problem method classification in optimization and control abstract there are many classifications in optimization and control continuous discrete integer problems convex or concave minimization steepest descent branch and bound path following space filling random search methods smooth or non smooth problems etc the aim of this short note is not to propose yet another classification but to clarify existing ambiguities and imprecisions in problem formulations and to emphasize certain fundamental distinctions in solution methods currently used

skew prime polynomial matrices the polynomial model approach abstract we examine the concept of skew primeness of polynomial matrices in terms of the associated polynomial model it is shown that skew primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism this basic result is then applied to the special case of nonsingular polynomial matrices we investigate the nonuniqueness of skew complements of a skew prime pair it is shown that the space of equivalence classes of skew complements is in bijective correspondence with a finite dimensional linear space finally the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly

global optimization in problems with uncertainties the gamma algorithm abstract problems with uncertainties can be viewed and formalized making use of multifunctions or general set valued functions a new concept of global optimality is proposed which allows us to solve global optimization problems with uncertainties in natural setting without imposing artificial constraints on uncertainties nor introducing a kind of partial ordering in order to apply conventional optimality concepts and optimization techniques nor considering solution in probability with the new concept deterministic optimization requires two optimization procedures a study of the subject is presented with many illustrative examples then a monotonic iterative algorithm is developed which renders approximate solutions with precision specified in advance a notion of piecewise continuous function of several variables is proposed and the method is then generalized for uncertain functions defined by a closed set valued function with piecewise continuous upper and lower boundaries the max2 f reduction precision and decomposition lemmas are proved to facilitate practical applications deferred deletions of sets with discontinuities are introduced and convergence theorem is proved for the modified algorithm

a novel 3 d cpfdtd scheme for modeling grid nonconformally aligned transmitter structures in this paper a novel three dimensional 3 d subcell algorithm for modeling metallic objects which are nonconformally oriented within the finite difference time domain fdtd grid is proposed the generally applicable algorithm is based on the contour path cp technique and enables in combination with an enhanced nonconformally aligned source model a more accurate spatial representation of antenna structures than the common staircase approach to evaluate the performance of the new algorithm the simulation results are compared with the classical staircase approach for dipole antennas as well as for a generic phone significant improvements were achieved for all antenna parameters from the near and far field i e feedpoint impedance radiation pattern and field distributions

two alternatives for the cubic algorithm abstract two alternatives are proved that allow application of the cubic algorithm for finding approximate solutions of the general non convex and non robust mathematical programming problem

a polynomial characterization of mathcal a mathcal b invariant and reachability subspaces based on the state space model of p fuhrmann a link is laid between the geometric approach to linear system theory as developed by w m wonham and a s morse and the approach based on polynomial matrices in particular polynomial characterizations of a b invariant and reachability subspaces are given these characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers finally constructive procedures and conditions are given for computing the supremal a b invariant subspace and reachability space and for checking the solvability of the exact model matching problem

locally conformal fdtd for anisotropic conductive interfaces locally conformal discretizations have been developed for the finite difference time domain fdtd method to mitigate staircasing errors that arise in the modeling of arbitrary e g curved slanted interfaces by regular orthogonal grids in particular locally conformal algorithms based on the computation of effective material parameters in partially filled fdtd cells are attractive because they are simple to implement and maintain the basic conditional stability of fdtd with only minor changes on the courant limit in this paper we propose a new locally conformal implementation for fdtd tailored for anisotropic conductive interfaces instead of employing arithmetic or geometrical averages we invoke the quasi static relationship between the current density and the electric field to derive the effective conductivities of partially filled fdtd cells

on the number of iterations of piyavskii s global optimization algorithm piyavskii s algorithm maximizes a univariate function f satisfying a lipschitz condition we compare the numbers of iterations needed by piyavskir s algorithm np to obtain a bounding function whose maximum value is within e of the unknown globally optimal value with that required by the best possible algorithm nb the main result is that np 2nb 1 and that this bound is sharp we also show that the number of iterations needed by piyavskii s algorithm to obtain a globally e optimal value together with a corresponding point npy satisfies npy 4nb 1 and that this bound is again sharp lower and upper bounds on nb are obtained as functions of f x e l0 and l1 where l0 is the smallest valid lipschitz constant and l1 the lipschitz constant used several properties of nb and of the distribution of evaluation points are deduced from these bounds

