l and r localized solvabilities of max separable interval linear equations and its applications in this paper we propose new types of solutions an l localized solution and an r localized solution to interval systems of max separable linear equations a x b where a a a and b b b we provide some small application examples to emphasize the importance of our proposed solutions we give necessary and sufficient conditions for checking l localized and r localized solvabilities and apply them to achieve the closed forms of the sets of all l localized solutions and r localized solutions respectively

numerical solution method for general interval quadratic programming abstract recently liu and wang described an interesting numerical method to a special class of interval quadratic programming where the linear term in objective function and constraints involved interval coefficients s t liu r t wang a numerical solution method to interval quadratic programming applied mathematics and computation 189 2007 1274 1281 in this paper we generalize liu and wang s method to general interval quadratic programming where all coefficients in the objective function and constraints are interval numbers although the interval quadratic programming model considered in this paper is more general the proposed method requires less computing compared with liu and wang s method numerical examples are presented to illustrate the whole idea

satisficing solutions and duality in interval and fuzzy linear programming in this paper we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions the necessary tools for dealing with such problems in this way we show that the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems we define the concept of duality and prove the weak and strong duality theorems further we define a class of interval linear programming problems as a special subclass of flp problems and apply the previous results to this special case

strong optimal solutions of interval linear programming abstract this paper considers optimal solutions of general interval linear programming problems the new concepts of optimal solutions are introduced in a unified framework some existed optimal solution concepts of interval linear program such as weak and strong optimal solutions are special cases in this framework necessary and sufficient conditions for checking optimality are developed also the features of the proposed methods are illustrated by some examples

discovering multiple lyapunov functions for switched hybrid systems in this paper we analyze local asymptotic stability of switched hybrid systems whose subsystems have polynomial vector fields by discovering multiple lyapunov functions in quadratic forms we start with an algebraizable sufficient condition for the existence of quadratic multiple lyapunov functions then since different discrete modes are considered we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step arriving at a set of semialgebraic sets which only involve the coefficients of the pre assumed multiple lyapunov function note that for each step we additionally use the information on the different discrete modes to optimize our intermediate computation results afterward we compute a sample point in the corresponding semialgebraic set for the coefficients resulting in a multiple lyapunov function finally we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier e

saint venant end effects in multilayered piezoelectric laminates the saint venant end effects and stress decay in multilayered laminates of piezoelectric materials are studied a state space approach is developed in the context of generalized plane strain by means of matrix algebra and transfer matrix the characteristics of the stress decay in 2 d piezoelectric strips and laminates are examined their elastic counterparts are included as special cases evaluation of the characteristic decay length for typical materials shows that the saint venant end effects are significant it also indicates that the analysis based on the plane strain assumption is misleading

cone constraints for solutions of homogeneous bilinear systems abstract sufficient conditions are derived under which all solutions of a homogeneous bilinear system remain within a cone for both the case where an initial state is prescribed and the case where the initial state lies within another prescribed cone this situation generalizes the classical uncontrolled case in which solutions of a linear system are constrained to lie in the cone consisting of the positive orthant the conditions are given in terms of an associated linear system originally utilized in realization theory and a particular representation of its inverse system in the case where the associated linear system has single input and output channels the conditions are also shown to be necessary

on the solution sets of particular classes of linear interval systems we characterize the solution set s of real linear system ax b by a set of inequalities if b lies between some given bounds b b and if the n n coefficient matrix a varies similarly between two bounds a and a in addition we restrict a to a particular class of matrices for instance the class of the symmetric the skew symmetric the persymmetric the toeplitz and the hankel matrices respectively in this way we generalize the famous oettli prager criterion numer math 6 1964 405 results by hartfiel numer math 35 1980 355 and the contents of the papers in r b kearfott v kreinovich eds applications of interval computations kluwer boston ma 1996 pp 61 79 and siam j matrix anal appl 18 1997 693

checking strong optimality of interval linear programming with inequality constraints and nonnegative constraints this paper considers optimal solutions of interval linear programming problems in a unified framework necessary and sufficient conditions for checking a b strong optimality and a b c strong optimality of interval linear programming with inequality constraints are developed the features of the proposed methods are illustrated by some examples

