Showing papers on "Solution set published in 2013"

Reaching an Optimal Consensus: Dynamical Systems That Compute Intersections of Convex Sets

Abstract: In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.

Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization

Abstract: Generalized Convexity and Generalized Monotonicity Elements of Convex Analysis Preliminaries and Basic Concepts Convex Sets Hyperplanes Convex Functions Generalized Convex Functions Optimality Criteria Subgradients and Subdifferentials Generalized Derivatives and Generalized Subdifferentials Directional Derivatives Gateaux Derivatives Dini and Dini-Hadamard Derivatives Clarke and Other Types of Derivatives Dini and Clarke Subdifferentials Nonsmooth Convexity Nonsmooth Convexity in Terms of Bifunctions Generalized Nonsmooth Convexity in Terms of Bifunctions Generalized Nonsmooth Convexity in Terms of Subdifferentials Generalized Nonsmooth Pseudolinearity in Terms of Clarke Subdifferentials Monotonocity and Generalized Monotonicity Monotonicity and Its Relation with Convexity Nonsmooth Monotonicity and Generalized Monotonicity in Terms of a Bifunction Relation between Nonsmooth Monotonicity and Nonsmooth Convexity Nonsmooth Pseudoaffine Bifunctions and Nonsmooth Pseudolinearity Generalized Monotonicity for Set-Valued Maps Nonsmooth Variational Inequalities and Nonsmooth Optimization Elements of Variational Inequalities Variational Inequalities and Related Problems Basic Existence and Uniqueness Results Gap Functions Solution Methods Nonsmooth Variational Inequalities Nonsmooth Variational Inequalities in Terms of a Bifunction Relation between an Optimization Problem and Nonsmooth Variational Inequalities Existence Criteria Extended Nonsmooth Variational Inequalities Gap Functions and Saddle Point Characterization Characterizations of Solution Sets of Optimization Problem and Nonsmooth Variational Inequalities Characterizations of the Solution Set of an Optimization Problem with a Pseudolinear Objective Function Characterizations of the Solution Set of Variational Inequalities Involving Pseudoaffine Bifunctions Lagrange Multiplier Characterizations of Solution Set of an Optimization Problem Nonsmooth Generalized Variational Inequalities and Optimization Problems Generalized Variational Inequalities and Related Topics Basic Existence and Uniqueness Results Gap Functions for Generalized Variational Inequalities Generalized Variational Inequalities in Terms of the Clarke Subdifferential and Optimization Problems Characterizations of Solution Sets of an Optimization Problem with Generalized Pseudolinear Objective Function Appendix A: Set-Valued Maps Appendix B: Elements of Nonlinear Analysis Index

Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

Abstract: The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov $n$-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

Boundedly rational user equilibria (BRUE): Mathematical formulation and solution sets

Abstract: Boundedly rational user equilibria (BRUE) represent traffic flow distribution patterns where travellers can take any route whose travel cost is within an ‘indifference band’ of the shortest path cost. Those traffic flow patterns satisfying the above condition constitute a set, named the BRUE solution set. It is important to obtain all the BRUE flow patterns, because it can help predict the variation of the link flow pattern in a traffic network under the boundedly rational behavior assumption. However, the methodology of constructing the BRUE set has been lacking in the established literature. This paper fills the gap by constructing the BRUE solution set on traffic networks with fixed demands connecting multiple OD pairs. According to the definition of the ɛ-BRUE, where ɛ is the indifference band for the perceived travel cost, we formulate the ɛ-BRUE problem as a nonlinear complementarity problem (NCP), so that a BRUE solution can be obtained by solving a BRUE-NCP formulation. To obtain the whole BRUE solution set encompassing all BRUE flow patterns, we firstly propose a methodology of generating various path combinations which may be utilized under the boundedly rational behavior assumption. We find out that with the increase of the indifference band, the path set that contains boundedly rational equilibrium flows will be augmented, and the critical values of indifference bands to augment these path sets can be identified by solving a family of mathematical programs with equilibrium constraints (MPEC) sequentially. After these utilized path sets are attained, the BRUE solution set can be obtained when we assign all traffic demands to these utilized paths. Various numerical examples are given to illustrate our findings.

