrices

A matrix A is called completely positive, if there exists an entrywise nonnegative matrix B such that A=BBT. These matrices play a major role in combinatorial and quadratic optimization. In this paper, we study the problem of finding a nonnegative factorization BBT of a given completely positive matrix A. We formulate this factorization problem as a nonconvex feasibility problem and develop a solution method based on alternating projections. A local convergence result can be shown for this algorithm. We also provide a heuristic extension which improves the numerical performance of the algorithm. Extensive numerical tests show that the factorization method is very fast in most of the test instances and performs better than existing algorithms.

The Douglas–Rachford algorithm for convex and nonconvex feasibility problems

The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the “eight queens puzzle” known as the “(m, n)-queens problem”, and the problem of constructing a probability distribution with prescribed moments.

Robust analysis for premium-reserve models in a stochastic nonlinear discrete-time varying framework

The effective management of uncertainty and complexity in premium pricing and reserve accumulation processes provide new challenges to the decision- and policy-makers. In this regard, the implementation of complex mathematical tools and advanced statistical techniques is highly acquired. Over the last three decades, methodologies and applications of robust control theory to finance and economics have received strong attention among researchers and practitioners. However, relatively scant research has been carried out so far in relation to insurance. This paper proposes a stochastic, nonlinear time-varying premium-reserve (P-R) system with Lipschitz-type conditions in discrete-time to explore P-R system’s stability, stabilization and H∞-control properties. In our case, as an extension of the classical quadratic condition, the one-side Lipschitz conditions are also considered and a nonconvex feasibility problem is formulated and solved. In addition, both robust stochastic stability and a pre-specified disturbance attenuation level can be guaranteed. Finally, numerical examples are presented to illustrate the applicability of theoretical treatment.

Robust Analysis for Premium-Reserve Models in a Stochastic Nonlinear Discrete-Time Varying Framework

The effective management of uncertainty and complexity in premium pricing and reserve accumulation processes provide new challenges to the decision- and policy- makers. In this regard, the implementation of complex mathematical tools and advanced statistical techniques is highly acquired. Over the last three decades, methodologies and applications of robust control theory to finance and economics have received strong attention among researchers and practitioners. However, relatively scant research has been carried out so far in relation to insurance. This paper proposes a stochastic, nonlinear time-varying premium-reserve (P-R) system with Lipschitz- type conditions in discrete-time to explore P-R system’s stability, stabilisation and Hoo-control properties. In our case, as an extension of the classical quadratic condition, the one-side Lipschitz conditions are also considered and a nonconvex feasibility problem is formulated and solved. In addition, both robust stochastic stability and a pre-specified disturbance attenuation level can be guaranteed. Finally, numerical examples are presented to illustrate the applicability of theoretical treatment.

Simultaneous stabilization of a collection of uncertain discrete-time systems with time-varying state-delay via discrete-time sliding mode control

This paper considers the simultaneous stabilization problem which concerns finding a single controller to simultaneously stabilize a finite collection of discrete-time nonlinear systems with time-varying state delay and norm-bounded uncertainties. For this purpose, a simultaneously stabilizing discrete-time sliding mode controller (DT-SMC) is synthesized. Based on the delay-dependent Lyapunov–Krasovskii functional method, sufficient conditions are derived which guarantee the existence of a common sliding surface in terms of a nonconvex feasibility problem. The cone complementarity linearization approach is exploited to transform the nonconvex conditions into a convex optimization problem. Finally, the robust simultaneously stabilizing DT-SMC is designed in order to draw the closed-loop trajectories of each system onto the common sliding surface in finite time and to maintain it thereafter. The effectiveness of the proposed method is investigated by simulation of a practical example.

Application of projection algorithms to differential equations: boundary value problems

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails. doi:10.1017/S1446181118000391

The Douglas–Rachford algorithm for a hyperplane and a doubleton

The Douglas--Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which --- somewhat surprisingly --- depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.

