cr structures with group action and extendability of cr functions part i presents results on local embedding of cr structures we consider an abstract cr manifold whose structure is invariant under a transversal lie group action we show that such a manifold can always be locally embedded in complex space as a generic submanifold the proof is based on selection of canonical coordinates and repeated use of the newlander nirenberg theorem 13 when the lie group is abelian the embedding can be given a particularly simple form let l 1 be the codimension of our submanifold called m throughout the paper it is then convenient to denote by n i the dimension of the ambient complex space and by zl z w 1 w t the complex coordinates we shall systematically write

a dynamical trajectory based methodology for systematically computing multiple optimal solutions of general nonlinear programming problems in this paper a novel dynamical trajectory based methodology is developed for systematically computing multiple local optimal solutions of general nonlinear programming problems with disconnected feasible components satisfying nonlinear equality inequality constraints the proposed methodology deterministic in nature exploits trajectories of two different nonlinear dynamical systems to find multiple local optimal solutions the methodology consists of two phases phase i starts from an arbitrary infeasible initial point and finds systematically multiple or all the disconnected feasible components phase ii finds an adjacent local optimal solution from a local optimum via a decomposition point thereby systematically locating multiple local optimal solutions which lie within each feasible component found in phase i by alternating between these two phases the methodology locates multiple or all the local optimal solutions which lie in all the disconnected feasible components a theoretical foundation for the proposed methodology is also developed the methodology is illustrated with a numerical example with promising results

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

a review of duality theory for linear programming over topological vector spaces abstract in this paper duality theory for infinite dimensional linear programs is discussed in a topological vector space setting infinite dimensional linear programs occur in many different areas and the duality theory of such problems has been discussed by a number of authors however the results are scattered in the literature and are proved in a variety of different settings the purpose of this paper is to bring together the main results on this subject and to present them in a unified setting and notation with some new simpler proofs

a new trajectory reversing method for the estimation of asymptotic stability regions a new method is presented for estimating regions of asymptotic stability for autonomous nonlinear systems the method applies to systems that can be proven to be non oscillatory using a suitable lyapunov technique it is especially effective for second order systems that possess several equilibrium states for which the regions of attraction must be found the determination of the regions of asymptotic stability requires the explicit computation of a lyapunov function and the determination by backward numerical integration of a limited number of trajectories the obtained estimates are close approximations of the exact regions of asymptotic stability of the system s stable equilibria

semi infinite transportation problems abstract a semi infinite transportation dual program pair is specified which involves general pairings of linear spaces stemming from an infinite number of destination requirements but where in the primal program least cost flows of goods are sought from only a finite number of origins to these destinations building on the work of m j todd solving the generalized market area problem management sci 24 1978 1549 1554 a finite dimensional dual unconstrained concave program is developed for the primal semi infinite program but without certain measure theoretic restrictions on the cost functions themselves optimality conditions for the dual program pair are specified involving generalized column number conditions which parellel but extend those of the classical finite dimensional transportation problem

the spectrum of the leray transform for convex reinhardt domains in c2 abstract the leray transform and related boundary operators are studied for a class of convex reinhardt domains in c 2 our class is self dual it contains some domains with less than c 2 smooth boundary and also some domains with smooth boundary and degenerate levi form l 2 regularity is proved and essential spectra are computed with respect to a family of boundary measures which includes surface measure a duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains many of these results are new even for the classical case of smoothly bounded strongly convex reinhardt domains

duality gap function in infinite dimensional linear programming abstract the concept of duality gap function in infinite dimensional linear programming is considered in this paper basic properties of the function and two theorems on its behavior are obtained by using duality theorems with interior conditions as illustrations for the results we investigate the parametric versions of an example due to d gale and parametric linear programs on spaces of continuous functions the notions of riemann stieltjes integral and function of bounded variation have been shown to be very useful for our investigations

separated continuous conic programming strong duality and an approximation algorithm motivated by recent applications in robust optimization and in sign constrained linear quadratic control we study in this paper a new class of optimization problems called separated continuous conic programming sccp focusing on a symmetric primal dual pair we develop a strong duality theory for the sccp our idea is to use discretization to connect the sccp and its dual to two ordinary conic programs we show if the latter are strongly feasible and with finite optimal values a condition that is readily verifiable then the strong duality holds for the sccp this approach also leads to a polynomial time approximation algorithm that solves the sccp to any required accuracy

the cauchy szegő projection for domains in mathbb c n with minimal smoothness we prove the lp bd regularity of the cauchy szeg projection also known as the szeg projection for bounded domains d cn which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class c2

on the number and types of unstable equilibria in nonlinear dynamical systems with uniformly bounded stability regions classical theorems on the complete characterization of stability boundaries provided little information about the existence and types of unstable equilibrium points on a stability boundary in this technical note we derive analytical results and sufficient conditions about the existence of all types of unstable equilibrium points in a dynamical system whose stability regions are uniformly bounded some applications of these analytical results are illustrated using electric power systems

neighboring stable equilibrium points in spatially periodic nonlinear dynamical systems theory and applications stability is of fundamental importance to the design and application of control systems in which stable equilibrium points and the neighboring points can have various interesting physical implications in the paper we derive a lower bound and an upper bound on the number of neighboring stable equilibrium points in the spatially periodic nonlinear dynamical systems it is shown that in such an n dimensional system there are at least 2n neighboring stable equilibrium points meanwhile an upper bound on the number of neighboring stable equilibrium points is derived some applications of these analytical results are illustrated

the cauchy integral in cn for domains with minimal smoothness abstract we prove l p b d regularity of the cauchy leray integral for bounded domains d c n whose boundary satisfies the minimal regularity condition of class c 1 1 together with a naturally occurring notion of convexity

on the convergence theory of trust region based algorithms for equality constrained optimization in a recent paper dennis el alem and maciel proved global convergence to a stationary point for a general trust region based algorithm for equality constrained optimization this general algorithm is based on appropriate choices of trust region subproblems and seems particularly suitable for large problems this paper shows global convergence to a point satisfying the second order necessary optimality conditions for the same general trust region based algorithm the results given here can be seen as a generalization of the convergence results for trust regions methods for unconstrained optimization obtained by more and sorensen the behavior of the trust radius and the local rate of convergence are analyzed some interesting facts concerning the trust region subproblem for the linearized constraints the quasi normal component of the step and the hard case are presented it is shown how these results can be applied to a class of discretized optimal control problems

a counterexample of a theorem by tsolas et al and an independent result by zaborszky et al with reply recent results on the composition of the stability boundary by n a tsolas et al ieee trans circuits syst vol cas 32 no 10 pp 1041 9 oct 1985 and j zaborszky et al proc ifac symp power syst power plant contr beijing china aug 1986 pp 597 603 claimed that it consisted of the stable manifolds of index one unstable equilibria and their closures a counterexample shows the potential existence of additional structures within the boundary this finding invalidates numerous theorems published in the literature the author concedes that the counterexample to the theorem in the paper by n a tsolas et al is unassailable and identifies the error in his proof

the leray transform factorization dual cr structures and model hypersurfaces in cp2 abstract we compute the exact norms of the leray transforms for a family s of unbounded hypersurfaces in two complex dimensions the s generalize the heisenberg group and provide local projective approximations to any smooth strongly c convex hypersurface s to two orders of tangency this work is then examined in the context of projective dual cr structures and the corresponding pair of canonical dual hardy spaces associated to s leading to a universal description of the leray transform and a factorization of the transform through orthogonal projection onto the conjugate dual hardy space