Showing papers on "Solution set published in 1998"

A Globally Convergent Inexact Newton Method for Systems of Monotone Equations

Abstract: We propose an algorithm for solving systems of monotone equations which combines Newton, proximal point, and projection methodologies. An important property of the algorithm is that the whole sequence of iterates is always globally convergent to a solution of the system without any additional regularity assumptions. Moreover, under standard assumptions the local superlinear rate of convergence is achieved. As opposed to classical globalization strategies for Newton methods, for computing the stepsize we do not use linesearch aimed at decreasing the value of some merit function. Instead, linesearch in the approximate Newton direction is used to construct an appropriate hyperplane which separates the current iterate from the solution set. This step is followed by projecting the current iterate onto this hyperplane, which ensures global convergence of the algorithm. Computational cost of each iteration of our method is of the same order as that of the classical damped Newton method. The crucial advantage is that our method is truly globally convergent. In particular, it cannot get trapped in a stationary point of a merit function. The presented algorithm is motivated by the hybrid projection-proximal point method proposed in [25].

A hybrid projection-proximal point algorithm.

Abstract: We propose a modification of the classical proximal point algorithm for finding zeroes of a maximal monotone operator in a Hilbert space. In particular, an approximate proximal point iteration is used to construct a hyperplane which strictly separates the current iterate from the solution set of the problem. This step is then followed by a projection of the current iterate onto the separating hyperplane. All information required for this projection operation is readily available at the end of the approximate proximal step, and therefore this projection entails no additional computational cost. The new algorithm allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems, which yields a more practical framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. Additionally, presented analysis yields an alternative proof of convergence for the exact proximal point method, which allows a nice geometric interpretation, and is somewhat more intuitive than the classical proof.

Hermite–Sinusoidal-Gaussian Beams in Complex Optical Systems

Abstract: Sinusoidal-Gaussian beams have recently been obtained as exact solutions of the paraxial wave equation for propagation in complex optical systems. Another useful set of beam solutions for Cartesian coordinate systems is based on Hermite–Gaussian functions. A generalization of these solution sets is developed here. The new solutions are referred to as Hermite–sinusoidal-Gaussian beams, because they are in the form of a product of Hermite-polynomial functions of either complex or real argument, sinusoidal functions of complex argument, and Gaussian functions of complex argument. These beams are valid for propagation through systems that can be represented in terms of complex beam matrices, and the previous beam solution sets are special cases of these more general results. Propagation characteristics and applications of these beams are discussed, including their use as a basis set for propagation of arbitrary electromagnetic beams.

Interaction of Design and Control: Optimization with Dynamic Models

Abstract: Process design is usually approached by considering the steady-state performance of the process based on an economic objective. Only after the process design is determined are the operability aspects of the process considered. This sequential treatment of the process design problem neglects the fact that the dynamic controllability of the process is an inherent property of its design. This work considers a systematic approach where the interaction between the steady-state design and the dynamic controllability is analyzed by simultaneously considering both economic and controllability criteria. This method follows a process synthesis approach where a process superstructure is used to represent the set of structural alternatives. This superstructure is modeled mathematically by a set of differential and algebraic equations which contains both continuous and integer variables. Two objectives representing the steady-state design and dynamic controllability of the process are considered. The problem formulation thus is a multiobjective Mixed Integer Optimal Control Problem (MIOCP). The multiobjective problem is solved using an ∈-constraint method to determine the noninferior solution set which indicates the trade-offs between the design and controllability of the process. The (MIOCP) is transformed to a Mixed Integer Nonlinear Program with Differential and Algebraic Constraints (MINLP/DAE) by applying a control parameterization technique. An algorithm which extends the concepts of MINLP algorithms to handle dynamic systems is presented for the solution of the MINLP/DAE problem. The MINLP/DAE solution algorithm decomposes the problem into a NLP/DAE primal and MILP master problems which provide upper and lower bounds on the solution of the problem. The MINLP/DAE algorithm is implemented in the framework MINOPT which is used as the computational tool for the analysis of the interaction of design and control. The solution of the MINLP/DAE problems is repeated with varying values of ∈ to generated the noninferior solution set. The proposed approach is applied to three design/control examples: a reactor network involving two CSTRs, an ideal binary distillation column, and a reactor/separator/recycle system. The results of these design examples quantitatively illustrate the trade-offs between the steady-state economic and dynamic controllability objectives.