further results on polynomial characterizations of f g invariant and reachability subspaces this paper is concerned with further development of the unification between polynomial matrix and geometric theories of linear systems following the work of emre and hautus equivalence between different polynomial characterizations of f g invariant and reachability subspaces is shown explicitly several new results are given which clarify the relations between the polynomial system matrix invariant subspaces and system zeros finally a polynomial characterization of and a constructive procedure to obtain the largest stabilizability subspace in ker h are given

finite difference time domain modeling of curved surfaces em scattering the finite difference time domain fdtd method is generalized to include the accurate modeling of curved surfaces this generalization the contour path cp method accurately models the illumination of bodies with curved surfaces yet retains the ability to model corners and edges cp modeling of two dimensional electromagnetic wave scattering from objects of various shapes and compositions is presented

variation free iterative method for global optimal control abstract a variation free method for global optimal control is proposed the method is based on a global parameter optimization algorithm and does not employ variational principles or variation based techniques

the cubic algorithm for global games with application to pursuit evasion games abstract new algorithms for solution of nonconvex global games defined over a cube are presented application to differential games and to pursuit evasion games is considered and extension is made for games over general compact and robust sets defined by constraints that may depend on time and on state variables the methods are deterministic and monotonically set convergent to deliver in the limit the unique exact full globally optimal minimax and maximin solutions for both players or the entire global saddle set if it exists a stopping rule is provided for determining approximate solutions with a given precision in a finite number of iterations a modification is considered to play on mistakes of the opponent on line replacement of cost functionals is possible in the course of iterations according to changing interests of the players no separability nor convexity concavity assumptions are imposed on the cost functional and variational methods are not used the ideas are illustrated by examples and application is made to determining the globally optimal closed loop strategies for the ship torpedo collision avoidance differential game with manoeuvrability constraints

global optimization of univariate lipschitz functions i survey and properties we consider the following global optimization problems for a univariate lipschitz functionf defined on an interval a b problem p find a globally optimal value off and a corresponding point problem p find a globally optimal value off and a corresponding point problem q localize all globally optimal points problem q find a set of disjoint subintervals of small length whose union contains all globally optimal points problem q find a set of disjoint subintervals containing only points with a globally optimal value and whose union contains all globally optimal points we present necessary conditions onf for finite convergence in problem p and problem q recall the concepts necessary for a worst case and an empirical study of algorithms i e those ofpassive and ofbest possible algorithms summarize and discuss algorithms of evtushenko piyavskii shubert timonov schoen galperin shen and zhu presenting them in a simplified and uniform way in a high level computer language we address in particular the problems of using an approximation for the lipschitz constant reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globally optimal value new algorithms for problems p and q and an extensive computational comparison of algorithms are presented in a companion paper

the beta algorithm abstract a set monotonic non gradient algorithm is proposed for finding global minima of general non convex mathematical programming problems the algorithm is based on the cubic algorithm e a galperin j math anal appl m 112 1985 635 640 equipped with a semi certain distinction operator and the marginal comparison constant generator

integral global optimization method for differential games with application to pursuit evasion games abstract a variation free method for global solution of differential games is proposed the method is based on the integral global optimization algorithm and it does not employ variation based techniques nor the notion of convexity the existence of a saddle point is not assumed a priori instead the method delivers the global minimax and maximin solutions for both players a necessary and sufficient global minimax condition is presented an application is made to the solution of nonlinear pursuit evasion games the ideas are illustrated by examples

an analytical expression for 3 d dyadic fdtd compatible green s function in infinite free space via z transform and partial difference operators fdtd solutions have their own properties distinct from the discrete samples of corresponding continuous wave solutions thus the discrete equivalent to the green s function is needed for applications like the one using a hybrid absorbing boundary condition which couples the fdtd algorithm with integral operators for nonconvex scatterers in this paper we propose a new closed form expression for the 3 d dyadic fdtd compatible green s function in infinite free space via a novel approach with the ordinary z transform along with the spatial partial difference operators the final expression involves a summation of standing wave modes with time varying coefficients the propagation of waves in the yee s grid can be interpreted by the selective property of the time varying coefficients which is very different from the conventional concept of a traveling wave the traditional dispersion analysis using plane waves for the fdtd algorithm in a source free region may not be applicable to explain the wave propagation phenomenon through our analytic expression because the corresponding z transform diverges for z on the unit circle