On a new class of differential variational inequalities and a stability result

Abstract: This paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. The purpose of this paper is two-fold. Firstly, we show that these differential variational inequalities, when considering slow solutions and the more general level of a Hilbert space, contain projected dynamical systems, another recent subclass of general differential inclusions. This relation follows from a precise geometric description of the directional derivative of the metric projection in Hilbert space, which is based on the notion of the quasi relative interior. Secondly we are concerned with stability of the solution set to this class of differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated set-valued maps and the constraint set. Here we impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty, and employ Mosco convergence as set convergence. Also as a consequence, we obtain a stability result for linear complementarity systems.

The optimal solution set of the interval linear programming problems

Abstract: Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ j , μ ij , μ i ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ ij = μ i = λ j = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ j , μ ij and μ i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.

Using Laplacian Eigenmap as Heuristic Information to Solve Nonlinear Constraints Defined on a Graph and Its Application in Distributed Range-Free Localization of Wireless Sensor Networks

Abstract: In this paper, we are concerned with the problem of nonlinear inequalities defined on a graph. The feasible solution set to this problem is often infinity and Laplacian eigenmap is used as heuristic information to gain better performance in the solution. A continuous-time projected neural network, and the corresponding discrete-time projected neural network are both given to tackle this problem iteratively. The convergence of the neural networks are proven in theory. The effectiveness of the proposed neural networks are tested and compared with others via its applications in the range-free localization of wireless sensor networks. Simulations demonstrate the effectiveness of the proposed methods.

A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems

Abstract: Generalized Nash equilibrium problems are important examples of quasi-equilibrium problems. The aim of this paper is to study a general class of algorithms for solving such problems. The method is a hybrid extragradient method whose second step consists in finding a descent direction for the distance function to the solution set. This is done thanks to a linesearch. Two descent directions are studied and for each one several steplengths are proposed to obtain the next iterate. A general convergence theorem applicable to each algorithm of the class is presented. It is obtained under weak assumptions: the pseudomonotonicity of the equilibrium function and the continuity of the multivalued mapping defining the constraint set of the quasi-equilibrium problem. Finally some preliminary numerical results are displayed to show the behavior of each algorithm of the class on generalized Nash equilibrium problems.

Solving Multi-Objective Economic Dispatch Problem Via Semidefinite Programming

Abstract: This paper presents a solution for multi-objective economic dispatch problem with transmission losses semidefinite programming (SDP) formulation. The vector objective is reduced to an equivalent scalar objective through the weighted sum method. The resulting optimization problem is formulated as a convex optimization via SDP relaxation. The convex optimization problem was solved to obtain Pareto-optimal solutions. The diversity of the solution set was improved by a nonlinear selection of the weight factor. Simulations were performed on IEEE 30-bus, 57-bus, and 118-bus test systems to investigate the effectiveness of the proposed approach. Solutions were compared to those from one of the well-known evolutionary methods. Results show that SDP has an inherently good convergence property and a lower but comparable diversity property.

Convergence of Regularized Time-Stepping Methods for Differential Variational Inequalities

Abstract: This paper provides convergence analysis of regularized time-stepping methods for the differential variational inequality (DVI), which consists of a system of ordinary differential equations and a parametric variational inequality (PVI) as the constraint. The PVI often has multiple solutions at each step of a time-stepping method, and it is hard to choose an appropriate solution for guaranteeing the convergence. In [L. Han, A. Tiwari, M. K. Camlibel and J.-S. Pang, SIAM J. Numer. Anal., 47 (2009) pp. 3768--3796], the authors proposed to use “least-norm solutions” of parametric linear complementarity problems at each step of the time-stepping method for the monotone linear complementarity system and showed the novelty and advantages of the use of the least-norm solutions. However, in numerical implementation, when the PVI is not monotone and its solution set is not convex, finding a least-norm solution is difficult. This paper extends the Tikhonov regularization approximation to the P$_0$-function DVI, whi...

On the Linear Convergence of a Proximal Gradient Method for a Class of Nonsmooth Convex Minimization Problems

Abstract: We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping, regularized by the sum of both � 1-norm and � 2-norm of the optimization variables. This class of problems arise naturally from applications in sparse group Lasso, which is a popular technique for variable selection. An effective approach to solve such problems is by the Proximal Gradient Method (PGM). In this paper we prove a local error bound around the optimal solution set for this problem and use it to establish the linear convergence of the PGM method without assuming strong convexity of the overall objective function.