numerical analysis of end effects in laminated piezoelectric circular cylinders abstract a method to quantify saint venant s principle for laminated piezoelectric circular cylinders is described an algebraic eigensystem is constructed based on homogeneous equations of equilibrium with displacement and electric potential voltage fields stated in exponential form the end effects for displacements and voltages as well as stresses and electric displacements of self equilibrated states are represented by the eigendata extracted from the eigensystem the real parts of the eigenvalues convey information on the inverse decay lengths and their corresponding eigenvectors are displacement and voltage distributions of the self equilibrated states stress and electric displacement eigenvectors are formed by appropriate differentiation of the eigenvectors for the displacement and voltage fields and through the coupled electromechanical constitutive laws the right and left eigenvectors which are obtained from the eigensystem and its adjoint are related through bi orthogonality relationships the end effects due to arbitrary displacements voltages tractions or electric displacements prescribed at the cylinder s ends are obtained by means of an expansion theorem based on these bi orthogonality relationships or by a least squares method a number of verification examples are provided to demonstrate that the present results compare well with known analytical solutions and numerical results obtained via three dimensional finite element analyses

radii of solvability and unsolvability of linear systems abstract we consider a problem of determining the component wise distance called the radius of a linear system of equations or inequalities to a system that is either solvable or unsolvable we propose explicit characterization of these radii and show relations between them then the radii are classified in the polynomial vs np hard manner we also present a generalization to an arbitrary linear system consisting from both equations and inequalities with both free and nonnegative variables eventually we extend the concept of the component wise distance to a non uniform one

providing a basin of attraction to a target region of polynomial systems by computation of lyapunov like functions in this paper we present a method for computing a basin of attraction to a target region for polynomial ordinary differential equations this basin of attraction is ensured by a lyapunov like polynomial function that we compute using an interval based branch and relax algorithm this algorithm relaxes the necessary conditions on the coefficients of the lyapunov like function to a system of linear interval inequalities that can then be solved exactly it iteratively refines these relaxations in order to ensure that whenever a nondegenerate solution exists it will eventually be found by the algorithm application of an implementation to a range of benchmark problems shows the usefulness of the approach

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

ae solutions and ae solvability to general interval linear systems abstract we consider linear systems of equations and inequalities with coefficients varying inside given intervals we define their solutions so called ae solutions and solvability so called ae solvability by using forall exists quantification of interval parameters we present an explicit description of the ae solutions and discuss complexity issues as well for ae solvability we propose a sufficient condition only but for a specific sub class of problems a complete characterization is developed moreover we investigate inequality systems for which ae solvability is equivalent to existence of an ae solution

an algorithm for computing the hull of the solution set of interval linear equations abstract described is a not a priori exponential algorithm which for each n n interval matrix a and for each interval n vector in a finite number of steps either computes the interval hull of the solution set of the system of interval linear equations a x b or finds a singular matrix s a

saint venant end effects for plane deformations of linear piezoelectric solids the recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity in this paper we investigate the decay of saint venant end effects for plane deformations of a piezoelectric semi infinite strip first of all we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid just as in the mechanical case not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non trivial state of plane deformation the governing system of four second order partial differential equations for the two in plane displacements and electric potential are overdetermined in general sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation the resulting traction boundary value problem with prescribed surface charge is an oblique derivative boundary value problem for a coupled elliptic system of three second order partial differential equations the special case of a piezoelectric material transversely isotropic about the poling axis is then considered thus the results are valid for the hexagonal crystal class 6mm the geometry is then specialized to be a two dimensional semi infinite strip and the poling axis is the axis transverse to the longitudinal direction we consider such a strip with sides traction free subject to zero surface charge and self equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity a formulation of the problem in terms of an airy type stress function and an induction function is adopted the governing partial differential equations are a coupled system of a fourth and third order equation for these two functions on seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second order ordinary differential equations this problem is the piezoelectric analog of the well known eigenvalue problem arising in the case of an anisotropic elastic strip it is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti symmetric deformations and electric fields assuming completeness of the eigenfunctions the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part numerical results are given for pzt 5h pzt 5 pzt 4 and ceramic b it is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials

new characterizations for the solution set to interval linear systems of equations new characterizations of the points from the solution set to interval linear systems of equations are proposed alternatives to the well known result by oettli and prager we also introduce recognizing functionals of the solution sets that determine at a given point aggregated quantitative measures of compatibility consistency between the interval data of the system

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

characterization of ae solution sets of parametric linear systems based on the techniques of convex sets abstract new approach to linear systems with uncertain parameters is proposed the notion of ae solution sets is extended to the case when the domain of parameters need not be an interval vector the tools from the theory of convex sets are applied to the characterization of ae solution sets the description of these sets by systems of inequalities is presented some examples are included