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

an implicit enumeration method for global optimization problems abstract this paper presents a new method for the global minimization of a function subject to linear constraints the feasible set is first covered by a union of simpler sets then some of them are eliminated using some global analytic information about the objective function e g lipschitz constants properties of the derivatives etc the remaining part of the cover is then refined and the process is repeated estimates for the computational complexity are given for some classes of problems numerical experience is also reported

causal factorization and linear feedback an algebraic framework for the investigation of linear dynamic output feedback is introduced pivotal in the present theory is the problem of causal factorization i e the problem of factoring two systems over each other through a causal factor the basic issues are resolved with the aid of the new concept of latency kernels

low frequency finite difference time domain modeling of a pec sphere based on a quasi analytical coupled dipole approximation a computational formulation is presented for the low frequency single cell finite difference time domain fdtd modeling of a perfectly electric conducting pec sphere the approach is based on the fact that the scattered field from electrically small objects can be expressed in terms of an electric and magnetic dipole these dipoles can be decomposed with respect to the dipole moments that can be defined along the discrete field components that comprise the cell wherein the pec sphere is inscribed the dipole moment components couple to each other and this mechanism is quantified by a quasi analytical coupled dipole approximation cda the quasi analyticity requires to substitute the involved dyadic green s function dgf terms in the cda formula by their numerically computed fdtd compatible equivalents the material properties of the equivalent electric and magnetic spheres are derived using the quasi analytical cda that leads to expressions that resemble the claussius mossotti mixing formula the theoretically derived results are supported by numerical simulations

global one dimensional optimization using smooth auxiliary functions in this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has lipschitzean first derivatives to solve this problem smooth auxiliary functions which are adaptively improved during the course of the search are constructed three new algorithms are introduced the first used the exact a priori known lipschitz constant for derivatives the second when this constant is unknown estimates it during the course of the search and finally the last method uses neither the exact global lipschitz constant nor its estimate but instead adaptively estimates the local lipschitz constants in different sectors of the search region during the course of optimization convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given 1998 the mathematical programming society inc published by elsevier science b v

the integer cubic algorithm abstract the integer and mixed cubic algorithms are proposed for full global solution of integer and mixed optimization problems the paper presents further development of the ideas in 1 4

disturbance decoupling problems via dynamic output feedback this paper considers the disturbance decoupling problems with or without internal stability and pole placement via dynamic output feedback using polynomial and rational matrix techniques we show that in all three problems considered the central solvability condition can be expressed as a two sided matching problem a bxc where a b and c are the polynomial system matrices of certain natural subsystems of the system model and x is to be determined over various subrings of the rational functions this matching problem can in turn be reduced to certain appropriate zero cancellation conditions on the polynomial system matrices a b and c

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

polynomial characterizations of h f invariant subspaces with applications abstract a direct study of h f invariant subspaces associated with the polynomial fractional system model z pq 1r w is undertaken the earlier results on the polynomial characterization of these subspaces are extended to cover the general case for full rank transfer matrices the smallest unobservability subspace containing im g and the largest reachability subspace in ker h are described in terms of factors of the polynomial system matrix a new polynomial model and its superspaces play the central role in all these characterizations the results obtained are applied to two measurement feedback problems in linear control theory yielding geometric interpretations for two rational matrix equations

the alpha algorithm and the application of the cubic algorithm in case of unknown lipschitz constant abstract a new algorithm for full global optimization of a lipschitzian function over an arbitrary bounded set in r n is presented a procedure is proposed to handle functions with unknown lip constants application to the cubic algorithm is given and illustrative examples are presented

global optimization to prescribed accuracy abstract programmable methods are outlined for rigorous global optimization to prescribed accuracy with references to enough detail to enable the implementation of the techniques on any computer some directions for future research are indicated

a 3 d tensor fdtd formulation for treatment of sloped interfaces in electrically inhomogeneous media a general integral finite difference time domain fdtd formulation on cubical grids for the modeling of electrically inhomogeneous media of arbitrary shape is derived the simple cubical structure is maintained and no modifications of the integral paths are necessary instead a three dimensional tensor relationship between the average electric flux density and electric field is determined beforehand and used during the simulation to account for discontinuities in the neighborhood of sloped interfaces the accuracy of the method is confirmed by comparisons with analytical solutions

autonomous subbehaviours and output nulling subspaces the aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours specifically given a behaviour we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems we use this to analyse the different module structures on some vectorial polynomial spaces subsequently we apply this analysis to the study and the polynomial characterization of output nulling subspaces controlled invariant subspaces and the natural feedback induced module structures in these spaces toeplitz operators and wiener hopf factorizations play an important role