Existence theorems of the variational-hemivariational inequalities

Abstract: This paper is devoted to the existence of solutions for the variational-hemivariational inequalities in reflexive Banach spaces. Using the notion of the stable $${\phi}$$ -quasimonotonicity and the properties of Clarke's generalized directional derivative and Clarke's generalized gradient, some existence results of solutions are proved when the constrained set is nonempty, bounded (or unbounded), closed and convex. Moreover, a sufficient condition to the boundedness of the solution set and a necessary and sufficient condition to the existence of solutions are also derived. The results presented in this paper generalize and improve some known results.

Analysing the performance of dynamic multi-objective optimisation algorithms

Abstract: Dynamic multi-objective optimisation problems (DMOOPs) have more than one objective, with at least one objective changing over time. Since at least two of the objectives are normally in conflict with one another, a single solution does not exist and the goal of the algorithm is to track a set of tradeoff solutions over time. Analysing the performance of a dynamic multi-objective optimisation algorithm (DMOA) is not a trivial task. For each environment (before a change occurs) the DMOA has to find a set of solutions that are both diverse and as close as possible to the optimal trade-off solution set. In addition, the DMOA has to track the changing set of trade-off solutions over time. Approaches used to analyse the performance of dynamic single-objective optimisation algorithms (DSOAs) and DMOAs do not provide any information about the ability of the algorithms to track the changing optimum. Therefore, this paper introduces a new approach to analyse the performance of DMOAs and applies this approach to the results obtained by five DMOAs. In addition, it compares the new analysis approach to another approach that does not take the tracking ability of the DMOAs into account. The results indicate that the new analysis approach provide additional information, measuring the ability of the algorithm to find good performance measure values while tracking the changing optima.

Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization Problems

Abstract: In many applications, it is required to optimize several conflicting objectives concurrently leading to a multobjective optimization problem (MOP). The solution set of a MOP, the Pareto set, typically forms a (k-1)-dimensional object, where k is the number of objectives involved in the optimization problem. The purpose of this chapter is to give an overview of recently developed set oriented techniques - subdivision and continuation methods - for the computation of Pareto sets \(\mathcal{P}\) of a givenMOP. All these methods have in common that they create sequences of box collections which aim for a tight covering of \(\mathcal{P}\). Further, we present a class of multiobjective optimal control problems which can be efficiently handled by the set oriented continuation methods using a transformation into high-dimensionalMOPs. We illustrate all the methods on both academic and real world examples.

A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria

Abstract: We propose a projection algorithm for solving an equilibrium problem (EP) where the bifunction is pseudomonotone with respect to its solution set. The algorithm is further combined with a cutting technique for minimizing the norm over the solution set of an EP whose bifunction is pseudomonotone with respect to the solution set.

Newton iterations in implicit time-stepping scheme for differential linear complementarity systems

Abstract: We propose a generalized Newton method for solving the system of nonlinear equations with linear complementarity constraints in the implicit or semi-implicit time-stepping scheme for differential linear complementarity systems (DLCS). We choose a specific solution from the solution set of the linear complementarity constraints to define a locally Lipschitz continuous right-hand-side function in the differential equation. Moreover, we present a simple formula to compute an element in the Clarke generalized Jacobian of the solution function. We show that the implicit or semi-implicit time-stepping scheme using the generalized Newton method can be applied to a class of DLCS including the nondegenerate matrix DLCS and hidden Z-matrix DLCS, and has a superlinear convergence rate. To illustrate our approach, we show that choosing the least-element solution from the solution set of the Z-matrix linear complementarity constraints can define a Lipschitz continuous right-hand-side function with a computable Lipschitz constant. The Lipschitz constant helps us to choose the step size of the time-stepping scheme and guarantee the convergence.

Outer enclosures to the parametric AE solution set

Abstract: We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by universally and existentially quantified parameters, and the former precede the latter. Based on a recently obtained explicit description of such solution sets, we present three approaches for obtaining outer estimations of parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Polynomially computable outer bounds for parametric AE solution sets are obtained by parametric AE generalization of a single-step Bauer---Skeel method. In the special case of parametric tolerable solution sets, we derive an enclosure based on linear programming approach; this enclosure is optimal under some assumption. The application of these approaches to parametric tolerable and controllable solution sets is discussed. Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods.

Certified Parallelotope Continuation for One-Manifolds

Abstract: Starting from an initial solution, continuation methods efficiently produce a sequence of points on a manifold typically defined as the solution set of an underconstrained system of equations. They have a wide range of applications ranging from curve plotting to polynomial root-finding by homotopy. However, classical methods cannot guarantee that the returned points all belong to the same connected component of the manifold, i.e., they may jump from one component to another. Trying to overcome this issue has given birth to several sophisticated heuristics on the one hand and to guaranteed methods based on rigorous computations on the other hand. In this paper we introduce a new rigorous predictor corrector continuation method based on interval computations. Its novelty lies in the fact that it uses parallelotopes as defined in A. Goldsztejn and L. Granvilliers, A new framework for sharp and efficient resolution of NCSP with manifolds of solutions, Constraints, 15 (2010), pp. 190--212, to enclose consecuti...

Risk-neutral second best toll pricing

Abstract: We propose a risk-neutral second best toll pricing (SBTP) scheme to account for the possible nonuniqueness of user equilibrium solutions. The scheme is designed to optimize for the expected objective value as the UE solution varies within the solution set. We show that such a risk-neutral scheme can be formulated as a stochastic program, which complements the traditional risk-prone SBTP approach and the risk-averse SBTP approach we developed recently. The proposed model can be solved by a simulation-based optimization algorithm that contains three major steps: characterization of the UE solution set, random sampling over the solution set, and a two-phase simulation optimization step. Numerical results illustrate that the proposed risk-neutral design scheme is less aggressive than the risk-prone scheme and less conservative than the risk-averse scheme, and may thus be more preferable from a toll designer’s point of view.

Generalized B-Well-Posedness for Set Optimization Problems

Abstract: This paper aims at studying the generalized well-posedness in the sense of Bednarczuk for set optimization problems with set-valued maps. Three kinds of B-well-posedness for set optimization problems are introduced. Some relations among the three kinds of B-well-posedness are established. Necessary and sufficient conditions of well-posedness for set optimization problems are obtained.

A Matrix Approach to Latticized Linear Programming With Fuzzy-Relation Inequality Constraints

Abstract: This paper investigates the latticized linear programming that is subject to the fuzzy-relation inequality (FRI) constraints with the max-min composition by using the semi-tensor product method, and proposes a matrix approach to this problem. First, the resolution of the FRI is studied, and it is proved that all the minimal solutions and the unique maximum solution are within the finite parameter set solutions. Based on this and using the semi-tensor product, solving FRIs is converted to solving a set of algebraic inequalities, and some new results on the resolution of FRIs are presented. Second, the latticized linear programming that is subject to the FRI constraints is solved by taking the following two key steps: 1) the optimal value is obtained by calculating the minimum value of the objective function among all the minimal solutions to the FRI constraints; and 2) the optimal solution set is obtained by solving the fuzzy-relation equation that is generated by letting the objective function equal to the optimal value. The study of illustrative examples shows that the new results that are obtained in this paper are very effective in solving the latticized linear programming subject to the FRI constraints.

Optimality conditions for a unified vector optimization problem with not necessarily preordering relations

Abstract: This paper studies a general vector optimization problem which encompasses those related to efficiency, weak efficiency, strict efficiency, proper efficiency and approximate efficiency among others involving non necessarily preordering relations. Based on existing results about complete characterization by scalarization of the solution set obtained by the same authors, several properties of (generalized) convexity and lower semicontinuity of the composition of the scalarizing functional and the objective vector function are studied. Finally, some optimality conditions are presented through subdifferentials in the convex and nonconvex case.

The effect of calmness on the solution set of systems of nonlinear equations

Abstract: We address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness. This property, also called upper Lipschitz-continuity in the literature, can be described by a local error bound and is being widely used as a regularity condition in optimization. Indeed, it is known to be significantly weaker than classic regularity assumptions that imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian of the function that defines the system of equations must be locally constant on the solution set. In addition, we prove that locally, the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications.

A smoothing Newton method for the second-order cone complementarity problem

Abstract: In this paper we introduce a new smoothing function and show that it is coercive under suitable assumptions. Based on this new function, we propose a smoothing Newton method for solving the second-order cone complementarity problem (SOCCP). The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. It is shown that any accumulation point of the iteration sequence generated by the proposed algorithm is a solution to the SOCCP. Furthermore, we prove that the generated sequence is bounded if the solution set of the SOCCP is nonempty and bounded. Under the assumption of nonsingularity, we establish the local quadratic convergence of the algorithm without the strict complementarity condition. Numerical results indicate that the proposed algorithm is promising.