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

ae solutions to interval linear systems over max plus algebra abstract this paper introduces a concept of ae solutions to interval max plus linear systems a rather general concept which includes many known concepts of solutions to interval systems weak solutions strong solutions tolerance solutions and control solutions as its special cases we state full characterizations of ae solutions for the interval max plus systems including both linear inequalities and linear equations moreover a kind of dependency characterized by double appearance of some sub matrix of the constraint matrix is discussed and we prove that which can be relaxed with no change in ae solvability

an algorithm for solving two sided interval system of max plus linear equations we study six types of solutions weak strong tolerance control left localized and right localized solutions to a two sided interval system of max plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions these conditions are in the form of two sided systems of max plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems therefore we could work on the interval inclusion linear systems instead of directly solving these six types of solutions themselves an optimization problem with the two sided interval system of max plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems

some results on the upper bound of optimal values in interval convex quadratic programming one of the fundamental problems in interval quadratic programming is to compute the range of optimal values for minimized problem with equality constraint computing the upper bound of the optimal values is known to be np hard one kind of the effective methods for computing the upper bound of interval quadratic programming is so called dual method based on the dual property of the problem to obtain the exact upper bound the dual methods require that the duality gap is zero however it is not an easy task to check whether this condition is true when the data may vary inside intervals in this paper we first present an easy and efficient method for checking the zero duality gap then some relations between the exact upper bound and the optimal value of the dual model considered in dual methods are discussed in detail we also report some numerical results and remarks to give an insight into the dual method s behavior

necessary and sufficient conditions of some strong optimal solutions to the interval linear programming abstract this paper considers optimal solutions of general interval linear programming problems necessary and sufficient conditions of a b strong and a b c strong optimal solutions to the interval linear programming with inequality constraints are proposed the features of the proposed methods are illustrated by some examples

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

new characterizations of tolerance control and localized solutions to interval system of linear equations abstract in this paper we propose new characterizations of tolerance control and localized solutions to interval system of linear equations and apply our proposed results to represent their solution sets in terms of level sets

optimistic optimization for model predictive control of max plus linear systems model predictive control for max plus linear discrete event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables which may be hard to solve efficiently an alternative approach is to transform the given problem into a mixed integer linear programming problem however the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon the focus of this paper is on making optimistic optimization suited to solve the given problem optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization a key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum we prove that optimistic optimization can be applied for the given problem by developing a dedicated semi metric and by proving it satisfies the necessary requirements for optimistic optimization moreover we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon hence using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

safe bounds in linear and mixed integer linear programming current mixed integer linear programming solvers are based on linear programming routines that use floating point arithmetic occasionally this leads to wrong solutions even for problems where all coefficients and all solution components are small integers an example is given where many state of the art milp solvers fail it is then shown how using directed rounding and interval arithmetic cheap pre and postprocessing of the linear programs arising in a branch and cut framework can guarantee that no solution is lost at least for mixed integer programs in which all variables can be bounded rigorously by bounds of reasonable size

generalized linear fractional programming under interval uncertainty abstract data in many real life engineering and economical problems suffer from inexactness herein we assume that we are given some intervals in which the data can simultaneously and independently perturb we consider a generalized linear fractional programming problem with interval data and present an efficient method for computing the range of optimal values the method reduces the problem to solving from two to four real valued generalized linear fractional programs which can be computed in polynomial time using an appropriate interior point method solver we consider also the inverse problem how much can data of a real generalized linear fractional program vary such that the optimal values do not exceed some prescribed bounds we propose a method for calculating often the largest possible ranges of admissible variations it needs to solve only two real valued generalized linear fractional programs we illustrate the approach on a simple von neumann economic growth model

revised cpa method to compute lyapunov functions for nonlinear systems the cpa method uses linear programming to compute continuous and piecewise affine lyapunov function for nonlinear systems with asymptotically stable equilibria in it was shown that the method always succeeds in computing a cpa lyapunov function for such a system the size of the domain of the computed cpa lyapunov function is only limited by the equilibrium s basin of attraction however for some systems an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori this is necessary if the equilibrium is not exponentially stable because the existence of a cpa lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in however if the equilibrium is exponentially stable then this was an artifact of the method in this paper we overcome this artifact by developing a revised cpa method we show that this revised method is always able to compute a cpa lyapunov function for a system with an exponentially stable equilibrium the only conditions on the system are that it is c and autonomous the domain of the cpa lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction whereas in a previous paper we have shown these results for planar systems in this paper we cover general n dimensional systems