APPLICATION OF PROJECTION ALGORITHMS TO DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm for a nonconvex setting

The Douglas-Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems which current theory cannot sufficiently explain. In this paper, we prove convergence of the Douglas-Rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.

A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems

In this paper, we propose a symmetric alternating method of multipliers for minimizing the sum of two nonconvex functions with linear constraints, which contains the classic alternating direction method of multipliers in the algorithm framework. Based on the powerful Kurdyka–Łojasiewicz property, and under some assumptions about the penalty parameter and objective function, we prove that each bounded sequence generated by the proposed method globally converges to a critical point of the augmented Lagrangian function associated with the given problem. Moreover, we report some preliminary numerical results on solving [Formula: see text] regularized sparsity optimization and nonconvex feasibility problems to indicate the feasibility and effectiveness of the proposed method.

Robust trading mechanisms over 0/1 polytopes

The problem of designing a trade mechanism (for an indivisible good) between a seller and a buyer is studied in the setting of discrete valuations of both parties using tools of finite-dimensional optimization. A robust trade design is defined as one which allows both traders a dominant strategy implementation independent of other traders’ valuations with participation incentive and no intermediary (i.e., under budget balance). The design problem which is initially formulated as a mixed-integer non-linear non-convex feasibility problem is transformed into a linear integer feasibility problem by duality arguments, and its explicit solution corresponding to posted price optimal mechanisms is derived along with full characterization of the convex hull of integer solutions. A further robustness concept is then introduced for a central planner unsure about the buyer or seller valuation distribution, a corresponding worst-case design problem over a set of possible distributions is formulated as an integer linear programming problem, and a polynomial solution procedure is given. When budget balance requirement is relaxed to feasibility only, i.e., when one allows an intermediary maximizing the expected surplus from trade, a characterization of the optimal robust trade as the solution of a simple linear program is given. A modified VCG mechanism turns out to be optimal.

Peaceman–Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.

Sufficient pruning conditions for MINLP in gas network design

One-quarter of Europe’s energy demand is provided by natural gas distributed through a vast pipeline network covering the whole of Europe. At a cost of 1 million Euros per kilometer the extension of the European pipeline network is already a multi-billion Euro business. Therefore, automatic planning tools that support the decision process are desired. We model the topology optimization problem in gas networks by a mixed-integer nonlinear program (MINLP). This gives rise to a so-called active transmission problem, a continuous nonlinear non-convex feasibility problem which emerges from the MINLP model by fixing all integral variables. We offer novel sufficient conditions for proving the infeasibility of this active transmission problem. These conditions can be expressed in the form of a mixed-integer program (MILP), i.e., the infeasibility of a non-convex continuous nonlinear program (NLP) can be certified by solving an MILP. This result provides an efficient pruning procedure in a branch-and-bound algorithm. Our computational results demonstrate a substantial speedup for the necessary computations.

Qualitative relational mapping and navigation for planetary rovers

This paper presents a novel method for qualitative mapping of large scale spaces which decouples the mapping problem from that of position estimation. The proposed framework makes use of a graphical representation of the world in order to build a map consisting of qualitative constraints on the geometric relationships between landmark triplets. This process allows a mobile robot to extract information about landmark positions using a set of minimal sensors in the absence of GPS. A novel measurement method based on camera imagery is presented which extends previous work from the field of Qualitative Spatial Reasoning. A Branch-and-Bound approach is taken to solve a set of non-convex feasibility problems required for generating off-line operator lookup tables and on-line measurements, which are fused into the map using an iterative graph update. A navigation approach for travel between distant landmarks is developed, using estimates of the Relative Neighborhood Graph extracted from the qualitative map in order to generate a sequence of landmark objectives based on proximity. Average and asymptotic performance of the mapping algorithm is evaluated using Monte Carlo tests on randomly generated maps, and a data-driven simulation is presented for a robot traversing the Jet Propulsion Laboratory Mars Yard while building a relational map. These results demonstrate that the system can be effectively used to build a map sufficiently complete and accurate for long-distance navigation as well as other applications.