The Linear l 1 Estimator and the Huber M-Estimator

Abstract: Relationships between a linear l1 estimation problem and the Huber M-estimator problem can be easily established by their dual formulations. The least norm solution of a linear programming problem studied by Mangasarian and Meyer [SIAM J. Control Optim., 17 (1979), pp. 745--752] provides a key link between the dual problems. Based on the dual formulations, we establish a local linearity property of the Huber M-estimators with respect to the tuning parameter $\gamma$ and prove that the solution set of the Huber M-estimator problem is Lipschitz continuous with respect to perturbations of the tuning parameter $\gamma$. As a consequence, the set of the linear l1 estimators is the limit of the set of the Huber M-estimators as $\gamma\to 0+. Thus, the Huber M-estimator problem has many solutions for small tuning parameter $\gamma$ if the linear l1 estimation problem has multiple solutions. A recursive version of Madsen and Nielsen's algorithm [SIAM J. Optim., 3 (1993), pp. 223--235] based on computation of the Huber M-estimator is proposed for finding a linear l1 estimator.

A Parametric Mixed-Integer Optimization Algorithm for Multiobjective Engineering Problems Involving Discrete Decisions

Abstract: An iterative, decomposition-based algorithm is proposed in this paper, for the solution of convex parametric MINLP problems and, in extension, the identification of the noninferior solution set in multiobjective problems involving continuous and discrete decisions. The parametric optimal solution is constructed via an upper- and lower-bound procedure. Parametric upper bounds are identified with the solution of parametric NLP problems, while the parametric lower bound is updated in each iteration via the solution of a parametric MILP Master problem, which involves only the binary variables of the initial problem. Convergence properties and computational requirements are discussed in example problems from process synthesis under uncertainty and simultaneous product/process design.

Global Error Bounds for Convex Multifunctions and Applications

Abstract: We give some results on the existence of global error bounds for convex multifunctions between normed linear spaces (until the present, only some results on local error bounds have been known in this general setting). As applications we obtain, among others, improvements of a theorem of Robinson on global error bounds for convex inequalities, of a result of Luo and Tseng on uniform boundedness of the Hoffman constants for linear inequalities and equalities, and of Lotov's result on pointwise Lipschitz continuity of the solution sets of linear inequalities, with respect to data perturbations.

Chapter 13 Π10 classes in mathematics

Abstract: Publisher Summary This chapter presents the applications of the ∏ 0 1 classes in a wide variety of fields including logic, nonmonotonic logic, algebra, combinatorics, orderings, and game theory. ∏ 0 1 Classes occur in many areas of recursive mathematic.. In each case, it shows that the set of solutions to a given recursive instance of the problem may be represented as an arbitrary (bounded, or recursively bounded) ∏ 0 1 class. For example, Bean observed that the set of κ-colorings of a recursive graph “G” is a recursively bounded ∏ 0 1 class. The chapter presents results on ∏ 0 1 classes and their members that can be applied to the mathematical problems. Through these results on various recursive problems in mathematics, the solution sets are shown to be represented by “r.b.” (Recursively bound) ∏ 0 1 classes and, in some cases, to represent all possible r.b. ∏ 0 1 classes (Certain problems are represented by bounded or unbounded represent all possible r.b. represent∏ 0 1 classes). The results are then applied to derive corollaries such as the existence of recursive instances of each such problem with no recursive solutions.

An Admissible Solution Approach to Inverse Electrocardiography

Abstract: The goal of the inverse problem of electrocardiography is noninvasive discrimination and characterization of normal/abnormal cardiac electrical activity from measurements of body surface potentials. Smoothing and attenuation in the torso volume conductor cause the problem to be ill posed. Standard regularized solutions employ an a priori constraint to achieve reliability and may be biased by the constraint chosen as well as the regularization parameter used to weight it. In this paper, we describe an approach that reformulates this inverse problem as the search for a solution that is a member of an admissible solution set; admissibility is defined in terms of the available constraints. In principle, this approach can utilize as many constraints as may be available, unlike standard techniques which do not easily permit the use of multiple constraints. No regularization parameter is required; instead, we need to choose the nature and size of the constraint sets. Constraints described include several spatial constraints, weighted constraints, and temporal constraints. We describe a solution approach based on iterative convex optimization, and the algorithm—the ellipsoid algorithm—which we have used. Accuracy and feasibility of the method are illustrated with simulation results using dipole sources and measured epicardial potentials. © 1998 Biomedical Engineering Society.

Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization

Abstract: In this paper, various necessary and sufficient conditions are given for the nonemptiness and compactness of the weakly efficient solution set of a convex vector optimization problem.

Multi-objective genetic programming for nonlinear system identification

Abstract: Genetic programming is applied to the identification of non-linear polynomial models This approach optimises multiple objectives simultaneously, and the solution set provides a trade-off between the complexity and the performance of the models This is achieved using the concept of the non-dominated or Pareto-optimal solutions The approach is tested on the simple Wiener model

On the generation of locally consistent solution spaces in mixed dynamic constraint problems

Abstract: The development of industrial products such as cars, mechanical tools as well as civil engineering structures includes the task of identifying components and arranging them in a product structure. This task is commonly called configuration. It can be formalized as a constraint satisfaction problem (CSP), which provides a concise mathematical model for combinatorial tasks. A CSP is defined by the variables of interest, each with a domain of possible values, and a set of constraints which restrict the allowed value combinations. The formulation of a configuration task as a CSP gives flexibility required to apply complex optimization criteria during search which often cannot be expressed by a single utility function. Solving a CSP corresponds to finding a consistent assignment to the variables that satisfies all constraints. Solution methods for CSPs encompass consistency and search techniques. Consistency techniques are preprocessing methods which remove inconsistent value combinations before search thus reducing the search space; Furthermore, a dynamic CSP model (DCSP) makes possible the introduction of new variables and constraints during problem resolution. However, CSP techniques have been weak for handling mixed (discrete-continuous) as well as dynamic problems. In this thesis, I present a new algorithm for solving DCSPs with mixed, continuous and discrete, constraints. It isolates and approximates the solution sets of a DCSP by local consistency techniques. To this purpose, local consistency techniques for continuous constraints had to be enhanced and integrated with existing discrete local consistency methods. Some advantages of this algorithm are: It can solve problems in which the existence of a variable depends on a decision taken over a continuous value domain. Different locally consistent solution spaces can be compared, thus providing additional information for decision making. Local consistency techniques are also applied to search for a single solution within the resulting locally consistent solution spaces. These techniques are particularly efficient for searching in continuous domains. The algorithm proposed is applied to a number of different problems such as conceptual bridge design, design of steel structures, configuration of trains, and industrial mixers.

A Comprehensive Survey of Linear Semi-Infinite Optimization Theory

Abstract: This paper reviews the lineax semi-infinite optimization theory as well as its main foundations, namely, the theory of linear semi-infinite systems. The first part is devoted to existence theorems and geometrical properties of the solution set of a linear semi-infinite system. The second part concerns optimality conditions, geometrical properties of the optimal set and duality theory. Finally, the third part analyzes the well-posedness of the linear semi-infinite programming problem and the stability (or continuity properties) of the feasible set, the optimal set and the optimal value mappings when all the data are perturbed.

The Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients

Abstract: A standard system of interval linear equations is defined by Ax = b, where A is an m × n coefficient matrix with (compact) intervals as entries, and b is an m-dimensional vector whose components are compact intervals. It is known that for systems of interval linear equations the solution set, i. e., the set of all vectors x for which Ax = b for some A ϵ A and b ϵ b, is a polyhedron. In some cases, it makes sense to consider not all possible A ϵ A and b ϵ b, but only those A and b that satisfy certain linear conditions describing dependencies between the coefficients. For example, if we allow only symmetric matrices A (aij = aji), then the corresponding solution set becomes (in general) piecewise-quadratic. In this paper, we show that for general dependencies, we can have arbitrary (semi)algebraic sets as projections of solution sets.

Structural and Stability Properties of P 0 Nonlinear Complementarity Problems

Abstract: We consider P0 nonlinear complementarity problems and study the connectedness and stability of the solutions by applying degree theory and the Mountain Pass Theorem to a smooth reformulation of the complementarity problem. We show that the solution set is connected and bounded if a bounded isolated component of the solution set exists and that a solution is locally unique if and only if it is globally unique. Furthermore, we prove that a solution is stable in Ha's sense if and only if it is globally unique, while the complementarity problem is stable if and only if the solution set is bounded.