interval strong solutions of interval systems of max plus linear equations abstract this paper considers the existence and uniqueness of interval strong solutions of interval systems of max plus linear equations a necessary and sufficient condition for the existence of interval strong solutions is presented the proof of the existence of interval strong solutions is constructive and results in a formula for computing such solutions a necessary and sufficient condition for the uniqueness of interval strong solutions is established by testing the unique solvability of a finite number of subsystems rather than all subsystems on this basis a polynomial algorithm is developed to verify the uniqueness of interval strong solutions

inverse linear programming with interval coefficients the paper deals with the inverse linear programming problem over intervals more precisely given interval domains for the objective function coefficients and constraint coefficients of a linear program we ask for which scenario a prescribed optimal value is attained using continuity of the optimal value function under some assumptions we propose a method based on parametric linear programming techniques we study special cases when the interval coefficients are situated in the objective function and or on the right hand sides of the constraints as well as the generic case when possibly all coefficients are intervals we also compare our method with the straightforward binary search technique finally we illustrate the theory by an accompanying numerical study called matrix casino showing some approaches to designing a matrix game with a prescribed game value

underapproximating backward reachable sets by semialgebraic sets underapproximations uas of backward reachable sets play an important role in controller synthesis and trajectory analysis for constrained nonlinear dynamical systems but there are few methods available to compute them given a nonlinear system a target region of simply connected compact type and a time duration we present a method using boundary analysis to compute an ua of the backward reachable set the ua is represented as a semialgebraic set formed by what we term polynomial mathtt level set functions the polynomial mathtt level set function is a semidefinite positive function with one real root such that the interior and closure of a semialgebraic set formed by it are both simply connected and have the same boundary the function can be computed by solving a convex program which is constructed based on sum of squares decomposition and linear interval inequalities we test our method on several examples and compare them with existing methods the results show that our method can obtain better estimations more efficiently in terms of time for these special examples

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

closed form of l localized solution set of max plus interval linear system and its application on optimization problem a closed form of the l localized solution set of an interval linear system with max plus algebra is a finite union of interval vectors this paper analyzes and proves some properties to reduce number of the union sets to be a smaller number of the union of interval vectors that are not subsets of each other this result helps simplify the feasible set of an optimization problem with max plus linear equation constraints

some comments on positive orthant controllability of bilinear systems consider the bilinear system dot x a ub x x in r n and u unrestricted the system has the property that any solution x t with x 0 geqq 0 i e all components of x nonnegative will remain in the positive orthant r n x in r n x geqq 0 for 0 leqq t 0 and detailed results are presented for the case n 2

interval systems of max separable linear equations an interval system of equations is weakly solvable if at least one of its subsystems is solvable and it is strongly solvable if all its subsystems are solvable we give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max plus and max min algebras

improving the modified interval linear programming method by new techniques we improve milp method proposed by zhou et al in 2009 solution region resulted from milp method is feasible and no completely optimal we propose two improved milp methods namely iilp and imilp solution region obtained through iilp and imilp methods is feasible and optimal in this study we consider interval linear programming ilp problems which are used to deal with uncertainties resulting from the range of admissible values in problem coefficients in most existing methods for solving ilp problems a part of the solution region is not feasible the solution set obtained through the modified ilp milp method is completely feasible i e it does not violate any constraints but is not completely optimal i e some points of the region are not optimal in this paper two new ilp methods and their sub models are presented these techniques improve the milp method giving a solution region that is not only completely feasible but also completely optimal

necessary and sufficient conditions for unified optimality of interval linear program in the general form abstract this paper considers optimal solutions of general interval linear programming problems the most general concepts of optimal solutions are introduced in a unified framework the existing optimal solution concepts of interval linear program such as weak and strong optimal solutions are all the special cases in this framework necessary and sufficient conditions for checking a class of optimality are developed according to the kkt conditions also the comparison among several conclusions using kkt conditions and tangent core technique in various forms of interval linear programming is proposed

estimating the robust domain of attraction for non smooth systems using an interval lyapunov equation the lyapunov equation allows finding a quadratic lyapunov function for an asymptotically stable fixed point of a linear system applying this equation to the linearization of a nonlinear system can also prove the exponential stability of its fixed points this paper proposes an interval version of the lyapunov equation which allows investigating a given lyapunov candidate function for non smooth nonlinear systems inside an explicitly given neighborhood leading to rigorous estimates of the domain of attraction eda of exponentially stable fixed points these results are developed in the context of uncertain systems experiments are presented which show the interest of the approach including with respect to usual approaches based on sum of squares for the computation of eda

