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

neutral einstein metrics in four dimensions in matsushita j math phys 22 979 982 1981 ibid 24 36 40 1983 for curvature endomorphisms for the pseudo euclidean space r2 2 an analog of the petrov classification as a basis for applications to neutral einstein metrics on compact orientable four dimensional manifolds is provided this paper points out flaws in matsushita s classification and moreover that an error in chern s pseudo riemannian geometry and the gauss bonnet formula acad brasileira ciencias 35 17 26 1963 and shiing shen chern selected papers springer verlag new york 1978 gauss bonnet formula for pseudo riemannian geometry was incorporated in matsushita s subsequent analysis a self contained account of the subject of the title is presented to correct these errors including a discussion of the validity of an appropriate analog of the thorpe hitchin inequality of the riemannian case when the inequality obtains in the neutral case the euler characteristic is nonpositive in contradistinction to matsushita

contour integrals for the ultrahyperbolic wave equation the penrose transform for the ultrahyperbolic wave equation is related to the problem of specifying data on an initial spacelike 2 plane it is shown that a smooth solution can always be represented locally by a twistor contour integral formula and this fact is used to rederive asgeirsson s mean value theorem as well as the classic result that it is not possible to specify arbitrary initial data on a non null 3 plane the global problem of constructing a twistor transform in the compactified space time with signature is also considered

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 statistical estimate of the structure of multi extremal problems the solution of ak extremal problem is defined as the set of pairs xi i i 1 k where xt isith local minimum and i is the volume of the set of attraction of this minimum a bayesian estimate ofk and 1 k is constructed

q deformed oscillator associated with the calogero model and its q coherent state we present the q deformed para bose oscillators associated with the two body calogero model the q deformed coherent state is also constructed and its resolution of unity is demonstrated

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

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

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

stochastic global optimization methods part 1 clustering methods in this stochastic approach to global optimization clustering techniques are applied to identify local minima of a real valued objective function that are potentially global three different methods of this type are described their accuracy and efficiency are analyzed in detail

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

dynamics of the q deformed lipkin model in this paper we investigate the dynamics of the lipkin meshkov glick model within the context of quantum algebras for this purpose we obtain the equations of motion with the help of the time dependent variational principle to analyse the results graphs for equi energies potentials and mean square deviation of the jz operator have been plotted modifications in the usual libration and rotational motions due to the introduction of quantum deformation are discussed

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

stochastic global optimization methods part 11 multi level methods in part ii of our paper two stochastic methods for global optimization are described that with probability 1 find all relevant local minima of the objective function with the smallest possible number of local searches the computational performance of these methods is examined both analytically and empirically

a spinor approach to walker geometry a four dimensional walker geometry is a four dimensional manifold m with a neutral metric g and a parallel distribution of totally null two planes this distribution has a natural characterization as a projective spinor field subject to a certain constraint spinors therefore provide a natural tool for studying walker geometry which we exploit to draw together several themes in recent explicit studies of walker geometry and in other work of dunajski 11 and plebanski 30 in which walker geometry is implicit in addition to studying local walker geometry we address a global question raised by the use of spinors

path integrals and supercoherent states for the real supergroup osp 1 2 r with both its compact and noncompact versions supercoherent states are introduced with a method close to the one by perelomov for the even subgroups su 2 or su 1 1 these states labeled by a complex c number and grassmann variable minimize the uncertainty of the quadratic casimir operator of the group a path integral formalism is developed for the transition amplitude between supercoherent states for a hamiltonian linear in the generators of the superalgebra which leads to a super riccati equation finally in the classical limit the canonical equations of motion are derived which involve a generalized super poisson bracket

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

a stochastic method for global optimization a stochastic method for global optimization is described and evaluated the method involves a combination of sampling clustering and local search and terminates with a range of confidence intervals on the value of the global optimum computational results on standard test functions are included as well

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

thorpe hitchin inequality for compact einstein 4 manifolds of metric signature and the generalized hirzebruch index formula it is proved that the euler characteristic and the hirzebruch index of a compact oriented einstein 4 manifold of metric signature satisfy an inequality which is well known as the thorpe hitchin inequality for the case of a riemannian metric to derive the inequality a generalized hirzebruch formula relating the index to the first pseudo pontrjagin number of the manifold is proved this formula may be contrasted with chern s generalized gauss bonnet formula for a pseudo riemannian manifold

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

nonlinear gravitons null geodesics and holomorphic disks we develop a global twistor correspondence for pseudo riemannian conformal structures of signature with self dual weyl curvature near the conformal class of the standard indefinite product metric on s 2 s 2 there is an infinitedimensional moduli space of such conformal structures and each of these has the surprising global property that its null geodesics are all periodic each such conformal structure arises from a family of holomorphic disks in cp3 with boundary on some totally real embedding of rp 3 into cp3 an interesting sub class of these

path integral formalism for suq 2 coherent states a path integral formulation in the representation of coherent states for the quantum group suq 2 is introduced an expression for the transition amplitude connecting two suq 2 coherent states is constructed and the corresponding canonical equations of motion are also derived

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

path integral formalism for osp 1 2 r coherent states a path integral formulation in the representation of coherent states for the non compact supergroup osp 1 2 r is introduced an expression for the transition amplitude connecting two osp 1 2 r coherent states is constructed and the corresponding canonical equations of motion derived a set of generalized poisson brackets is introduced and interpreted

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

on euler characteristics of compact einstein 4 manifolds of metric signature in the present paper the euler characteristic is studied for a compact oriented einstein 4 manifold of signature for such a manifold there are three types of normal forms of the curvature tensor it is shown that for each type the euler characteristic is nonnegative and even several new inequalities are obtained concerning the euler characteristic and the volume of the manifold the second betti number is even and if it is zero then the first betti number also vanishes the arguments developed here are based upon the famous work of chern about the gauss bonnet formula for a pseudo riemannian manifold