Discretization results for the Huff and Pareto-Huff competitive location models on networks

Abstract: In this paper we prove that there always exists a finite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an e-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.

Initial value problem for maximally nonlocal actions

Abstract: We study the initial value problem for actions which contain non-trivial functions of integrals of local functions of the dynamical variable. In contrast to many other non-local actions, the classical solution set of these systems is at most discretely enlarged, and may even be restricted, with respect to that of a local theory. We show that the solutions are those of a local theory whose (spacetime constant) parameters vary with the initial value data according to algebraic equations. The various roots of these algebraic equations can be plausibly interpreted in quantum mechanics as different components of a multi-component wave function. It is also possible that the consistency of these algebraic equations imposes constraints upon the initial value data which appear miraculous from the context of a local theory.

Frontier and Closure of a Semi-Pfaffian Set

Abstract: For a semi-Pfaffian set, i.e., a real semianalytic set defined by equations and inequalities between Pfaffian functions in an open domain G , the frontier and closure in G are represented as semi-Pfaffian sets. The complexity of this representation is estimated in terms of the complexity of the original set.

Geometric Set Systems

Abstract: Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higher-dimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the Vapnik-Chervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the tools described might be useful in other areas of mathematics too.

Pareto-optimality in classical inventory problems

Abstract: In this paper the inventory problem with backorders both deterministic and stochastic is studied using trade-off analysis in the context of vector optimization theory. The set of Pareto-optimal solutions is geometrically characterized in both the constrained and unconstrained cases. Moreover, a new way of utilizing Pareto-optimality concepts to handle classical inventory problems with backorders is derived. A new analysis of these models is done by means of a trade-off analysis. New solutions are shown, and an error bound for total inventory cost is provided. Other models such as multi-item or stochastic lead-time demand inventory problems are addressed and their Pareto-optimal solution sets are obtained. An example is included showing the additional applicability of this kind of analysis to handle parametric problems. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 83–98, 1998

A Non-Interior Predictor-Corrector Path-Following Method for LCP

Abstract: In a previous work the authors introduced a non—interior predictor-corrector path following algorithm for the monotone linear complementarity problem The method uses Chen—Harker—Kanzow—Smale smoothing techniques to track the central path and employs a refined notion for the neighborhood of the central path to obtain the boundedness of the iterates under the assumption of monotonicity and the existence of a feasible interior point With these assumptions, the method is shown to be both globally linearly convergent and locally quadratically convergent In this paper it is shown that this basic approach is still valid without the monotonicity assumption and regardless of the choice of norm in the definition of the neighborhood of the central path Furthermore, it is shown that the method can be modified so that only one system of linear equations needs to be solved at each iteration without sacrificing either the global or local convergence behavior of the method The local behavior of the method is further illuminated by showing that the solution set always lies in the interior of the neighborhood of the central path relative to the affine constraint In this regard, the method is fundamentally different from interior point strategies where the solution set necessarily lies on the boundary of the neighborhood of the central path relative to the affine constraint Finally, we show that the algorithm is globally convergent under a relatively mild condition

On Effectively Computing the Analytic Center of the Solution Set by Primal-Dual Interior-Point Methods

Abstract: The computation of the analytic center of the solution set can be important in linear programming applications where it is desirable to obtain a solution that is not near the relative boundary of the solution set. In this work we discuss the effective computation of the analytic center solution by the use of primal-dual interior-point methods. A primal-dual interior-point algorithm designed for effectively computing the analytic-center solution is proposed, and theory and numerical results are presented.

Sup-T Equations: State of the Art

Abstract: The study of fuzzy relational equations is one of the most appealing subjects in fuzzy set theory, both from a mathematical and a systems modelling point of view. The basic fuzzy relational equations are the sup-T equations, with T a t-norm. In this overview paper, we deal with these equations in a general lattice-theoretic framework. We consider t-norms on bounded ordered sets, and in particular on complete lattices. We then solve sup-T equations on distributive, complete lattices of which all elements are either join-irreducible or join-decomposable. Solution sets are represented by means of root systems. Some additional necessary and sufficient solvability conditions are listed. Also systems of sup-T equations are discussed. The theoretical results presented are applied to the real unit interval and to the real unit hypercube. In the latter case, particular attention is paid to pointwise extensions of t-norms defined on the real unit interval and the corresponding residual operators.