weak and strong solvability of interval linear systems of equations and inequalities abstract we consider weak and strong solvability of general interval linear systems consisting of mixed equations and inequalities with mixed free and sign restricted variables we generalize the well known weak solvability characterizations by oettli prager for equations and gerlach for inequalities to a unified framework in the same manner we extend strong solvability theorems to general interval linear systems next we propose a sufficient condition for checking strong solvability we give an application in linear programming with interval data by means of weak and strong solvability we determine limits of the optimal values for any form of the problem setting

farkas type conditions of general interval linear systems for ae solvability abstract we present farkas type necessary and sufficient conditions for ae solvability of two specific general interval linear systems we generalize the farkas type theorems of interval linear systems to a unified framework moreover each particular previously published result of farkas type condition of interval linear systems is a special case of our general approach

bracketing backward reach sets of a dynamical system in this paper we present a new method for bracketing i e characterising from inside and from outside backward reach set of the target region t of a continuous time dynamical system the princip

exact three dimensional analysis for static torsion of piezoelectric rods based on the theory of elasticity exact analytical and numerical solutions of piezoelectric rods under static torsion are studied in this paper direct solution method is used the main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading it is shown that previous assumptions are valid only in some types of piezoelectric materials while in other types these assumptions lead to considerable deviations from accurate modeling the present analytical solutions are compared with three dimensional finite element analysis results and absolute agreements are found at the end of this article torsional rigidity shape effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

checking solvability of systems of interval linear equations and inequalities via mixed integer programming abstract this paper deals with the problems of checking strong solvability and feasibility of linear interval equations checking weak solvability of linear interval equations and inequalities and finding control solutions of linear interval equations these problems are known to be np hard we use some recently developed characterizations in combination with classical arguments to show that these problems can be equivalently stated as optimization tasks and provide the corresponding linear mixed 0 1 programming formulations

localized solutions to interval linear equations in this paper we propose a new type of solutions localized solutions to interval linear equations ax b the new solutions are based on l localized solutions and r localized solutions the concepts of the three solutions are given and the characteristics of the new solutions to the interval linear equations are described by means of nonlinear inequality as is similar to the well known oettli prager inequality which is related to the mid point and interval radius some necessary conditions of the three types of solutions to the equations are given

computation of lyapunov functions for smooth nonlinear systems using convex optimization linearly parameterized non quadratic lyapunov functions for smooth nonlinear systems are computed numerically using large scale linear or quadratic programming

saint venant end effects in piezoceramic materials the stress decay rate and characteristic decay length for anisotropic materials and fiber composites have been intensively investigated however those for piezoelectric materials have not been studied in this paper we examine the saint venant end effects of piezoceramic materials by considering the problem of a semi infinite piezoceramic strip polarized in the thickness direction and with applied voltages on the upper and lower surfaces it is assumed that the gradient of electric potential in the axial direction is much smaller than that in the thickness direction thus the governing equations in terms of the airy stress function and electric potential function can be uncoupled the governing equation in terms of the airy stress function involves only two non dimensional parameters after non dimensionalization finally the stress decay rates and characteristic decay lengths for a variety of piezoceramic materials are computed numerically and their variation with the two non dimensional parameters is presented

discrete maximum principle for higher order finite elements in 1d we formulate a sufficient condition on the mesh under which we prove the discrete maximum principle dmp for the one dimensional poisson equation with dirichlet boundary conditions discretized by the hp fem the dmp holds if a relative length of every element k in the mesh is bounded by a value h rel p 0 9 1 where p 1 is the polynomial degree of the element k the values h rel p are calculated for 1 p 100

new method for computing the upper bound of optimal value in interval quadratic program we consider the interval quadratic programming problems the aim of this paper is to present a new method to compute the upper bound of the optimal values under weaker conditions moreover we discuss the relations between the new method and previous results the features of the proposed methods are illustrated by some examples

possibilistic linear regression with fuzzy data tolerance approach with prior information abstract we introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data both input and output the method is very general the only assumption is that cuts of the fuzzy data are efficiently computable we take into account possible prior restrictions of the parameters space we assume that the restrictions are given by linear and quadratic constraints the method for construction of the possibilistic regression coefficients is in a reduction of the fuzzy valued model to an interval valued model on a given cut which is further reduced to a linear time method i e running in time o n p computing with endpoints of the intervals here n is the number of observations and p is the number of regression parameters the speed of computation makes the method applicable for huge datasets unlike various approaches based on mathematical programming formulations the tolerance based construction preserves central tendency of the resulting regression coefficients in addition we prove further properties if inputs are crisp and outputs are fuzzy then the construction preserves piecewise linearity and convex shape of fuzzy numbers we derive an o n 2 p algorithm for enumeration of breakpoints of the membership function of the estimated coefficients similar results are also derived for the fuzzy input and output model we illustrate the theory for the case of triangular and asymmetric gaussian fuzzy inputs and outputs of an inflation consumption model

interval linear systems as a necessary step in fuzzy linear systems this article clarifies what it means to solve a system of fuzzy linear equations relying on the fact that they are a direct extension of interval linear systems of equations already studied in a specific interval mathematics literature we highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals each of which based on a generalization of scalar equality to intervals each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals we also consider the smallest multidimensional intervals containing the solution sets we propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals this unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

an interval approach to compute invariant sets this paper proposes an original interval based method to compute an outer approximation of all invariant sets such as limit cycles of a continuous time nonlinear dynamic system which are included inside a prior set of the state space contrary to all other existing approaches our method has the following properties first it is guaranteed a solution cannot be lost second it is applicable to a large class of systems without any specific assumption such as the knowledge of a lyapunov function or any partial linearity and third there is no need to integrate the system

solvability and feasibility of interval linear equations and inequalities abstract this paper considers solvability and feasibility of interval linear equations and inequalities the new concepts of solvability and feasibility are introduced in a unified framework some existing concepts such as weak solvability strong solvability weak feasibility and strong feasibility are special cases in this framework necessary and sufficient conditions for checking new types of solvability and feasibility are developed

controllable solution set to interval static systems for interval systems of algebraic equations the concept of a controllable solution set is introduced and its practical interpretation is given the main results of the work concern interval linear systems and afford simple sufficient criteria for the controllable solution set to be nonempty and for the intersection of the controllable solution set with some orthants to be empty

a numerical solution method to interval quadratic programming quadratic programming has been widely applied to solving real world problems the conventional quadratic programming model requires the parameters to be known constants in the real world however the parameters are seldom known exactly and have to be estimated this paper discusses the interval quadratic programming problems where the cost coefficients constraint coefficients and right hand sides are represented by interval data since the parameters are interval valued the objective value is interval valued as well a pair of two level mathematical programs is formulated to calculate the upper bound and lower bound of the objective values of the interval quadratic program based on the duality theorem and by applying the variable transformation technique the pair of two level mathematical programs is transformed into conventional one level quadratic program solving the pair of quadratic programs produces the interval of the objective values of the problem an example illustrates the whole idea and sheds some light on interval quadratic programming

on max plus linear dynamical system theory a class of timed discrete event systems can be modeled by using timed event graphs a class of timed petri nets that can have its firing dynamic described by using an algebra called max plus algebra for this kind of systems it may be desirable to enforce some timing constraints in steady state in this paper this problem is called a max plus regulation problem in this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems the obtained controller is a simple linear static state feedback and can be computed using efficient pseudo polynomial algorithms simulation results will illustrate the applicability of the proposed methodology

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

ae and ea robustness of interval circulant matrices in max min algebra abstract max min algebra is defined as a linearly ordered set with two binary operations classical addition and multiplication are replaced by maximum and minimum respectively a square matrix is robust if its eigenspace is achieved starting at arbitrary vector and it is x robust if its eigenspace is achieved starting at each vector from a given interval vector x the ea and ae robustness and x robustness of interval circulant matrices over max min algebra are defined polynomial algorithms for checking these types of robustness and x robustness are given

flow functions control flow functions and the reach control problem the paper studies the reach control problem rcp to make trajectories of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope we introduce the notion of a flow function which provides the analog of a lyapunov function for the equilibrium stability problem a flow function comprises a scalar function that decreases along closed loop trajectories and its existence is a necessary and sufficient condition for closed loop trajectories to exit the polytope it provides an analysis tool for determining if a specific instance of rcp is solved without the need for calculating the state trajectories of the closed loop system results include a variant of the lasalle principle tailored to rcp an open problem is to identify suitable classes of flow functions we explore functions of the form v x max v i x and we give evidence that these functions arise naturally when rcp is solved using continuous piecewise affine feedbacks next we introduce the notion of a control flow function it is shown that the artstein sontag theorem of control lyapunov functions has direct analogies to rcp via control flow functions

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs